Thèse de doctorat -

École Doctorale de Sciences Mathématiques de Paris Centre (ED 386)Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR 7586)

Thèse de doctorat

del'Université Sorbonne Paris Cité

Préparée à l'Université Paris Diderot

Anneaux de Grothendieck quantiques, algèbresamassées et catégorie O a�ne quantique

Présentée par

Léa Bittmann

Discipline : Mathématiques

Dirigée par David Hernandez

Soutenue publiquement le 16 juillet 2019 devant le jury composé de :

M. David Hernandez PR - Université Paris Diderot DirecteurM. Bernhard Keller PR - Université Paris Diderot ExaminateurM. Hiraku Nakajima PR - University of Tokyo ExaminateurM. Pierre-Guy Plamondon MCF - Université de Paris Sud RapporteurMme. Michela Varagnolo MCF - Université de Cergy-Pontoise Examinatrice

Au vu des rapports de :

Mme. Vyjayanthi Chari University of California, RiversideM. Pierre-Guy Plamondon Université de Paris Sud

This thesis was supported by the European Research Council (ERC) under the Euro-pean Union's Framework Programme H2020, with ERC Grant Agreement number 647353Qa�ne.

Institut de Mathématiques deJussieu-Paris Rive Gauche(CNRS � UMR 7586)Université Paris Diderot (Paris 7)Bâtiment Sophie Germain - BoîteCourrier 70128, place Aurélie Nemours75205 Paris Cedex 13France

École Doctorale de Sciences Mathé-matiques de Paris CentreBoîte Courrier 2904, place Jussieu75252 Paris Cedex 05France

Résumé/Abstract

Résumé

L'objectif de cette thèse est de construire et d'étudier une structure d'anneau de Gro-thendieck quantique pour une catégorie O de représentations de la sous-algèbre de BorelUqpbq d'une algèbre a�ne quantique Uqpgq.

On s'intéresse dans un premier lieu à la construction de modules standards asympto-tiques pour la catégorie O, qui sont des analogues des modules standards existant dans lacatégorie des représentations de dimension �nie de Uqpgq. Une construction complète deces modules est proposée dans le cas où l'algèbre de Lie simple sous-jacente g est sl2.

Ensuite, nous dé�nissons un tore quantique qui étend le tore quantique contenant l'an-neau de Grothendieck quantique de la catégorie des représentations de dimension �nie.Nous utilisons pour cela des notions liées aux algèbres amassées quantiques. Dans le mêmeesprit, nous proposons une construction d'une structure d'algèbre amassée quantique surl'anneau de Grothendieck quantique KtpC

�Z q d'une sous-catégorie monoïdale C�

Z de lacatégorie des représentations de dimension �nie.

Puis, nous dé�nissons un anneau de Grothendieck quantique KtpO�Z q d'une sous-catégorie O�Z de la catégorie O, comme une algèbre amassée quantique. Nous établissonsensuite que cet anneau de Grothendieck quantique contient celui de la catégorie des repré-sentations de dimension �nie. Ce résultat est montré directement en type A, puis en touttype simplement lacé en utilisant la structure d'algèbre amassée quantique de KtpC

�Z q.

En�n, nous dé�nissons des pq, tq-caractères pour des représentations simples de di-mension in�nie remarquables de la catégorie O. Ceci nous permet d'écrire des versionst-déformées de relations importantes dans l'anneau de Grothendieck classique de la caté-gorie O�Z , qui ont des liens avec les systèmes intégrables quantiques associés.

Mots-clefs

Groupes quantiques, théorie des représentations, algèbres a�nes quantiques, catégorieO, anneaux de Grothendieck quantiques, algèbres amassées quantiques.

iv Résumé/Abstract

Quantum Grothendieck rings, cluster algebras

and quantum a�ne category O

Abstract

The aim of this thesis is to construct and study some quantum Grothendieck ringstructure for the category O of representations of the Borel subalgebra Uqpbq of a quantuma�ne algebra Uqpgq.

First of all, we focus on the construction of asymptotical standard modules, analogs inthe context of the category O of the standard modules in the category of �nite-dimensionalUqpgq-modules. A construction of these modules is given in the case where the underlyingsimple Lie algebra g is sl2.

Next, we de�ne a new quantum torus, which extends the quantum torus containingthe quantum Grothendieck ring of the category of �nite-dimensional modules. In order todo this, we use notions linked to quantum cluster algebras. In the same spirit, we builda quantum cluster algebra structure on the quantum Grothendieck ring of a monoidalsubcategory C�

Z of the category of �nite-dimensional representations.With this quantum torus, we de�ne the quantum Grothendieck ring KtpO�Z q of a sub-

category O�Z of the category O as a quantum cluster algebra. Then, we prove that thisquantum Grothendieck ring contains that of the category of �nite-dimensional representa-tion. This result is �rst shown directly in type A, and then in all simply-laced types usingthe quantum cluster algebra structure of KtpC

�Z q.

Finally, we de�ne pq, tq-characters for some remarkable in�nite-dimensional simple rep-resentations in the category O�Z . This enables us to write t-deformed analogs of importantrelations in the classical Grothendieck ring of the category O, which are related to thecorresponding quantum integrable systems.

Keywords

Quantum groups, representation theory, quantum a�ne algebras, category O, quantumGrothendieck rings, quantum cluster algebra.

Remerciements

Il est maintenant temps de remercier toutes les personnes qui m'ont aidée, soutenue,ont été là au bon moment pendant ma thèse ou dans les années qui ont menées à ma thèse.À tous ceux que j'ai oublié : vous étiez les personnes les plus importantes, merci mille fois.

En tout premier lieu, je tiens à remercier mon directeur de thèse, David Hernandez.Je le remercie tout d'abord pour avoir accepté, en 2014, d'encadrer mon stage de Master2, puis de m'avoir soutenu dans mon projet de passer l'agrégation l'année suivante. Ces5 années d'encadrement, sous une forme ou une autre, auront été pour moi extrêmementformatrices. Sous sa tutelle, j'ai appris ce qu'était la recherche en mathématiques, et j'aimême pu contribuer de ma modeste pierre à cet édi�ce. Je le salue pour sa patience in�nie,et pour son soutien durant les (nombreuses) phases de doutes qui ont parsemées cette thèse.Je le remercie pour toutes ces enrichissantes discussions, mathématiques ou plus générales.

The next person I would like to thank is Vyjayanthi Chari. First, for welcoming me inRiverside in Spring 2018. I was very honored to be there and to have the opportunity todiscuss with you during that time. I also want to thank you for your formidable supportand your patience during my applications to di�erent job positions. Lastly, I thank you foraccepting to review this thesis, even if you could not be there in person for the defense.

J'aimerais aussi remercier Pierre-Guy Plamondon pour avoir accepté de rapporter cettethèse. Je sais que ce sont des heures qui ont été prises sur des visites de Kyoto, et je lui ensuis d'autant plus reconnaissante.

Merci également à Bernhard Keller et Michela Varagnolo de me faire l'honneur de fairepartie de mon jury de thèse. Plus particulièrement, je remercie Bernhard Keller pour sonsoutien et pour m'avoir donné accès aux toutes dernières fonctionnalités de sa merveilleuseapplication de mutation des carquois, sans laquelle le Chapitre 3 ne serait pas du tout lemême ; je remercie Michela Varagnolo pour m'avoir invitée, puis encouragée, à parler demes travaux au séminaire d'équipe, c'était un moment important de ma jeune carrière.

Additionally, I would like to thank Hiraku Nakajima for accepting to take part in myjury. It is a great honor for me to have you there, as so much of this thesis is based on yourfundamental work.

Going back in time a little bit, I would like to thank Pavel Etingof for welcoming meat MIT in 2015, and advising me for my Master's degree internship. I also thank him forhis continuing help and support.

During the last three years, I had the chance to go to several conferences and meet anddiscuss with di�erent mathematicians, and I thank all those who gave me some of theirtime. In particular, I want to thank Jan Grabowski and Bernard Leclerc for their help andtheir willingness to answer my questions. I also would like to thank Hironori Oya for our

vi Remerciements

useful discussions.To close the part dedicated to the mathematicians, I would like to thank Anton Mellit

for giving me this wonderful opportunity to pursue my researches in Vienna next year.Je tiens à remercier le personnel administratif de l'IMJ-PRG, de l'UFR de mathé-

matiques et de l'École Doctorale, avec en particulier Emilie Jacquemot, Claire Lavollay,Nadine Fournaiseau, Evariste Ciret, Sakina Madel-Kawami et Amina Hariti, pour m'avoirguidée dans les méandres administratifs de la thèse avec patience et gentillesse.

Viens maintenant le moment de remercier mes collègues en thèse à l'IMJ-PRG (etε-voisinage). Ce sont eux et elles qui ont vécu au plus près les aléas journaliers de ces 3dernières années et qui ont fait du bureau un endroit où j'avais envie d'aller le matin. Je vaiscommencer chronologiquement, en remerciant les Vieux, pour leur accueil au laboratoire.Les Vieux sont ceux et celles qui étaient en train de �nir quand je suis arrivée, et plusgénéralement les amateurs de ping-pong par canicule. Je remercie en particulier Aurélien,dont le thé froid nous a bien manqué ces deux dernières années, Baptiste, pour son accueiltoujours chaleureux et son enthousiasme toujours débordant (ou peut-être une seule deces choses), Charles F., David, Kévin D., Marco, et son amour presque romantique pourl'univers Marvel, Martin, la gentillesse personni�ée, et Victoria. Vient ensuite le tour desUn Peu Moins Vieux, dont certains sont aussi déjà partis, mais reviennent encore de tempsen temps, je les remercie d'avoir montré la voie dans les di�érentes étapes de la thèse,dans les hauts et les bas de la dernière année. Je pense en particulier à Alexandre V., etsa connaissance encyclopédique des jeux de société, qui les explique mieux que quiconque,Antoine J., le bobo du 11ième qui écoute de la bonne musique, Elie G., avec qui il m'estmaintenant impossible de jouer à un jeu basé sur le mensonge, Esther et ses histoiresrocambolesques, Juan-Pablo, et le bon temps passé au Brésil, Kévin B., Noé, Omar, et ses"j'ai une question", qui mènent toujours vers des discussions interminables, Pooneh, Reda,pour la team Représentations, et Siarhei. On en arrive naturellement à la fournée 2016(la meilleure), je remercie en particulier Andreas, le viking o�ciel du bureau, Grégoire,Nicolas, pour son soutien dans la déprime de la 3ème année et pour ses passions portéesà l'extrême, Parisa, Rahman, Sacha, le maitre des clés et de l'administration (mais pas dugoûter ...) et Théophile, le bon Vosgien ami des bêtes (en�n des chats et des ratons-laveurssurtout). On continue avec les Jeunes, qui assurent bien la relève. Je remercie Alex, quipour moi sera toujours méchant, Alexandre L., Amandine, qui reprendra magni�quementle �ambeau de la représentativité féminine, Andrei, Charles V., mon cher compagnon depromo, Colin, pour m'avoir un jour dit "tiens, et si on devenait des amis ?" , Corentin M. etses super expériences pour illustrer la physique à des collégiens, Daniele, Elie C. mon biengrand petit frère de thèse, Fatna, Jérémie, membre de la secte du Magic Clipper, Kévin M.,à qui je pardonne presque d'avoir introduit les échecs au bureau, Leonardo, Mario, pouravoir bien mieux représenté les doctorants au comité parité que moi, Matthieu, Maud, poursa façon se raconter les histoires, Pierre et toutes ses new meta, Tancrède, spécialiste dejeux de mots foireux (mais on ne choisit pas sa famille !), et Wille, mon adjoint responsablefournitures, toujours motivé, surtout pour aller au Japon. Pour ses blagues vaseuses et sonsoutien contre les Vosgiens, merci à Jérôme, qui transcende les années. Je remercie aussiAmélie, pour les discussions ASOIAF passionnantes et passionnées, et Marie C., pour lesbrunchs et les jeux, passés ou à venir (oui tu es dans la section du labo) et Patteo, pourles câlins pendant la période de rédaction.

Je remercie tous les Arêcheois de c÷ur (avec techniquement quelques non-Arêcheois,mais on vous aime quand même), pour le ski, les weekends, les bou�es, les jeux, les pique-niques, les soirées meufs etc. En espérant n'oublier personne, merci à Alice M., Anne-

Remerciements vii

Fleur, Antoine P., Aurore, Aymeric, Babeth, Claire, Corentin C., Delphin, Emilie, François,Hélène, Jacko, Julie, Michel, Olivier, Raphaël, Rémi, Salomé, Sophie, Tristan, Vincent etYaël.

Merci également à l'8-team, Alice L. et Guillaume, pour avoir sauvé le monde avec moià de nombreuses reprises, et pour s'être échappés avec succès de bateaux, musées, prisons,et pièces en tous genres. Toujours 8 !

Je remercie tous mes anciens colocataires, en particuliers les originaux, les Gens deCompagnie, pour ne pas m'avoir laissée seule une seule minute ces 5 dernières années, etpour m'avoir fourni un safe space que je peux emmener partout avec moi ; merci à Andreea,pour toutes ces soirées canap', à Ken pour ta générosité, et à Laurent pour m'avoir sortieet t'être bien occupé de moi.

Merci à Marc et Marie d'avoir été là depuis le début (des maths en tous cas), et d'êtretoujours là des années plus tard, je vous aime. Merci bien évidemment à Anne, ma fannuméro 1, pour ton soutien in�ni (et pour tes relectures), si tu n'existais pas il faudraitt'inventer.

Je remercie également tous les professeurs de mathématiques, ou autre, qui m'ontsoutenue pendant ma scolarité, et qui m'ont permis d'en arriver jusqu'ici. Merci aussià ceux qui n'ont pas cru en moi, ils m'ont poussée à leur prouver qu'ils avaient tort.

Merci à Georges R.R. Martin de ne pas avoir annoncé la sortie de Winds of Winterpendant ma rédaction de thèse.

En�n, je remercie ma famille pour tout le soutien qu'elle m'a apporté au cours desannées. Je remercie en particulier mes parents, ma mère, pour m'avoir donné le goût dela curiosité scienti�que dès mon plus jeune âge, et mon père pour avoir toujours suivi meschoix de près et m'avoir toujours remarquablement bien guidée. Merci à Lisa et Simon,parce qu'on a toujours besoin d'un grand frère. Merci également à Amaury, pour avoirouvert la voie pour les Bittmann dans la recherche mathématique. Mes pensées vont àceux qui nous ont quittés, mais aussi à aux petits nouveaux qui arrivent !

En conclusion, merci à Virgile, TMTC.

� Say what you know, do what you must,

come what may. �

So�a Kovalevskaya�

Contents

Résumé/Abstract iii

Remerciements v

Introduction 5I.1 Contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I.1.1 Spectres des systèmes intégrables quantiques . . . . . . . . . . . . . 5I.1.2 Algèbres a�nes quantiques . . . . . . . . . . . . . . . . . . . . . . . 6I.1.3 Représentations de dimension �nie des algèbres a�nes quantiques . . 8I.1.4 Anneaux de Grothendieck et q-caractères pour les représentations de

dimension �nie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9I.1.5 Anneaux de Grothendieck quantiques . . . . . . . . . . . . . . . . . 11I.1.6 Catégorie O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13I.1.7 Algèbres amassées . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

I.2 Résultats de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16I.2.1 Modules standards asymptotiques . . . . . . . . . . . . . . . . . . . . 16I.2.2 Tore quantique étendu . . . . . . . . . . . . . . . . . . . . . . . . . . 18I.2.3 L'anneau de Grothendieck quantique KtpC

�Z q comme algèbre amas-

sée quantique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20I.2.4 Anneau de Grothendieck quantique pour la catégorie O� et calcul

des pq, tq-caractères . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23I.2.5 Relations remarquables dans l'anneau de Grothendieck quantique . . 25I.2.6 Pistes d'ouverture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Asymptotics of standard modules of quantum a�ne algebras 291.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.1.1 The quantum a�ne algebra and its Borel subalgebra . . . . . . . . . 311.1.2 Representations of the Borel algebra . . . . . . . . . . . . . . . . . . 341.1.3 A q-character theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2 Limits of q-characters of standard modules . . . . . . . . . . . . . . . . . . . 391.2.1 Standard modules for �nite dimensional representations . . . . . . . 391.2.2 Limits of q-character . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.2.3 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.3 Asymptotical standard modules . . . . . . . . . . . . . . . . . . . . . . . . . 431.3.1 Construction of the U¥0

q pbq-module. . . . . . . . . . . . . . . . . . . 43

2 Contents

1.3.2 Construction of induced modules . . . . . . . . . . . . . . . . . . . . 481.4 Decomposition of the q-character of asymptotical standard modules . . . . . 52

1.4.1 Partial order on P r` . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.4.2 Decomposition for MpΨΨΨ�11 q . . . . . . . . . . . . . . . . . . . . . . . 53

1.4.3 General decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2 Quantum Grothendieck rings as quantum cluster algebras. 552.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 Category O of representations of quantum loop algebras . . . . . . . . . . . 58

2.2.1 Quantum loop algebra and Borel subalgebra . . . . . . . . . . . . . . 582.2.2 Highest `-weight modules . . . . . . . . . . . . . . . . . . . . . . . . 592.2.3 De�nition of the category O . . . . . . . . . . . . . . . . . . . . . . . 602.2.4 Connection to �nite-dimension UqpLgq-modules . . . . . . . . . . . . 602.2.5 Categories O� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2.6 The category O�Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.7 The Grothendieck ring K0pOq . . . . . . . . . . . . . . . . . . . . . . 622.2.8 The q-character morphism . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3 Quantum tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3.1 The torus Yt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.3.2 The torus Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 Quantum Grothendieck rings . . . . . . . . . . . . . . . . . . . . . . . . . . 682.4.1 The �nite-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 682.4.2 Compatible pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.4.3 De�nition of KtpO�Z q . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.5 Properties of KtpO�Z q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.1 The bar involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.2 Inclusion of quantum Grothendieck rings . . . . . . . . . . . . . . . . 752.5.3 pq, tq-characters for positive prefundamental representations . . . . . 79

2.6 Results in type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.6.1 Proof of the conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 802.6.2 A remarkable subalgebra in type A1 . . . . . . . . . . . . . . . . . . 81

3 Quantum cluster algebra algorithm and pq, tq-characters 853.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.2 Finite-dimensional representations of quantum a�ne algebras . . . . . . . . 88

3.2.1 Quantum a�ne algebra . . . . . . . . . . . . . . . . . . . . . . . . . 883.2.2 Finite-dimensional representations . . . . . . . . . . . . . . . . . . . 893.2.3 q-characters and truncated q-characters . . . . . . . . . . . . . . . . 893.2.4 Cluster algebra structure . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3 Quantum Grothendieck rings . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.1 Quantum torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.2 Commutative monomials . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.3 Quantum Grothendieck ring KtpCZq . . . . . . . . . . . . . . . . . . 933.3.4 pq, tq-characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.5 Truncated pq, tq-characters and quantum Grothendieck ring KtpC

�Z q 95

3.3.6 Quantum T -systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.4 Quantum cluster algebra structure . . . . . . . . . . . . . . . . . . . . . . . 96

3.4.1 A compatible pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Contents 3

3.4.2 The quantum cluster algebra At . . . . . . . . . . . . . . . . . . . . 983.5 Quantum cluster algebras and quantum Grothendieck ring . . . . . . . . . . 99

3.5.1 A note on the bar-involution . . . . . . . . . . . . . . . . . . . . . . 1003.5.2 A sequence of vertices . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5.3 Truncated pq, tq-characters as quantum cluster variables . . . . . . . 1013.5.4 Proof of Theorem 3.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.6 Application to the proof of an inclusion conjecture . . . . . . . . . . . . . . 1073.6.1 The quantum Grothendieck ring KtpO�Z q . . . . . . . . . . . . . . . . 1073.6.2 Intermediate quantum cluster algebras . . . . . . . . . . . . . . . . . 1073.6.3 Inclusion of quantum Grothendieck rings . . . . . . . . . . . . . . . . 111

3.7 Explicit computation in type D4 . . . . . . . . . . . . . . . . . . . . . . . . 115

4 Further directions 1274.1 Further results in the sl2 case . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.1.1 pq, tq-characters for negative prefundamental representations . . . . . 1274.1.2 Quantum Grothendieck ring for the category O�Z . . . . . . . . . . . 1294.1.3 pq, tq-characters of simple modules . . . . . . . . . . . . . . . . . . . 1304.1.4 Quantum Wronskian relation . . . . . . . . . . . . . . . . . . . . . . 1334.1.5 pq, tq-characters of asymptotical standard modules . . . . . . . . . . 134

4.2 Leads for other types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.2.1 The (q,t)-characters of the negative prefundamental representations . 1364.2.2 Quantum Grothendieck ring for the category O�Z . . . . . . . . . . . 1364.2.3 Extension to non-simply-laced types . . . . . . . . . . . . . . . . . . 137

Appendix

A Cartan data and quantum Cartan data 141A.1 Root data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Quantum Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.3 Height function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.4 In�nite quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

B Cluster algebras and quantum cluster algebras 145B.1 Cluster algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.2 Compatible pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146B.3 De�nition of quantum cluster algebras . . . . . . . . . . . . . . . . . . . . . 147B.4 Laurent phenomenon and quantum Laurent phenomenon . . . . . . . . . . . 149B.5 Specializations of quantum cluster algebras . . . . . . . . . . . . . . . . . . 149B.6 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliographie 153

Introduction

I.1 Contexte

I.1.1 Spectres des systèmes intégrables quantiques

Les motivations originelles des questions étudiées dans ce travail de thèse se situent enmécanique statistique et dans l'étude des systèmes intégrables quantiques.

Le modèle qui nous intéresse particulièrement est le modèle à 6 sommets, aussi appelémodèle de la glace. Il a été introduit par Pauling en 1935 pour décrire certaines structurescristallines, en particulier celle des cristaux de glace. Ce système est un important objetd'étude, pour les physiciens autant que pour les mathématiciens (voir par exemple [Bax82,Chapitre 8] et [Res10]).

Le système est modélisé par un réseau à deux dimensions pour lequel chaque sommetest relié à exactement quatre autres sommets. Un état du système consiste en un choixd'orientation de chacune des arêtes du réseau. De plus, on impose une condition appeléecondition de la glace : les arêtes sont orientées de façon à ce que chaque sommet aitexactement deux �èches entrantes et deux �èches sortantes. Ainsi, chaque sommet a sixcon�gurations possibles, d'où le nom de ce modèle.

Figure 1 � Con�gurations possibles

L'état physique du système, c'est-à-dire le choix d'orientation de chaque arête, est en-codé par une fonction Z appelée fonction de partition. Les systèmes considérés comportentgénéralement un nombre gigantesque de particules. Ainsi, une manière d'accéder à cettefonction de partition est d'étudier le comportement moyen, ou macroscopique, du système.La fonction de partition peut être écrite comme la trace d'un opérateur T appelé matricede transfert :

Z � TrW pT q,

où W est l'espace des états. Donc, dans le but d'étudier Z, on cherche à obtenir les valeurs

6 Introduction

propres de l'opérateur T :TrW pT q �

¸j

λj .

En 1967, à l'aide d'une technique appelée Ansatz de Bethe, Lieb [Lie67] a résolu lemodèle à six sommets en construisant explicitement les vecteurs propres. Puis, en 1972,Baxter [Bax72] a résolu algébriquement le modèle à huit sommets, qui est une généralisationdu modèle à six sommets, a�ranchi de la condition de la glace. Il a montré que les valeurspropres de la matrice de transfert avaient une forme particulière :

λj � ApzqQjpzq

2q

Qjpzq�Dpzq

Qjpzq�2q

Qjpzq, (I.1.1.1)

où z et q sont des paramètres du système, les fonctions Apzq et Dpzq sont universelles, etle seul terme dépendant de la valeur propre est Qjpzq, qui est une fonction polynomiale.La relation (I.1.1.1) est appelée relation de Baxter. Elle sera importante dans la suite.

L'approche de Baxter pour comprendre ce modèle est basée sur une famille commutativede matrices de transfert. La condition de commutation de deux de ces matrices pouvantêtre traduite par une relation, appelée par Baxter "star-triangle relation".

�

Figure 2 � Relation de Yang-Baxter

Cette relation est plus connue sous le nom de Relation de Yang-Baxter. Les travaux deBaxter ont mené à d'importantes généralisations des techniques de Bethe, et ont permisde mettre en évidence la richesse de la structure algébrique de ce modèle. Quand Sklya-nin [Skl79], simultanément avec Kulish et Reshetikhin [KRS81], ont mis au jour le lienentre la relation de Yang-Baxter dans le modèle à six sommets et la quanti�cation de sys-tèmes intégrables classiques, cela a ouvert la voie pour de fructueuses recherches dans ledomaine des systèmes intégrables quantiques dans les décennies qui ont suivi (voir [JM95]ou [KBI93] pour des vues d'ensemble). C'est dans la continuité de ces questions que lesgroupes quantiques émergeront.

I.1.2 Algèbres a�nes quantiques

Dans les années 1980, Drinfeld [Dri87] et Jimbo [Jim85] ont indépendamment dé�ni,pour toute algèbre de Lie simple de dimension �nie g, une famille d'algèbres de Hopf Uqpgq,dépendant d'un paramètre q P C�, appelées algèbres enveloppantes universelles quanti�ées.Il s'agit de q-déformations de l'algèbre enveloppante universelle Upgq de l'algèbre de Lie g.

Une des caractéristiques principales de ces types de groupes quantiques réside dansle fait qu'à toute représentation de dimension �nie V de Uqpgq, l'on puisse associer unopérateur R P EndpV b V q qui satisfait l'équation de Yang-Baxter :

R12R13R23 � R23R13R12, (I.1.2.1)

I.1. Contexte 7

où R12 � R b Id P EndpV b V b V q, et R13 et R23 sont dé�nis de manière similaire. Larelation (I.1.2.1) peut aussi être représentée par le diagramme de la Figure 2.

Intéressons-nous maintenant à l'algèbre de Kac-Moody a�ne (non-twistée) g corres-pondant à l'algèbre de Lie simple g, il s'agit d'une extension centrale g � Lpgq ` Cc del'algèbre des lacets

Lpgq :� gb Crt�1s.

Kac [Kac68] et Moody [Moo69] ont montré indépendamment que l'algèbre g avait uneprésentation analogue à la Serre des algèbres simples complexes de dimension �nie. Ceciqui permet d'appliquer la construction générique des groupes quantiques de Drinfeld etJimbo à g. L'algèbre enveloppant q-déformée obtenue s'appelle l'algèbre a�ne quantiqueUqpgq.

Dans [Dri88], Drinfeld donna une nouvelle réalisation de l'algèbre a�ne quantique parun procédé d'a�nisation quantique de l'algèbre enveloppante universelle quanti�ée de l'al-gèbre de Lie simple g correspondante à g. À l'origine de cette nouvelle présentation setrouvent des théories de Lusztig sur l'action des groupes de tresses sur l'algèbre envelop-pante quanti�ée. Cette présentation implique une in�nité de générateurs, semblables auxgénérateurs de l'algèbre des lacets ; si I � t1, . . . , nu avec n le rang de l'algèbre de Liesimple g, ces derniers s'écrivent :

x�i,r pi P I, r P Zq, φ�i,�m pi P I,m ¥ 0q, k�1i pi P Iq, (I.1.2.2)

où x�i,r est l'équivalent quantique de l'élément x�1i b tr de l'algèbre des lacets Lpgq.

Drinfeld énonça l'existence d'un isomorphisme entre cette nouvelle présentation et laprésentation classique, et Beck [Bec94] donna un isomorphisme explicite entre les deuxprésentations (précisé plus récemment par Damiani dans [Dam15]). Ainsi, les deux procé-dés, d'a�nisation puis de quanti�cation, ou de quanti�cation, puis d'a�nisation quantiquecommutent. Ceci se résume par le diagramme commutatif suivant :

gquanti�cation

vva�nisation

''Uqpgqa�nisationquantique

((

gquanti�cation

ww

Uqpgq

.

De plus, Beck [Bec94] explicita une décomposition triangulaire de Uqpgq, grâce à lanouvelle présentation de Drinfeld :

Uqpgq � Uqpgq� b Uqpgq0 b Uqpgq�, (I.1.2.3)

où Uqpgq� (resp. Uqpgq0) est la sous-algèbre de Uqpgq engendrée par les générateurs x�i,r(resp. φ�i,�m, k

�1i ) donnés en (I.1.2.2).

Drinfeld montra [Dri85] que l'algèbre a�ne quantique possède une R-matrice univer-selle R. Il s'agit d'une solution de l'équation de Yang-Baxter (I.1.2.1), appartenant à unecomplétion du produit tensoriel de deux copies de Uqpgq :

Rpzq P Uqpgqb Uqpgqrrzss. (I.1.2.4)

Une formule explicite pour la R-matrice universelle a été obtenue par d'autres auteurs,d'une part par Khoroshkin et Tolstoy [KT94] et par une autre méthode par Levendorsky-Soibelman-Stukopin [LSS93] pour g � sl2, puis Damiani dans le cas général [Dam98].

8 Introduction

La théorie des représentations des algèbres a�nes quantiques est très riche et a été inten-sivement étudiées ces 30 dernières années, avec des liens vers de très nombreux domainesdes mathématiques (géométrie, topologie, combinatoire, etc) et de la physique (théorieconforme des champs, systèmes intégrables, etc).

I.1.3 Représentations de dimension �nie des algèbres a�nes quantiques

Rappelons ici quelques résultats importants de l'étude des représentations de dimen-sion �nie des algèbres a�nes quantiques, qui est un domaine de recherche très fertile,comportant encore de nombreuses questions ouvertes.

Soit C la catégorie des représentations de dimension de l'algèbre a�ne quantique Uqpgq.Comme Uqpgq est une algèbre de Hopf, le produit tensoriel de deux de représentations deUqpgq possède aussi une structure de Uqpgq-module, et C est une catégorie tensorielle.

Une question importante du domaine est la description et la classi�cation des repré-sentations simples de la catégorie C , c'est-à-dire des représentations irréductibles de di-mension �nie de Uqpgq. Quand g � sl2, cette question a été complètement résolue parChari et Pressley dans [CP91], qui ont donné une description explicite des représenta-tions irréductibles. Pour les autres types, cette question est beaucoup plus di�cile, et resteencore majoritairement ouverte, bien qu'elle ait été considérablement étudiée. Parmi lesnombreuses références sur le sujet, on peut en particulier citer [CP95a], [CP91], [CP95b],[CP96], [GV93], [Vas98], [KS95], [AK97], [FR96], et [Nak04], mais cette liste n'est bienévidemment pas exhaustive.

En comparaison, dans la limite classique q Ñ 1, la classi�cation des représentationssimples de dimension �nie de g est beaucoup plus accessible. Pour tout a P C�, considéronsle morphisme d'évaluation eva : g Ñ g, dé�ni en évaluant les polynômes de Laurentde l'algèbre des lacets Lpgq en a. Le tiré en arrière par eva de n'importe quel g-moduleirréductible V pλq de plus haut poids λ est un g-module irréductible noté Vapλq. Alors,toutes les représentations irréductibles (à isomorphisme près) de g sont de la forme

Va1pλ1q b Va2pλ2q b � � � b Vakpλkq,

avec ai � aj pour i � j [Cha86]. Dans le cas quantique, le morphisme d'évaluation eva nes'étend pas en un morphisme Uqpgq Ñ Uqpgq si g n'est pas de type A.

Néanmoins, les représentations de dimension �nie de Uqpgq ont tout de même uneclassi�cation qui peut être vue comme un analogue quantique de la classi�cation de Cartandes représentations simples de dimension �nie de g par leur plus haut poids.

La partie Uqpgq0 de la décomposition triangulaire (I.1.2.3) joue le rôle de la sous-algèbrede Cartan. Cette algèbre, appelée sous-algèbre de Drinfeld-Cartan, agit comme une sous-algèbre commutative de Uqpgq sur les représentations de dimension �nie. Un vecteur v d'unUqpgq-module est dit de `-poids ΨΨΨ s'il appartient à un sous-espace de Jordan pour l'actionsimultanée des générateurs tφ�i,�m, k

�1i uiPI,m¥0 de la sous-algèbre de Drinfeld-Cartan. Un

`-poids est donc une famille pΨ�i,�mqm¥0, que l'on écrit par sa fonction génératrice :

ΨΨΨ � pΨipzqqiPI , où Ψipzq �¸m¥0

Ψi,�mz�m.

Un vecteur v d'un Uqpgq-module est dit de plus haut `-poids s'il est de `-poids et s'ilest annulé par les générateurs x�i,r de la partie positive de la décomposition triangulaire(I.1.2.3).

I.1. Contexte 9

Chari et Pressley [CP95b] ont montré le résultat fondamental suivant : toute représen-tation irréductible de Uqpgq est engendrée par un vecteur de plus haut `-poids, de plus,une représentation irréductible engendrée par un vecteur de plus haut `-poids ΨΨΨ est dedimension �nie si et seulement si celui-ci est de la forme :

ΨΨΨ � pΨipzqqiPI , Ψipzq � qdegpPiqi

Pipzq�1i q

Pipzqiq, (I.1.3.1)

où les Pi sont des polynômes de coe�cients constants égaux à 1, appelés polynômes deDrinfeld, et qi :� qdi , où D � diagpd1, . . . , dnq est l'unique matrice diagonale telle queB � DC est symétrique et les di sont des entiers positifs premiers entre eux.

En particulier, pour tout i P I, et tout a P C�, si on considère le n-uplet de polynômes :

p1, . . . , 1, 1� azloomooni

, 1, . . . , 1q, (I.1.3.2)

la représentation irréductible de dimension �nie associée, dite représentation fondamentale,est de plus haut poids ωi comme Uqpgq-module, on la note Vipaq. En type A, il s'agit d'unereprésentation d'évaluation. Pour les autres types, comme mentionné précédemment iln'existe pas de morphisme d'évaluation, et en particulier Vipaq est en général plus grandque V pωiq comme Uqpgq-module. Il s'agit là d'une di�érence fondamentale entre le casclassique et le cas quantique, et un des points de di�culté, mais aussi d'intérêt, de l'étudedes représentations des algèbres a�nes quantiques.

La description des représentations simples de Uqpgq donnée par Chari et Pressley nousdonne une paramétrisation e�cace de ces modules, malheureusement elle ne donne que peud'informations vis-à-vis de leur structure. Par exemple, leur dimension n'est pas connue engénéral.

I.1.4 Anneaux de Grothendieck et q-caractères pour les représentationsde dimension �nie

Notons que la catégorie C n'est pas semi-simple. Elle admet un anneau de GrothendieckK0pC q, qui est dé�ni de la manière suivante (voir [EGNO15] pour plus de détails). Pourtout objet simple (représentation irréductible) Xi de la catégorie C et tout objet Y , onnote rY : Xis la multiplicité de Xi dans la suite de composition de Y (la catégorie C estlocalement �nie).

On dé�nit le groupe de Grothendieck comme le groupe abélien libre engendré par lesclasses d'isomorphismes rXis des objets simples (représentations irréductibles) de C . Achaque objet X de C , on associe sa classe rXs dans le groupe de Grothendieck, dé�nie par

rXs �¸i

rX : XisrXis.

Comme la catégorie C est abélienne, pour chaque suite exacte courte 0 Ñ X Ñ Y Ñ Z Ñ0, on a la relation rY s � rXs � rZs dans le groupe de Grothendieck. Comme de plus lacatégorie C est linéaire et monoïdale, et que le foncteur produit tensoriel est biexact, alorsce produit tensoriel induit une loi multiplicative associative sur le groupe de Grothendieckde la catégorie C . celle-ci est dé�nie par, pour tous objets simples Xi et Xj de C :

rXis � rXjs �¸k

rXi bXj : XksrXks

10 Introduction

L'anneau de Grothendieck K0pC q est l'anneau formé par le groupe de Grothendieck munide cette multiplication.

Dans [FR96] et [FR99], Frenkel et Reshetikhin ont dé�ni des W-algèbres déformées.Dans le cas classique (q � 1), les W-algèbres permettent de décrire le centre de l'algèbreenveloppante Upgq comme une intersection de noyaux d'opérateurs dits "d'écrantage". Pourle cas quantique, l'anneau de Grothendieck K0pC q de la catégorie C peut être vu commeune W-algèbre q-déformée.

Dans ce contexte, ils ont pu dé�nir un morphisme d'anneaux injectif, appelé morphismede q-caractère sur l'anneau de Grothendieck de la catégorie C vers un anneau de polynômesde Laurent en un nombre in�ni de variables :

χq : K0pC q Ñ Y � ZrY �1i,a siPI,aPC� . (I.1.4.1)

La construction de ce morphisme en elle-même est basée sur l'existence de la R-matriceuniverselle R évoquée en (I.1.2.4). Pour toute représentation de dimension �nie pV, ρV q deUqpgq, on peut construire une matrice de transfert associée à V de la façon suivante :

tV pzq � TrV�ρV pzq b IdqpR

�P Uqpgqrrzss, (I.1.4.2)

où V pzq est une version twistée de V , et z est un paramètre formel.Remarquons que ces matrices de transfert sont celles évoquées en Section I.1.1 dans la

construction de Baxter. En particulier, elles commutent, pour tous V, V 1, z, z1,�tV pzq, tV 1pz

1q�� 0. (I.1.4.3)

Si g � sl2 et V est une représentation irréductible de dimension 2, et W est un produittensoriel de N copies de V , alors tV agissant sur W devient précisément l'opérateur deBaxter.

Le morphisme de q-caractère est construit à partir des opérateurs tV ainsi qu'un ana-logue du morphisme d'Harish-Chandra.

De façon cruciale, le morphisme obtenu rend compte de la décomposition des repré-sentations en espaces de `-poids. Précisons cela. Frenkel et Reshetikhin ont montré [FR99]que les `-poids des représentations de dimension �nie de Uqpgq étaient de la forme

ΨΨΨ � pΨipzqqiPI , Ψipzq � qdegpQiq�degpRiqi

Qipzq�1i qRipzqiq

QipzqiqRipzq�1i q

, (I.1.4.4)

où les Qi et Ri sont des polynômes de coe�cients constants égaux à 1, généralisant (I.1.3.1).À chaque paire de n-uplet de polynômes ppQiqiPI , pRiqiPIq, décomposés en

Qipzq �ki¹r�1

p1� airzq, Ripzq �li¹s�1

p1� biszq,

on associe le monôme de Laurent¹iPI

�ki¹r�1

Yi,air

li¹s�1

Y �1i,bis

�.

En résumé la variable Yi,a correspond au `-poids

Yi,aú

��1, . . . , 1, qi1� aq�1

i z

1� aqizloooooomooooooni

, 1, . . . , 1

�� . (I.1.4.5)

I.1. Contexte 11

Ainsi, la classi�cation des modules simples dans la catégorie C de Chari-Pressley peut setraduire de la façon suivante : les Uqpgq-modules simples de dimension �nie sont indexéspar les monômes en les variables pYi,aqiPI,aPC� . Pour tout monôme m �

±rk�1 Yik,ak , on

note Lpmq le module simple de plus haut `-poids m par la correspondance (I.1.4.5).Ensuite, il a été montré ([FR99], [FM01]) que le q-caractère d'un Uqpgq-module donnait

les dimensions de ces espaces de `-poids :

χqpV q �¸

ΨΨΨ `-poids

dimpVΨΨΨqrΨΨΨs P Y,

où rΨΨΨs désigne le monôme de Laurent associé au `-poids ΨΨΨ.Par exemple, pour g � sl2, la représentation fondamentale V1paq est de dimension 2 et

son q-caractère est :χqpV1paqq � Y1,a � Y �1

1,aq2. (I.1.4.6)

Malheureusement, il n'existe pas de formule explicite, équivalente à la formule de Weyl,donnant les q-caractères en règle générale. À la place, Frenkel et Mukhin [FM01] proposentun algorithme permettant de les calculer, communément appelé algorithme de Frenkel-Mukhin. Ils ont prouvé que cet algorithme permettait de calculer les q-caractères des re-présentations fondamentales, en revanche il a été démontré [NN11] qu'il ne permet pas decalculer les q-caractères de toutes les représentations irréductibles.

I.1.5 Anneaux de Grothendieck quantiques

Dans cette section, sauf mention explicite du contraire, les algèbres de Lie sont consi-dérées simplement lacées : de type A, D ou E.

Nakajima [Nak01b] proposa un nouvel algorithme, celui-ci permettant de calculer lesq-caractères de tous les modules simples. Cet algorithme se base sur la connaissance desq-caractères des représentations fondamentales, par l'algorithme de Frenkel-Mukhin, et surla contribution de la géométrie, plus particulièrement les variétés de carquois.

Les variétés de carquois de Nakajima sont des variétés algébriques complexes associéesà des carquois, elles forment en elles-mêmes une direction importante de recherche (voir[Nak96], [Sch08], [Gin12], [MO12] et le livre [Kir16]). Le premier lien entre représentationsde carquois et algèbres de Kac-Moody date des années 90 avec les travaux de Ringel[Rin90]. Celui-ci réalisa la partie positive Uqpnq de l'algèbre enveloppante quanti�ée Uqpgqd'une algèbre de Lie g de dimension �nie comme une algèbre de Hall associée à un carquoisqui n'est autre qu'une version orientée du diagramme de Dynkin de g. Peu de temps après,Lusztig [Lus91] combina les travaux de Ringel avec la théorie des faisceaux pervers pourconstruire une base canonique de Uqpnq.

Dans l'approche de Nakajima, des représentations de l'algèbre enveloppante complètesont construites géométriquement, ce qui généralise les constructions en termes de l'al-gèbre de convolution en K-théorie équivariante des variétés de drapeaux partiels en typeA [GV93], [Vas98]. Ces variétés sont en e�et des cas particuliers de variétés de carquois as-sociées à n'importe quelle algèbre de Kac-Moody symétrique [Nak94], [Nak98]. Ceci menaà une construction géométrique de représentations de l'algèbre enveloppante quanti�éevia l'algèbre de convolution en K-théorie équivariante des variétés de carquois [VV99],[Nak01a].

A l'aide de cette approche géométrique, Nakajima [Nak01a] dé�nit une nouvelle basede l'anneau de Grothendieck K0pC q de la catégorie C des représentations de dimension�nie de Uqpgq, formées par les modules standards. Comme la base des modules simples,

12 Introduction

celle-ci est indexée par les monômes en les variables pYi,aqiPI,aPC� . Les modules standards,notés Mpmq, sont les équivalents pour Uqpgq des modules de Weyl. On peut aussi les voircomme des produits tensoriels de modules fondamentaux [VV02a], si m est un monômem �

±rk�1 Yik,ak , alors

Mpmq �râ

k�1

LpYik,akq. (I.1.5.1)

Ensuite, Nakajima montra que la matrice de passage entre la base des modules simpleset celle des modules standards était triangulaire supérieure, pour un certain ordre partielsur les `-poids. Pour tout monôme m, dans l'anneau de Grothendieck de C ,

rMpmqs � rLpmqs �¸

m1 m

Pm1,mrLpm1qs. (I.1.5.2)

En fait, Nakajima montre une version t-déformée de la décomposition (I.1.5.2) :

rMpmqst � tα

�rLpmqst �

¸m1 m

Pm1,mptqrLpm1qst

�, (I.1.5.3)

où α P Z et les Pm1,mptq P Zrts sont des polynômes sans coe�cients constants.Cette relation a lieu dans une version t-déformée de l'anneau de Grothendieck de la

catégorie C , introduite dans [VV03] comme une déformation plate de l'anneau de Gro-thendieck, en termes de faisceaux pervers. Cet anneau de Grothendieck quantique, que l'onnotera KtpC q, est contenu dans un tore quantique Yt,

KtpC q � Yt. (I.1.5.4)

Le tore quantique Yt est une t-déformation non-commutative de l'anneau des `-poids Y de(I.1.4.1), il s'agit d'une Zrt�1s-algèbre engendrée par les variables Y �1

i,a , et des relations det-commutation de la forme suivante :

Yi,a � Yj,b � tNi,jpa,bqYj,b � Yi,a. (I.1.5.5)

Ce tore quantique se projette par l'évaluation en t � 1 sur l'anneau de polynômes deLaurent qui contient l'image du morphisme de q-caractère par (I.1.4.1) :

evt�1 : Yt Ñ Y.

On peut noter que l'anneau de Grothendieck quantique est noté KtpC q alors qu'il fautl'évaluer en t � 1 pour retrouver l'anneau de Grothendieck classique, noté K0pC q.

Géométriquement, les t-déformations des classes des modules standards rMpmqst dansla relation (I.1.5.3) sont interprétées comme des faisceaux constants et les classes des mo-dules simples rLpmqst comme des complexes de cohomologie d'intersection (IC sheaves),d'où une t-graduation qui arrive naturellement. Cette interprétation géométrique a permisà Nakajima de montrer que les coe�cients des polynômes Pm1,mptq étaient positifs et quel'évaluation à t � 1 donnait la multiplicité du module dans le module standard :

Pm1,mp1q � rMpmq, Lpm1qs.

C'est la relation (I.1.5.3) qui donne un algorithme pour obtenir les q-caractères de tousles modules simples de la catégorie C . En e�et, les classes rLpmqst et rMpmqst dans l'anneaude Grothendieck quantique peuvent être vues comme des t-déformations des q-caractères,

I.1. Contexte 13

c'est pour cela qu'elles sont souvent appelées pq, tq-caractères. Tout d'abord, ce sont lespq, tq-caractères des représentations fondamentales qui sont dé�nis, grâce à une version t-déformée de l'algorithme de Frenkel-Mukhin, puis par multiplication et par (I.1.5.1), ceuxdes modules standards. Par construction, ils véri�ent

rMpmqstt�1ÝÝÑ χqpMpmqq.

Ensuite, les pq, tq-caractères des modules simples sont dé�nis par un algorithme àla Kazhdan-Lusztig : il s'agit de l'unique famille d'éléments de l'anneau de Grothen-dieck quantique qui véri�ent la propriété de décomposition (I.1.5.3), ainsi qu'une cer-taine propriété d'invariance. De façon remarquable, les éléments obtenus sont aussi dest-déformations des q-caractères [Nak01b] :

rLpmqstt�1ÝÝÑ χqpLpmqq.

Ainsi, cette méthode fournit un algorithme permettant de calculer les pq, tq-caractères, etdonc les q-caractères, de toutes les représentations simples de dimension �nie. En revanche,cet algorithme ne donne toujours pas de formule explicite, et peut être très complexe àmettre en ÷uvre. On remarque que la première étape de cet algorithme commence par lecalcul des pq, tq-caractères des représentations fondamentales. En type E8, par exemple, lecalcul du pq, tq-caractère de la cinquième représentation fondamentale prend 350h sur unsuper-ordinateur et l'écriture du résultat nécessite 180Go [Nak10].

Par la suite, un certain nombre de ces résultats ont pu être étendus au cas non-simplement lacé (quand g est de type B, C, F ou G). Hernandez [Her04] a dé�ni al-gébriquement des pq, tq-caractères pour les représentations fondamentales en tout type. Saméthode est aussi basée sur une t-déformation de l'algorithme de Frenkel-Mukhin, dont ilfaut prouver dans ce cas qu'il n'échoue pas. Dans le cas simplement lacé, on retrouve parcette méthode les pq, tq-caractères de Nakajima. En revanche, sans l'apport de la géométrie,la positivité des coe�cients dans la décomposition (I.1.5.3) n'est pas garantie et reste pourl'instant au stade de conjecture [Her04], sauf en type B où elle a récemment été prouvéepartiellement par Hernandez-Oya [HO19].

I.1.6 Catégorie O

Le contexte de la catégorie O de représentations dé�nie par Hernandez et Jimbo [HJ12]est une direction naturelle de généralisation de ces résultats.

En regardant le résultat de classi�cation des représentations de Uqpgq irréductiblesde dimension �nie (cf (I.1.3.1)), il est en e�et naturel de se demander quelles sont lesreprésentations de plus haut `-poids dont le plus haut `-poids est formé plus généralementde fractions rationnelles.

Pour répondre à cette question il faut s'intéresser aux représentations de la sous-algèbrede Borel Uqpbq de Uqpgq, pour la présentation de Drinfeld-Jimbo, il s'agit aussi d'une algèbrede Hopf. La décomposition triangulaire de (I.1.2.3) passe à cette sous-algèbre de Borel etla notion de `-poids aussi.

La catégorie O, appelé ici a�ne quantique, est dé�nie comme la catégorie des Uqpbq-modules qui sont sommes de leurs espaces de poids (comme Uqpgq-modules), dont cesespaces de poids sont de dimension �nie, et dont les poids sont dans une union �nie decônes, pour l'ordre partiel de Chevalley sur les poids.

Une représentation irréductible de plus haut `-poids de Uqpbq est alors dans la catégorieO si et seulement si son plus haut `-poids ΨΨΨ � pΨipzqqiPI est formé de fractions rationnelles

14 Introduction

ne s'annulant pas en 0, et n'ayant pas de pôle en 0 [HJ12]. En général, ces représentationsirréductibles ne sont pas de dimension �nie, mais en tant que Uqpgq-modules, ont desespaces de poids de dimension �nie [HJ12].

Cette catégorie possède deux familles de représentations irréductibles particulières : lesreprésentations préfondamentales, positives et négatives. Pour tout i P I et tout a P C�,soit L�i,a la représentation simple de plus haut `-poids

ΨΨΨ�1i,a :� p1, . . . , 1, p1� azq�1looooomooooon

i

, 1, . . . , 1q.

Contrairement à l'écriture (I.1.3.2), il s'agit du `-poids et non des polynômes de Drinfeld.On remarque le lien suivant entre les Yi,a et les ΨΨΨ�1

i,a :

Yi,a � qiΨΨΨi,aq�1i

ΨΨΨ�1i,aqi

, @i P I, a P C�. (I.1.6.1)

Néanmoins, là n'est pas la première occurrence des représentations préfondamentales.Elles sont apparues pour g � sl2 sous le nom de représentations de q-oscillation dans lestravaux de Bazhanov-Lukyanov-Zamolodchikov [BLZ99] en théorie conforme des champs,qui les ont construites explicitement. Puis, elles ont été dé�nies pour g � sl3 par Bazhanov-Hibberd-Khoroshkin [BHK02] et pour g � sln et i � 1 par Kojima [Koj08].

La catégorie O est une catégorie monoïdale. Les objets de la catégorie O ne sont pasnécessairement de longueur �nie, mais sa sous-catégorie formée des objets de longueur �nien'est pas stable par produit tensoriel (voir [BJM�09, Lemme C.1]). Néanmoins, commepour la catégorie O d'une algèbre de Kac-Moody, la multiplicité d'un module simple dansun module donné est bien dé�nie (voir [Kac90, Section 9.6]). On s'intéresse à son anneaude Grothendieck, qui est formé de sommes formelles

χ �¸ΨΨΨ

λΨΨΨrLpΨΨΨqs P K0pOq,

qui véri�ent les hypothèses de construction de la catégorieO (espaces de poids de dimension�nie et poids dans une union �nie de cônes).

Hernandez et Jimbo [HJ12] ont montré que le morphisme de q-caractère de Frenkel-Reshetikhin, donné par la décomposition en espaces de `-poids, s'étendait aux représenta-tions de la catégorie O :

χqpV q �¸ΨΨΨ

dimpVΨΨΨqrΨs P E`,

où E` est un anneau commutatif étendant Y � ZrY �1i,a siPI,aPC� .

Par exemple, le q-caractère de la représentation préfondamentale positive L�i,a est :

χqpL�i,aq � rΨΨΨi,asχi, (I.1.6.2)

où χi est le caractère de la représentation L�i,a (pour l'action de Uqpgq), et ne dépend pasde a P C� (voir [FH15]).

De plus, l'application ainsi dé�nie est un morphisme d'anneaux injectif sur l'anneau deGrothendieck de la catégorie O :

χq : K0pOq Ñ E`.

Cette catégorie de représentations permet de donner des interprétations catégoriquesà certaines relations liées aux systèmes intégrables quantiques. Dans [BLZ99], la matrice

I.1. Contexte 15

de transfert de la représentation de "q-oscillation" mentionnée plus haut est identi�ée àl'opérateur étudié par Baxter dans [Bax72] (voir Section I.1.1). Cela permet d'interpréterla relation de Baxter (I.1.1.1) comme une relation dans l'anneau de Grothendieck de lacatégorie O.

En particulier, ce point de vue apporté par la catégorie O a permis à Hernandez-Frenkel[FH15] de démontrer une conjecture de Frenkel-Reshetikhin [FR99] sur les valeurs propresdes matrices de transfert des représentations de dimension �nies de Uqpgq. Pour toute re-présentation de dimension �nie V de Uqpgq, si l'on remplace chaque variable Yi,a dans son q-caractère par le quotient des classes des représentations préfondamentales rL�

i,aq�1s{rL�i,aqs,

multiplié par la classe rω1s de la représentation de dimension 1 de Uqpbq, sur laquelle lesgénérateurs k�1

j de l'algèbre de Cartan agissent par le poids fondamental ωi, alors le ré-sultat obtenu est égal à la classe de la représentation V dans l'anneau de GrothendieckK0pOq. Par exemple, pour g � sl2, la cas de la relation de Baxter est le suivant : soitV1paq � LpY1,aq, une représentation fondamentale de Uqpgq, de dimension 2, dont le q-caractère a été rappelé en (I.1.4.6). La relation suivante est satisfaite dans l'anneau deGrothendieck K0pOq :

rLpY1,aqs � rL�1,aqs � rω1srL

�1,aq�1s � r�ω1srL

�1,aq3

s. (I.1.6.3)

Il s'agit d'une version catégori�ée de la relation de Baxter (I.1.1.1).Cette dernière relation (I.1.6.3) a aussi une toute autre interprétation : comme relation

d'échange dans une algèbre amassée.

I.1.7 Algèbres amassées

Les algèbres amassées ont été introduites par Fomin et Zelevinsky au début des an-nées 2000 dans une série d'articles [FZ02], [FZ03a], [BFZ05], [FZ07], en lien avec les basescanoniques duales des groupes quantiques. Il s'agit par dé�nition d'objets fortement combi-natoires, mais des connections avec des domaines très variés des mathématiques ont ensuiteété trouvées.

Par exemple dans [FZ03c], Fomin et Zelevinsky les utilisent pour démontrer une conjec-ture de Zamolodchikov sur certains systèmes dynamiques discrets appelé Y -systèmes, pro-venant d'une version thermodynamique de l'Ansatz de Bethe. Ces Y -systèmes sont forte-ment liés [KNS94] à un autre système de relations fonctionnelles liées aux représentationsdes algèbres a�nes quantiques, appelés T -systèmes. En lien plus direct avec les questionsqui nous intéressent, Hernandez et Leclerc ont introduit la notion de catégori�cation monoï-dale d'une algèbre amassée dans [HL10]. Dans ce travail, ils mettent en valeur une structured'algèbre amassée pour l'anneau de Grothendieck d'une sous-catégorie monoïdale de C .

Par la suite, cette approche a été généralisée, par les mêmes auteurs d'une part ([HL13],[HL16a], [HL16b]), mais aussi par d'autres, notoirement par Nakajima [Nak11], Qin [Qin17],Kang-Kashiwara-Kim-Oh [KKKO18].

La structure d'algèbre amassée qui est utilisée plus particulièrement dans ce travail dethèse est la suivante. Considérons la sous-catégorieO� (resp.O�) [HL16b] de la catégorieOformées des représentations dont les constituants simples ont des plus hauts `-poids qui sontdes monômes en les variables Yi,a etΨΨΨi,a (resp.ΨΨΨ

�1i,a ). Il s'agit de sous-catégories monoïdales

de O [HL16b]. Soient O�Z les sous-catégories monoïdales de O�, où l'on a restreint lesvariables Yi,a et ΨΨΨ�1

i,a des plus hauts `-poids à une certaine famille dénombrable. Ce typede restriction avait déjà été considéré par les mêmes auteurs dans [HL10] au sujet de lacatégorie C , elle contient en fait toute l'information importante de la catégorie complète.

16 Introduction

Hernandez et Leclerc ont alors montré que, pour toute algèbre de Lie simple de dimen-sion �nie g, l'anneau de Grothendieck de la catégorie O�Z de représentations de Uqpgq avaitune structure d'algèbre amassée :

ApΓqb E �ÝÑ K0pO�Z q, (I.1.7.1)

où Γ est un carquois in�ni basé sur le diagramme de Dynkin de g, E correspond à l'imagede K0pOq par le morphisme de caractère, et le produit tensoriel est complété pour ad-mettre des sommes formelles (dont les espaces de poids sont de dimension �nie). De plus,l'isomorphisme (I.1.7.1) est obtenu en prenant comme graine initiale les représentationspréfondamentales positives. Un énoncé symétrique existe pour la catégorie O�Z .

Notons que la relation de Baxter catégori�ée (I.1.6.3) correspond à la relation d'échanged'une mutation au sommet pi, aq du carquois Γ.

Nous n'entrerons pas dans les détails techniques concernant les algèbres amassées danscette introduction, mais toutes les notions que nous utilisons dans ce domaine sont rappe-lées en Annexe B.

I.2 Résultats de la thèse

L'objectif de cette thèse est de construire et d'étudier une structure d'anneau de Gro-thendieck quantique pour la catégorie O de représentations de l'algèbre de Borel Uqpbq.

Nous présentons en premier lieu dans le Chapitre 1 des résultats directement liés àcette question principale, concernant la construction d'équivalents des modules standardsdans le contexte de la catégorie O. Ensuite, dans le Chapitre 2, nous entamons la construc-tion de l'anneau de Grothendieck quantique en commençant par dé�nir un tore quantiqueTt étendant le tore quantique Yt. Nous utilisons pour cela des notions liées aux algèbresamassées quantiques. Dans cet esprit, nous construisons dans le Chapitre 3 une structured'algèbre amassée quantique sur l'anneau de Grothendieck quantique de la catégorie C�

Z ,une sous-catégorie monoïdale de C . Puis, nous dé�nissons dans le Chapitre 2 l'anneau deGrothendieck quantiqueKtpO�Z q de la catégorieO

�Z comme une algèbre amassée quantique.

Pour que ces résultats soient cohérents, il nous faut montrer que cet anneau de Grothen-dieck quantique contient celui de la catégorie C�

Z . Ce résultat est montré directement entype A dans le Chapitre 2, et en tout type simplement lacé dans le Chapitre 3, en utilisantla structure d'algèbre amassée quantique de KtpC

�Z q. Subséquemment, dans le Chapitre 2,

sont dé�nis des pq, tq-caractères pour les représentations fondamentales positives, ce quipermet en particulier d'écrire une version t-déformée de la relation de Baxter (I.1.6.3).En�n, dans le Chapitre 4, nous présentons certains résultats supplémentaires obtenus dansle cas où l'algèbre de Lie simple de dimension �nie sous-jacente g est sl2, ainsi que di-vers prolongements possibles, notamment pour généraliser nos résultats en tout type nonnécessairement simplement lacé.

I.2.1 Modules standards asymptotiques

Une première question qui vient naturellement lorsque l'on cherche à généraliser l'ap-proche de Nakajima pour calculer les pq, tq-caractères des modules simples par un algo-rithme à la Kazhdan-Lusztig est celle de la dé�nition d'analogues des modules standardsdans le contexte de la catégorie O. En e�et, pour pouvoir mettre en place un algorithmesimilaire pour calculer les pq, tq-caractères des représentations simples de la catégorie O,il faudrait expliciter une famille de modules qui engendreraient l'anneau de Grothendieck

I.2. Résultats de la thèse 17

de la catégorie O, et dont les pq, tq-caractères seraient faciles à calculer. La dé�nition demodules standards par produit tensoriel de modules fondamentaux (I.1.5.1) est particuliè-rement propice pour ce dernier point. L'approche de Nakajima et de Varagnolo-Vasserotpour dé�nir ces modules standards pour la catégorie C est basée sur une constructiongéométrique, qui n'existe malheureusement pas dans le contexte de la catégorie O, il nousfaut donc proposer une approche plus ad hoc.

Lorsque l'on s'intéresse aux premières motivations derrière l'introduction de la catégorieO et des représentations préfondamentales, on trouve le résultat suivant de Nakajima[Nak03a], démontré dans les cas non-simplement lacés par Hernandez dans [Her06].

Pour tout k P N�, a P C� et i P I, considérons le module de Kirillov-Reshetikhin (oumodule KR) :

Wpiqk,a � LpYi,aYi,aq2i

� � �Yi,aq2k�2i

q. (I.2.1.1)

Alors le q-caractère normalisé du module de Kirillov-Reshetikhin est un polynôme en desvariables A�1

j,b , et la suite des q-caractères normalisés des modules W piq

N,aq�2N�2i

a une limite

quand k Ñ �8 comme série formelle en ces variables. De plus, dans [HJ12], Hernandezet Jimbo montrent que cette limite est égale au q-caractère normalisé de la représentationpréfondamentale négative ΨΨΨ�1

i,aqi. Ce résultat vient du fait que, par (I.1.6.1), le `-poids Ψ�1

i,a

peut être vu comme un produit in�ni de Yj,b normalisés :

Ψ�1i,a � lim

kÑ�8Yi,aq�1

iYi,aq�3

i� � � Yi,aq�2k�1

i,

où Yi,a �Yi,aqi

.En se basant sur la même idée, nous cherchons donc à construire un module standard

MpΨΨΨ�1i,a q comme produit tensoriel in�ni de modules fondamentaux :

LpYi,aq�1iq b LpYi,aq�3

iq b � � � b LpYi,aq�2k�1

iq b � � �ùMpΨΨΨ�1

i,a q.

La première étape est de véri�er que cette dé�nition peut faire sens du point de vuedes q-caractères, car le but de la dé�nition de ces modules standard asymptotiques est decalculer des q-caractères.

On répond à cette question par la positive comme premier résultat du Chapitre 1.

Théorème 1. (Théorème 1.2.2) La suite des q-caractères normalisés d'un produit tensorield'un nombre croissant de modules fondamentaux converge comme série formelle en lesvariables A�1

j,b :

χq

�LpYi,aq�1

iq b LpYi,aq�3

iq b � � � b LpYi,aq�2N�1

iq

NÑ�8ÝÝÝÝÝÑ χ8i,a P ZrrA�1

j,b ss.

La deuxième étape est de donner un sens du côté de la théorie des représentations àcette limite, donc de construire un Uqpbq-module MpΨ�1

i,a q dont le q-caractère normaliséserait χ8i,a.

C'est ce qui est proposé dans la suite du Chapitre 1. Pour des raisons techniques, lereste de ce chapitre est concentré sur le cas où g � sl2. Tout d'abord, nous donnons unsens au produit tensoriel in�ni :

T � LpYq�1q b LpYq�3q b LpYq�5q b LpYq�7q b � � � .

Celui-ci est construit de façon "localement �nie", de manière à ce qu'il soit de façon natu-relle une limite d'une suite croissante de modules standards.

18 Introduction

On construit ensuite explicitement une action de l'algèbre Uqpbq¥0, qui correspondau produit tensoriel de la partie positive de Uqpbq et de la partie Drinfeld-Cartan, dans ladécomposition triangulaire (I.1.2.3), sur l'espace vectoriel T (Proposition 1.3.3). On montreensuite qu'il s'agit d'un bon candidat pour notre module standard asymptotique, avec lerésultat suivant.

Proposition 2. (Proposition 1.3.7) Le Uqpbq¥0-module T est engendré par ses vecteurs de`-poids, ses espaces de `-poids sont de dimension �nie, et son q-caractère véri�e

χqpT q � rΨΨΨ�11 sχ81,1.

Malheureusement, cette action ne s'étend pas naturellement à l'algèbre Uqpbq entière,et pour obtenir un Uqpbq-module ayant le bon q-caractère, il est nécessaire de passer parun procédé d'induction :

T c :� T bUqpbq¥0 Uqpbq�.L'argument clé à cette étape est de montrer que cette induction ne produit pas de

nouveaux vecteurs de `-poids, qui changeraient le q-caractère. C'est e�ectivement ce quenous montrons en Proposition 1.3.18. Ce qui amène naturellement au

Théorème 3. (Théorème 1.3.19) L'espace vectoriel T c est un Uqpbq-module dont les es-paces de `-poids sont de dimension �nies et son q-caractère véri�e :

χqpTcq � rΨΨΨ�1

1 sχ81,1.

Ce résultat est bien sûr étendu aux `-poids négatifs plus généraux que ΨΨΨ�11 , mais la

construction centrale est celle de MpΨΨΨ�11 q.

Dans la dernière section du Chapitre 1, nous nous proposons d'étudier les propriétés dedécomposition et de multiplicativité du q-caractère sur ces nouveaux modules standardsasymptotiques. En particulier, nous démontrons que χqpT cpΨΨΨ

�11 qq a une décomposition en

somme de modules simples de la forme de (I.1.5.2) :

Théorème 4. (Théorème 1.4.3)

χqpTcpΨΨΨ�1

1 qq � χqpL�1 q �

¸ΨΨΨ ΨΨΨ�1

1

Pm1,mχqpLpΨΨΨqq,

pour un ordre partiel qui étend celui considéré par Nakajima.

I.2.2 Tore quantique étendu

Comme expliqué précédemment, l'autre pan de ce travail est la construction concrèted'un anneau de Grothendieck quantique pour la catégorie O�Z , et l'étude de sa structure.

La première étape d'un travail dans cette direction consiste à la mise en place d'untore quantique étendu, contenant le tore quantique Yt de (I.1.5.4), et pouvant contenir lespq, tq-caractères des représentations préfondamentales.

Dans la suite, l'algèbre de Lie g simple de dimension �nie à l'origine de l'algèbre deBorel Uqpbq dont nous étudions les représentations, est supposée simplement lacée.

Comme mentionné précédemment, le fait de s'intéresser à la sous-catégorie monoïdaleO�Z de O� n'est que peu restrictif car celle-ci est assez riche pour contenir toute l'informa-tion importante de la catégorie O�. Dorénavant, les `-poids considérés seront les

ΨΨΨ�1i,r � ΨΨΨ�1

i,qr ,

Y �1j,s � Y �1

j,qs ,


où pi, rq P I et pj, sq P J , et I et J sont des sous-ensembles de I � Z dénombrables.Pour pouvoir construire des pq, tq-caractères pour la catégorie OZ dans ce nouveau tore

quantique, celui-ci doit contenir les `-poids préfondamentaux ΨΨΨ�1i,r , ainsi que des sommes

(potentiellement in�nies) et des produits de ceux-ci.On commence donc par former le tore quantique Tt comme une Zrt�1s-algèbre engen-

drée par les ΨΨΨ�1i,r et par des relations de t-commutations de la forme :

ΨΨΨi,r �ΨΨΨj,s � tFijps�rqΨΨΨj,s �ΨΨΨi,r, @ppi, rq, pj, sq P Iq. (I.2.2.1)

Bien évidemment, la loi de t-commutation F doit être choisie de façon cohérente avec lesrelations de t-commutation sur Yt (I.1.5.5), et avec le lien entre les variables Yi,r et les ΨΨΨi,r

(I.1.6.1).La relation suivante est donc imposée sur la fonction F :

Nijpr, sq � 2Fijps� rq � Fijps� r � 2q � Fijps� r � 2q, @ppi, rq, pj, sq P Iq. (I.2.2.2)

Il semblerait donc qu'il existe un certain degré de liberté dans le choix de fonction det-commutation F . Cependant, nous voulons que certaines relations particulières soientvéri�ées dans notre anneau de Grothendieck quantique, comme par exemple une versionquanti�ée de la relation de Baxter (I.1.6.3).

Comme expliqué en Section I.1.7, la relation de Baxter est en fait une relation d'échangecorrespondant à une mutation dans une algèbre amassée. Les algèbres amassées possèdentdes t-déformations non-commutatives naturelles en algèbres amassées quantiques, intro-duites par Berenstein et Zelevinsky dans [BZ05]. Pour pouvoir dé�nir une telle algèbreamassée quantique, l'on doit disposer d'une part d'un carquois Γ, qui correspond à ladonnée d'une algèbre amassée classique, et d'autre part d'une matrice anti-symétriqueΛ � pΛijq encodant les relatons de t-commutation entre les variables initiales :

Xi �Xj � tΛijXj �Xi.

De plus, ces deux objets doivent véri�er une condition de compatibilité : pΓ,Λq doit formerce qu'on appelle une paire compatible (voir Chapitre 2, Section B.2).

La structure d'algèbre amassée de l'anneau de Grothendieck de la catégorie O�Z de(I.1.7.1) est construite sur un carquois in�ni Γ, en prenant comme graine initiale les classesdes représentations préfondamentales positives rL�i,rs. De plus, par (I.1.6.2) nous savonsque les q-caractères de ces représentations sont égaux, à un inversible près, à leur plushaut `-poids rΨΨΨi,rs. Ainsi, pour pouvoir obtenir une structure d'algèbre amassée quantiquequi t-déforme la structure d'algèbre amassée de K0pO�Z q, il faut nécessairement véri�er lacompatibilité entre le carquois Γ et la fonction de t-commutation entre les ΨΨΨi,r.

Nous �xons donc une famille de fonctions F � pFijqi,jPI telle que la relation (I.2.2.2)soit véri�ée, et que la "matrice I � I" donnée par

Λppi, rq, pj, sqq � Fijps� rq,

forme une paire compatible avec le carquois Γ de (I.1.7.1), utilisé pour dé�nir une structured'algèbre amassée sur l'anneau de Grothendieck K0pO�Z q.

Proposition 5. (Proposition 2.4.10) Soit B est la matrice d'adjacence du carquois Γ, alorspBTΛq forme une paire compatible, plus précisément,

pBTΛqppi, rq, pj, sqq �

"�2 si pi, rq � pj, sq,

0 sinon, (I.2.2.3)

20 Introduction

Notons que le carquois Γ est un carquois in�ni, ce qui ne correspond pas à la dé�nitiond'algèbre amassée quantique telle qu'elle apparait dans [BZ05]. Cependant, nous montronsque nous pouvons en fait considérer une suite croissante de carquois �nis ΓN , qui formentdes paires compatibles pΓN ,ΛN q, où ΛN est la restriction de Λ à la partie �nie du carquois.Comme les variables d'amas sont toujours obtenues par un nombre �ni de mutations, ceprocédé permet e�ectivement d'obtenir l'algèbre amassée quantique de "rang in�ni" commelimite d'algèbres amassées quantiques de rangs �nis.

On conclue la construction du tore quantique étendu en posant :

Tt :� TtbZrt�1{2sE , (I.2.2.4)

où le produit tensoriel est complété pour admettre certaines sommes in�nies, comme dans(I.1.7.1).

Par construction, on a donc naturellement une inclusion de tores quantiques :

Proposition 6. (Proposition 2.3.3) Il existe un morphisme injectif de Zrt�1{2s-algèbres

J : Yt Ñ Tt. (I.2.2.5)

I.2.3 L'anneau de Grothendieck quantique KtpC�Z q comme algèbre amas-

sée quantique

Cette construction de tore quantique étendu a introduit une nouvelle donnée au pro-blème : si les anneaux de Grothendieck des catégories de représentations d'algèbres a�nesquantique peuvent avoir des structures d'algèbres amassées, alors leurs anneaux de Gro-thendieck quantiques peuvent avoir des structures d'algèbres amassées quantiques !

Revenons au cas des représentations de dimension �nies. Dans ce cas, l'anneau de Gro-thendieck est déjà connu, ainsi que la structure d'algèbre amassée, nous pouvons doncvéri�er que celle-ci se déforme bien en une algèbre amassée quantique, et que nous retrou-vons par ce moyen la structure d'anneau de Grothendieck quantique. Il s'agit là du travailprésenté dans la première moitié du Chapitre 3.

Tout d'abord, intéressons-nous à la catégorie C des représentations de dimension �niede l'algèbre a�ne quantique Uqpgq. Nous pouvons comme précédemment se restreindre àl'étude d'une sous-catégorie tensorielle CZ, engendrée par les modules simples dont les plushauts `-poids sont des monômes en les pYi,rqpi,rqPJ . Hernandez et Leclerc ont considéré dans

[HL16a] la catégorie C�Z , sous-catégorie monoïdale C�

Z de CZ, où nous avons restreint lesplus hauts `-poids des modules simples aux monômes en les Yi,r, avec r ¤ 0. Ils ont montréque l'anneau de Grothendieck de cette catégorie avait une structure d'algèbre amassée,basée sur un carquois "semi-in�ni" G�, qui correspond à la moitié du carquois in�ni Γ. Ilsmontrent de plus que les q-caractères de certains modules de Kirillov-Reshetikhin (et doncen particulier de certaines représentations préfondamentales) s'obtiennent comme variablesd'amas à partir de la graine initiale

ui,r �¹

r�2k¤0

Yi,r�2k, (I.2.3.1)

et de suites �nies de mutations. L'idée est de voir ces variables d'amas comme des q-caractères tronqués, c'est-à-dire où l'on n'a gardé que les variables Y �1

i,r , avec r ¤ 0. Lesvariables d'amas initiales sont e�ectivement des q-caractères tronqués, où il ne reste duq-caractère que le terme de plus haut `-poids. Ensuite, une suite particulière de mutations


est explicitée, telle qu'à chaque étape, la relation d'échange correspondant à la mutationest une relation de T -systèmes, comme mentionnés en Section I.1.7. Ces relations de T -systèmes sont des relations véri�ées dans l'anneau de Grothendieck, elles ont été démontréespar Nakajima [Nak04] en type ADE, et par Hernandez [Her06] en tout type. Ceci permetde montrer que toutes les variables d'amas apparaissant suite à ces mutations successivessont des q-caractères tronqués, et qu'en appliquant ce procédé un certain nombre de fois,les q-caractères tronqués deviennent complets. Notons que les résultats de [HL16a] ontété énoncés ici dans le cas simplement lacé, mais sont en fait véri�és dans les cas non-simplement lacés.

Dans notre travail, nous nous proposons de démontrer que, dans le cas où g est simple-ment lacé, l'anneau de Grothendieck quantique de la catégorie C�

Z a une structure d'algèbreamassée quantique, en t-déformant cette structure d'algèbre amassée.

On commence par montrer que si nous considérons le tore quantique T � Yt, engendrépar les variables u�1

i,r données en (I.2.3.1), alors si L est la matrice donnant les relations det-commutation entre ces variables

ui,r � uj,s � tLppi,rq,pj,sqquj,s � ui,r, (I.2.3.2)

(qui est entièrement déterminée par la famille de fonctions N de (I.1.5.5)), celle-ci formeune paire compatible, au sens des algèbres amassées quantiques, avec le carquois G�.

Proposition 7. (Proposition 3.4.1) Si B� est la matrice d'adjacence du carquois G�, alorscelle-ci forme une paire compatible avec la fonction L :

pBT�Lqppi, rq, pj, sqq �

"�2 si pi, rq � pj, sq,

0 sinon.

Ainsi, nous pouvons dé�nir à partir de la donnée initiale pG�, Lq, une algèbre amasséequantique :

At :� AtpG�, Lq.

Il s'agit d'une sous Zrt�1{2s-algèbre du tore quantique T .Il reste ensuite à montrer que cette algèbre amassée quantique est isomorphe à l'anneau

de Grothendieck KtpC�Z q de la catégorie C�

Z . Pour cela, l'idée est d'adapter en versionquantique les arguments présentés dans [HL16a], en gérant les di�cultés apportées par lanon-commutativité.

Premièrement, les relations de T -systèmes ont des analogues dans l'anneau de Gro-thendieck quantique, appelés T -systèmes quantiques, qui ont été démontrés dans les cassimplement lacés par Nakajima dans [Nak03a]. Nous remarquons en Section 3.3.6 queces relations s'étendent aux pq, tq-caractères tronqués (la troncature est faite de manièreidentique à la troncature des q-caractères). Nous constatons aussi qu'en prenant commegraine initiale les variables ui,r, celles-ci sont aussi égales aux pq, tq-caractères tronquésdes modules de Kirillov-Reshetikhin correspondants. Ainsi, en considérant la même suitede mutations quantiques dans le carquois G� que Hernandez et Leclerc, nous obtenons lerésultat suivant :

Proposition 8. (Proposition 3.5.2) Les variables d'amas qui apparaissent dans cette suitede mutations sont des pq, tq-caractères tronqués de modules de Kirillov-Reshetikhin. Deplus, en appliquant cette suite de mutations un certain nombre de fois, nous obtenons despq, tq-caractères complets.

22 Introduction

Un élément important qui di�ère de la preuve de [HL16a] est le fait que pour vé-ri�er que les relations d'échange de la suite de mutations quantiques sont e�ectivementdes T -systèmes quantiques, il faut montrer que les puissances en t qui apparaissent sontles mêmes. Pour cela, nous utilisons le fait que les variables d'amas quantiques et lespq, tq-caractères des modules simples sont des éléments de leurs tores quantiques respectifsinvariants par des involutions barres, et que ces involutions correspondent à la même ap-plication, à l'identi�cation des tores quantiques près. Il est intéressant de noter ici que cerésultat peut aussi être obtenu en utilisant la positivité des variables d'amas quantiquesdémontrée par Davison dans [Dav18], mais qu'il s'agit d'un résultat beaucoup plus fort, etqui n'est �nalement pas nécessaire dans ce cas.

Ensuite, il ne reste qu'à montrer le résultat suivant :

Théorème 9. (Théorème 3.4.6) L'identi�cation entre les variables d'amas initiales et lespq, tq-caracyères tronqués des modules de Kirillov-Reshetikhin donne un isomorphisme

At�ÝÑ KtpC

�Z q. (I.2.3.3)

Là encore, il s'agit en partie d'une généralisation de la preuve de [HL16a, Théorème5.1]. En revanche, comme les variables véri�ent des relations de t-commutation, les identi-�cations des variables doivent se faire avec précaution, nous démontrons donc dans ce butle Lemme 3.5.4.

L'inclusion KtpC�Z q � At a essentiellement déjà été démontrée ci-dessus. En e�et, l'an-

neau de Grothendieck quantique KtpC�Z q est engendré par les pq, tq-caractères tronqués des

représentations préfondamentales, et nous avons déjà vu que ceux-ci s'obtenaient commevariables d'amas quantiques dans l'algèbre amassée quantique At.

Pour l'inclusion inverse, il s'agit de montrer que toutes les variables d'amas quantiquessont des éléments de l'anneau de Grothendieck quantique. Nous procédons par récurrencesur la longueur des chemins dans le graphe d'échange de l'algèbre amassée quantique. Lesvariables d'amas initiales sont e�ectivement dans l'anneau de Grothendieck quantique,comme démontré en Proposition 3.5.2. Pour l'induction, nous utilisons la caractérisationdes éléments de l'anneau de Grothendieck quantique comme zéros d'une famille d'opéra-teurs appelés opérateurs d'écrantage. Ce point de la preuve est particulièrement délicat àgénéraliser par rapport à ce qui a été fait dans [HL16a]. En fait, Hernandez a dé�ni dans[Her03] des t-déformations de ces opérateurs d'écrantage. Il s'agit d'une famille d'opéra-teurs

Si,t : Yt Ñ Yi,t, (I.2.3.4)

où Yi,t est un certain Yt-module, qui sont des dérivations, et tels que

KtpCZq �£iPI

kerpSi,tq. (I.2.3.5)

Le fait que ces opérateurs soient des dérivations permet de transmettre naturellement lapropriété d'appartenance à l'anneau de Grothendieck quantique aux nouvelles variablesd'amas crées par mutations. En e�et, si la nouvelle relation d'échange est de la forme

Z � Z1 � tαM1 � tβM2,

où nous supposons par récurrence que la variable d'amas Z, ainsi que les monômes d'amasM1 etM2 sont dans l'anneau de Grothendieck quantique, alors l'application des opérateursd'écrantage Si,t permet d'obtenir

0 � Z � Si,tpZ1q.


Pour en déduire que la nouvelle variable d'amas Z1 est aussi dans le noyau de l'opérateurnous utilisons le fait que le module Yi,t est libre. Dans le cas non t-déformé, la liberté de cemodule est donnée dans la construction [FR99] ; en revanche dans notre situation elle n'estpas évidente car ce module est dé�ni comme un quotient. Néanmoins, nous le montronsdans le Lemme 3.3.4 de la Section 3.3.3. On peut noter que la caractérisation (I.2.3.5)de l'anneau de Grothendieck quantique comme intersection des noyaux des opérateursd'écrantage est valable pour l'anneau de Grothendieck quantique KtpCZq et non pour celuiqui nous intéresse KtpC

�Z q. Pour outrepasser cette di�culté, il nous faut au préalable se

ramener au cas de KtpCZq en appliquant à notre algèbre amassée des suites in�nies demutations, qui sont en fait localement des suites �nies.

En conclusion, nous avons montré que cette structure d'algèbre amassée quantique,construite à partir de la structure d'algèbre amassée de l'anneau de Grothendieck K0pC

�Z q,

et des relations de t-commutations du tore quantique Yt, est égale à l'anneau de Grothen-dieck quantique KtpC

�Z q.

L'avantage de cette approche est qu'elle fournit un algorithme permettant de calculerles pq, tq-caractères des modules de Kirillov-Reshetikhin, et donc en particulier des modulesfondamentaux. Dans le Chapitre 3, nous nous sommes intéressés aux nombre précis d'étapesqui étaient nécessaires pour obtenir ces pq, tq-caractères complets et il semble clair quele nombre d'étapes de calcul est largement inférieur à celui de l'algorithme de Frenkel-Mukhin (voir Remarque 3.6.10). De plus, cet algorithme ne nécessite pas de garder enmémoire toutes les étapes de calcul précédentes, mais uniquement la graine courante, cequi pourrait permettre de palier aux problèmes de mémoire rencontrés lorsque l'on essaiede calculer explicitement ces pq, tq-caractères en dimension élevée (voir [Nak10]).

Notons que Qin a obtenu dans [Qin17] des résultats parallèles à ceux présentés ici.Il a démontré que pour certaines sous-catégories de C engendrées par un nombre �nide modules fondamentaux, leur anneau de Grothendieck quantique avait une structured'algèbre amassée quantique. Dans notre travail, nous présentons une preuve plus directede ce résultat, en explicitant une suite de mutation permettant d'obtenir les pq, tq-caractèresdes modules fondamentaux.

I.2.4 Anneau de Grothendieck quantique pour la catégorie O� et calculdes pq, tq-caractères

On a donc montré que pour les représentations de dimension �nie, l'anneau de Grothen-dieck quantique avait une structure d'algèbre amassée quantique. De plus, nous avons déjàutilisé en Section I.2.2 l'apport des algèbres amassées quantiques pour �xer les fonctionsde t-commutation entres les variables ΨΨΨ�1

i,r . Tout cela nous encourage à construire l'anneaude Grothendieck quantique KtpO�Z q comme une algèbre amassée.

D'autant plus que, dans le cas de la catégorie O, les approches usuelles ne fonctionnentpas. En e�et, l'approche géométrique est inenvisageable, car il n'existe pas de constructiongéométrique des représentations de la catégorie O. Une autre approche usuelle est devoir l'anneau de Grothendieck quantique comme un espace invariant par des symétries deWeyl. Par exemple, la caractérisation (I.2.3.5) pour la catégorie CZ est ce qui a permis àHernandez de dé�nir un anneau de Grothendieck quantique et des pq, tq-caractères pour lescas non-simplement lacés dans [Her04]. Malheureusement, il n'existe pas de caractérisationsemblable pour la catégorie O, et ainsi l'approche par les algèbres amassées quantiquessemble être celle qui fonctionne.

Nous avons déjà présenté le fait que le carquois in�ni Γ et la fonction de t-commutation,

24 Introduction

encodée dans la matrice I � I anti-symétrique B formaient une paire compatible. Nousposons donc en Dé�nition 2.4.15,

KtpO�Z q :� AtpΓ, BqbZrt�1{2sE � Tt, (I.2.4.1)

où le produit tensoriel est complété comme en (I.1.7.1) et en (I.2.2.4).Bien sûr, il est important de véri�er que cette dé�nition d'anneau de Grothendieck

quantique pour la catégorie O�Z est cohérente avec les structures préexistantes. En parti-culier, nous savons que la catégorie O�Z contient la catégorie CZ, il faut donc déjà voir queKtpO�Z q contient l'anneau de Grothendieck quantique KtpCZq.

Pour cela, nous utilisons la structure d'algèbre amassée quantique de KtpC�Z q que nous

venons d'étudier. Nous montrons qu'en appliquant la même suite de mutations qui per-mettent dans KtpC

�Z q d'obtenir les pq, tq-caractères (non tronqués) des représentations fon-

damentales dans l'algèbre amassée KtpO�Z q, nous obtenons encore les pq, tq-caractères desreprésentations fondamentales. Ainsi, ces pq, tq-caractères sont des variables d'amas dansl'algèbre a�ne quantique KtpO�Z q et nous en déduisons le résultat principal du Chapitre 3 :

Théorème 10. (Théorème 3.6.5) L'anneau de Grothendienck quantique KtpO�Z q contientcelui de la catégorie C�

Z :KtpC

�Z q � KtpO�Z q. (I.2.4.2)

Pour pouvoir obtenir ce résultat, il s'agit de voir l'algèbre amassée quantique AtpG�, Lqcomme une sous-structure de AtpΓ,Λq. Sur ce point, les algèbres amassées quantiques sontplus délicates à gérer que leurs homologues classiques. La bonne dé�nition de sous-structurepour les algèbres amassées quantiques a été donnée par Grabowski et Gratz dans [GG18]et elle nécessite que les carquois soient seulement connectés par coe�cients, ce qui signi�eque les sommets du plus petit carquois qui sont reliés à des sommets qui ne sont quedans le plus grand carquois correspondent à des variables gelées. Nous constatons que lescarquois Γ et G� ne sont pas uniquement connectés par coe�cients. Nous introduisonsdonc un carquois intermédiaire Γ� en ajoutant des variables gelées à G�. Les carquois Γet Γ� sont alors uniquement connectés par coe�cients, et AtpΓ�,Λq est une sous-algèbreamassée quantique de AtpΓ,Λq.

De plus, l'algèbre amassée quantique AtpΓ�,Λq peut être obtenue comme une versiontwistée deAtpG�, Lq, dans les sens des twists apparaissant dans les travaux de Grabowski etLaunois [GL14]. Nous en déduisons que les variables d'amas de AtpΓ�,Λq sont des variablesd'amas de AtpG�, Lq, à multiplication par les variables gelées qui ont été ajoutées près. Lacombinaison de ces deux approches permet de faire un lien entre AtpG�, Lq et AtpΓ,Λq etde voir les pq, tq-caractères des représentations fondamentales comme des variables d'amasde AtpΓ,Λq et de montrer l'inclusion (I.2.4.2).

Dans le Chapitre 2, cette inclusion avait été présentée sous forme de conjecture, maisdémontrée lorsque l'algèbre de Lie simple de dimension �nie g est de type A. Dans ce cas,les arguments utilisés ne sont pas exactement les mêmes. En type A, il est connu que les es-paces de `-poids des modules fondamentaux sont de dimension 1 (voir [FR96, Section 11] etles références qui y sont mentionnées). De plus, nous savons par [HL16b] que dans l'algèbreamassée (classique) ApΓq, il existe des suites de mutations permettant d'obtenir les classesdes représentations fondamentales comme variables d'amas. Dans la Proposition 2.5.9 duChapitre 2, nous constatons que les variables d'amas quantiques obtenues en appliquantla même suite de mutations dans l'algèbre amassée quantique AtpΓ,Λq sont des élémentsinvariants par l'involution barre, qui redonnent les q-caractères des représentations fon-damentales en les évaluant en t � 1, et dont la décomposition sur la base des monômes


commutatifs est à coe�cients positifs. En type A, cela su�t à montrer que ces variablesd'amas quantique s'identi�ent aux pq, tq-caractères des représentations fondamentales :

Théorème 11. (Théorème 2.6.1) Les variables d'amas obtenues en appliquant cette suitede mutations sont les pq, tq-caractères des représentations fondamentales.

On remarque que dans ce cas, nous utilisons de façon cruciale le résultat de positivité desvariables d'amas quantiques de Davison [Dav18]. De plus, cette preuve ne fonctionne pasen dehors du type A, où des multiplicité apparaissent et où les coe�cients des q-caractèresdes modules fondamentaux ont plusieurs t-déformations possibles.

En particulier, en Section 3.7, nous calculons explicitement le pq, tq-caractère de lareprésentation fondamentale LpY2,�6q, quand g est de type D4. Il s'agit de l'exemple leplus simple où une multiplicité apparait (voir Remarque 3.7.1).

En�n, à partir de cette donnée, nous pouvons construire des pq, tq-caractères pour lesreprésentations préfondamentales positives dans la catégorieO�Z . En e�et, si nous suivons lamême logique, les pq, tq-caractères doivent être obtenus dans l'algèbre amassée quantiqueAtpΓ,Λq par les mêmes mutations utilisées pour obtenir les classes des représentationspréfondamentales dans l'algèbre amasséeApΓq. Or, dans l'isomorphisme (I.1.7.1), les classesdes représentations préfondamentales sont prises comme variables d'amas initiales, à unpoids près :

zi,r � rL�i,qr s b

��r

2ωi

�.

Pour obtenir leurs q-caractères, nous multiplions ces variables par un poids inversible :

χqpL�i,qrq � rΨΨΨi,rsχi P E`,

en utilisant la notation rΨΨΨi,rs � zi,r�r2ωi

�.

Le même procédé dans l'anneau de Grothendieck quantiqueKtpO�Z q donne donc commepq, tq-caractères (voir Dé�nition 2.5.3.1) :

rLi,qr st :� rΨΨΨi,rs b χi P Tt. (I.2.4.3)

I.2.5 Relations remarquables dans l'anneau de Grothendieck quantique

Après avoir dé�ni l'anneau de Grothendieck quantique KtpO�Z q , ainsi que les pq, tq-caractères des représentations préfondamentales positives rLi,qr s, il est possible de travailleravec ces objets. Par exemple, nous pouvons écrire des relations qui sont véri�ées dans cetanneau. De plus, la Zrt�1{2s-algèbre KtpO�Z q a été dé�nie comme une algèbre amasséequantique, une structure qui est riche en relations.

On peut commencer par écrire une relation d'échange provenant d'une mutation quel-conque à n'importe quel sommet du carquois Γ. Sur la base des monômes commutatifs,pour tout pi, rq P I, il existe α, β P 1

2Z tels que :

zp1qi,r � zi,r � tαzi,r�2

¹j�i

zj,r�1 � tβzi,r�2

¹j�i

zj,r�1. (I.2.5.1)

Lorsque g � sl2, cette relation a une traduction particulièrement intéressante en termesde pq, tq-caractères. Dans ce cas, (I.2.5.1) devient :

zp1q2r � z2r � t1{2z2r�2 � t�1{2z2r�2. (I.2.5.2)

26 Introduction

Or, dans ce cas, c'est cette relation d'échange qui permet d'obtenir le pq, tq-caractèrede la représentation fondamentale, par le raisonnement expliqué précédemment,

zp1q2r � J prLpY2r�1qstq.

Donc (I.2.5.2) se traduit par le résultat suivant :

Proposition 12. (Proposition 2.6.3) Pour tout r P Z, on a, dans l'anneau de Grothendieckquantique KtpO�Z q,

rLpY2r�1qst � rL2rst � t�1{2rω1srL2r�2st � t1{2r�ω1srL2r�2st. (I.2.5.3)

Il s'agit d'une version quantique de la relation de Baxter (I.1.6.3), qui était une versioncatégori�ée de la relation originelle de Baxter (I.1.1.1).

Quand g � sl2, nous remarquons une structure particulièrement intéressante dans unesous-algèbre de KtpO�Z q. Ce sont les résultats détaillés dans la Section 2.6.2. Si nous nousintéressons à une sous-algèbre amassée quantique AtpΓ1,Λ1q de AtpΓ,Λq de type �ni, alorsnous pouvons �xer des variables d'amas particulières

E,F,K,K 1 P AtpΓ1,Λ1q,

qui engendrent l'algèbre amassée AtpΓ1,Λ1q et qui permettent de dé�nir un isomorphsimeentre AtpΓ1,Λ1q et un quotient du double de Drinfeld de paramètre �t1{2.

Le double de Drinfeld D2 que nous considérons est le double de Drinfeld, au sens de[Dri87], de la sous-algèbre de Borel du groupe quantique (non-a�nisé) Uqpsl2q. Il s'agit dela Cpqq-algèbre engendrée par E,F,K,K 1 et par les relations

KE � q2EK, K 1E � q�2EK 1

KF � q�2FK, K 1F � q2FK 1

KK 1 � K 1K, et rE,F s � pq � q�1qpK �K 1q.

L'élément de Casimir est alors :

Cq :� EF � qK � q�1K 1.

Proposition 13. (Proposition 2.6.6) L'algèbre amassée quantique ApΛ1,Γ1q est isomorph-sime au quotient du double de Drinfeld par l'élément de Casimir :

ApΛ1,Γ1q�ÝÑ D2{C�t1{2 . (I.2.5.4)

Ce résultat est à rapprocher du travail de Schrader et Shapiro [SS19], où des liens sontexplicités entre des variétés d'amas et le groupe quantique Uqpsln�1q.

Il est intéressant de noter que l'isomorphisme (I.2.5.4) fait intervenir le groupe quan-tique Uqpsl2q entier. Il existe de nombreux résultats de ce type, comme par exemple dans[GLS13], ou [HL15], mais qui ne réalisent que la partie positive du groupe quantique.

Dans le Chapitre 4, nous présentons quelques pistes d'ouvertures des travaux présentésdans cette thèse. En particulier, la Section 4.1 présente des résultats plus avancés dans lecas où g � sl2 que nous espérons généraliser par la suite. Nous trouvons dans cette sectiond'autres relations importantes qui peuvent être écrites en type A1.

Pour ce type, nous dé�nissons un pq, tq-caractère pour les représentations préfondamen-tales négatives de la façon suivante (voir (4.2.1.2)) :

rL�q2rst :� rΨΨΨ�1

q2rs

�1�

�8

m�1

m¹l�0

A�1q2r�2l

�P Tt.


Ce qui nous permet d'écrire une nouvelle version de la relation de Baxter quanti�ée (I.2.5.3),en termes de représentations préfondamentales négatives.

Proposition 14. (Proposition 4.1.4) Pour tout r P Z :

rLpYq2r�1qst � rLq2r st � t1{2rω1srL�q2r�2st � t�1{2r�ω1srL

�q2r�2st.

On peut aussi former une relation entre les pq, tq-caractères des représentations préfon-damentales positives et négatives. Cette relation, appelée Relation Wronskien quantique, ades généralisation à d'autres types sous le nom de systèmes QQ. Ce type de relation estfortement liée aux systèmes intégrables quantiques dont il a été question en Section I.1.1car elles permettent d'encoder en une relation le système d'équations de l'Ansatz de Bethe.

La version t-déformée de la relation de Wronskien quantique est la suivante :

Proposition 15. (Proposition 4.1.11) Pour tout r P Z,

rL�q2rst � rL

�q2rst � t�1r�α1srL

�q2r�2st � rL

�q2r�2st � χ1.

I.2.6 Pistes d'ouverture

Dans le Chapitre 4, en plus de résultats supplémentaires dans le cas g � sl2, sontprésentées quelques pistes de généralisations de certains de ces résultats.

En particulier, nous proposons en Section 4.2.1 une dé�nition des pq, tq-caractères desreprésentations préfondamentales négatives, comme des éléments du tore quantique Tt.Ces pq, tq-caractères sont vus comme des limites de pq, tq-caractères de modules de Kirillov-Reshetikhin (voir (4.2.1.2)).

De plus, nous présentons en Section 4.2.2 une méthode pour dé�nir un anneau deGrothendieck quantique KtpO�Z q pour la catégorie O�Z , basée elle-aussi sur une structured'algèbre amassée quantique, ainsi que des arguments en faveur de cette méthode.

En�n, en Section 4.2.3, nous évoquons la généralisation de nos résultats aux algèbresde Lie g non-simplement lacées.

Chapter 1Asymptotics of standard modules ofquantum a�ne algebras

This chapter is an adapted version of [Bit18], published in Algebras and RepresentationTheory.

Abstract. We introduce a sequence of q-characters of standard modules of a quan-

tum a�ne algebra and we prove it has a limit as a formal power series. For g � sl2, weestablish an explicit formula for the limit which enables us to construct corresponding

asymptotical standard modules associated to each simple module in the category O of

a Borel subalgebra of the quantum a�ne algebra. Finally, we prove a decomposition

formula for the limit formula into q-characters of simple modules in this category O.

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.1.1 The quantum a�ne algebra and its Borel subalgebra . . . . . . . . . 311.1.2 Representations of the Borel algebra . . . . . . . . . . . . . . . . . . 341.1.3 A q-character theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2 Limits of q-characters of standard modules . . . . . . . . . . . . . . . . . . . 391.2.1 Standard modules for �nite dimensional representations . . . . . . . 391.2.2 Limits of q-character . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.2.3 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.3 Asymptotical standard modules . . . . . . . . . . . . . . . . . . . . . . . . . 431.3.1 Construction of the U¥0

q pbq-module. . . . . . . . . . . . . . . . . . . 431.3.2 Construction of induced modules . . . . . . . . . . . . . . . . . . . . 48

1.4 Decomposition of the q-character of asymptotical standard modules . . . . . 521.4.1 Partial order on P r

` . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.4.2 Decomposition for MpΨΨΨ�1

1 q . . . . . . . . . . . . . . . . . . . . . . . 531.4.3 General decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Introduction

The quantized universal enveloping algebra of a �nite-dimensional simple Lie algebrag was introduced independently by Drinfeld [Dri87] and Jimbo [Jim85] in 1985. It is aq-deformation of the universal enveloping algebra Upgq, for q a generic complex number,and it has a Hopf algebra structure. If g is an untwisted a�ne Kac-Moody algebra, thenits universal enveloping algebra also admits a q-deformation, called the quantum a�ne

30 Chapter 1. Asymptotics of standard modules

algebra Uqpgq. The quantum a�ne algebra can also be obtained as a quantum a�nizationof the quantized universal enveloping algebra of the corresponding simple Lie algebra.Both processes (a�nization then quantization and quantization then quantum a�nization)commute ([Dri88, Bec94]). The quantum a�ne algebra has a presentation with the so-called "Drinfeld-Jimbo generators", and from this presentation one can de�ne its Borelsubalgebra Uqpbq � Uqpgq.

Both the quantum a�ne algebra and the Borel algebra are Hopf algebras, thus theircategories of �nite-dimensional representations are monoidal categories. In [FR99], Frenkeland Reshetikhin de�ned the q-character χq, which is an injective ring morphism on theGrothendieck ring of the category of �nite-dimensional representations of Uqpgq, mimickingthe usual character morphism. For every �nite-dimensional representation V of Uqpgq, onecan consider its q-character χqpV q. It describes the decomposition of V into `-weight spacesthe action of the `-Cartan subalgebra of Uqpgq. These `-weight are more suited than theweights spaces to the study of �nite-dimensional Uqpgq-modules (for detailed results, see[FM01] for example).

Hernandez and Jimbo introduced in [HJ12] a category O of representations of the Borelalgebra Uqpbq. Objects in this category O are sums of their `-weight spaces and have �nite-dimensional weight spaces, with weights in a �nite union of certain cones. The category Ocontains the �nite-dimensional representations, as well as some other remarkable represen-tations, called prefundamental representations. For g � sl2, the latter appeared naturally,under the name q-oscillator representations in Conformal Field Theory (for example in[BLZ99]). In general, the prefundamental representations are constructed as limits ofparticular sequences of simple �nite-dimensional modules [HJ12]. Note that the prefunda-mental representations were used in [FH15] to prove the Frenkel-Reshetikhin's conjectureon the spectra of quantum integrable systems [FR99] (generalizing certain results of Baxter[Bax72] on the spectra of the 6 and 8-vertex models).

The q-character morphism extends to a well-de�ned injective ring morphism on theGrothendieck ring of the category O ([HJ12]).

In the category O, objects do not necessarily have �nite lengths (see [BJM�09, LemmaC.1], the tensor product of some prefundamental representations presents an in�nite �l-tration of submodules), thus the category O is not a Krull-Schmidt category. However, asin the case of the classical category O of representations of a Kac-Moody Lie algebra, themultiplicity of a simple module in a module is well-de�ned. As such, all the elements ofthe Grothendieck ring of O can be written as (in�nite) sums of classes of simple modules.

There is a second basis of the Grothendieck ring of �nite-dimensional Uqpbq-modulesformed by the standard modules, introduced in [Nak01a]. Both simple and standard mod-ules are indexed by the same set: the monomials m in the ring ZrYi,asiPI,aPC� . In [Nak01a],Nakajima showed that for the simply laced types, the multiplicities of the simple modules inthe standard modules can be realized as dimensions of certain varieties. Nakajima showedthat the transition matrix between theses two bases is upper triangular:


m1 m

Pm1,mrLpm1qs, (1.0.0.1)

for some partial order ¤ on the monomials in ZrYi,asiPI,aPC� , the exact de�nition of theorder is recalled in (1.2.2.3), in the proof of Theorem 1.2.2.

Moreover, the coe�cients Pm1,m P Z are non-negative. Moreover, using t-deformationsof the q-characters, Nakajima showed ([Nak04]) that the coe�cients Pm1,m can be computedas the evaluation at t � 1 of some polynomials (analogs of the Kazhdan-Lusztig polynomials

1.1. Background 31

for Weyl groups). With this type of formula, one can hope to deduce from the q-charactersof the standard modules the q-characters of the simple modules, which are not known inthe general case.

One would want to have the same type of decomposition in the category O, in whichthe simple modules also form a basis. For that, the �rst step is to build analogs of thestandard modules corresponding to each simple module in the category O.

In order to do that, let us recall that the q-characters of the prefundamental represen-tations can be obtained as limits (as formal power series) of sequences of q-characters ofsimple modules.

In the present Chapter, we establish that a limit of a particular sequence of q-charactersof standard modules exists. We conjecture that this limit is the q-character of a certainasymptotical standard module. In the case g � sl2 we give an explicit formula for the limitand we prove this conjecture. In this case, the simple modules are known for certain naturalmonoidal subcategories (see [HL16b, Theorem 7.9]), thus we can work the other way anddeduce from the simple modules information about the standard modules. Furthermore,will we show at the end of the chapter that the q-characters of these asymptotical standardmodules satisfy a decomposition formula of the type (1.0.0.1).

This chapter is organized as follows. The �rst section presents the background for ourwork. It gathers the de�nitions and already known properties of the objects we study. Westart with the de�nitions of the quantum a�ne algebra and the Borel algebra. Section 1.1.2presents some results on the representations of the Borel algebra, speci�cally the categoryO of Hernandez-Jimbo. Finally, we present the q-character theory for the category O.

In Section 1.2, we recall the de�nitions of standard modules for �nite-dimensionalrepresentations, and study more precisely their q-characters, which will be the base forthe construction of the asymptotical standard modules. We prove the convergence, asa formal power series, of some sequences of normalized q-character of standard modules(Theorem 1.2.2). Then we conjecture (Conjecture 1.2.7) that these limits are q-charactersof certain asymptotical standard modules.

From Section 1.3 on, we focus on the case where g � sl2. Section 1.3 tackles the tech-nical part of the construction of the standard modules, with the two main theorems. First,we build a U¥0

q pbq-module T , with �nite-dimensional `-weight spaces, such that its normal-ized q-character is the limit obtained from the result of Theorem 1.2.2 (Theorem 1.3.12).Then we induce from T a Uqpbq-module with �nite-dimensional `-weight spaces and thesame q-character as T (Theorem 1.3.19).

Finally, in Section 1.4 we present some results for the decomposition of the q-charactersof our asymptotical standard modules. The last theorem is a result of the type of (1.0.0.1):the limit obtained in Theorem 1.2.2, which from Theorem 1.3.19 can be realized as the q-character of a Uqpbq-module, admits a decomposition into a sum of q-characters of simplemodules in the category O. Moreover, the coe�cients are non-negative integers (Theo-rem 1.4.3).

1.1 Background

1.1.1 The quantum a�ne algebra and its Borel subalgebra

Let us start by recalling the de�nitions of the two main algebras we study: the quantuma�ne algebra Uqpgq and its Borel subalgebra for the Drinfeld-Jimbo generators Uqpbq.


Quantum a�ne algebra

Let g be an untwisted a�ne Kac-Moody algebra, with C � pCi,jq0¤i,j¤n its Cartanmatrix. Let 9g be the associated �nite-dimensional simple Lie algebra, let I � t1, . . . , nube the vertices of its Dynkin diagram and 9C � pCi,jqi,jPI its Cartan matrix.

Let pαiqiPI , pα_i qiPI and pωiqiPI be the simple roots, the simple coroots and the funda-mental weights of 9g, respectively. We use the usual lattices Q �

ÀiPI Zαi, Q� �

ÀiPI Nαi

and P �À

iPI Zωi. Let PQ � P b Q, endowed with the partial ordering: ω ¤ ω1 if andonly if ω1 � ω P Q�. Let D � diagpd0, d1, . . . , dnq be the unique diagonal matrix such thatB � DC is symmetric and the pdiq0¤i¤n are relatively prime positive integers.

Fix an non-zero complex number q, which is not a root of unity, and h P C such thatq � eh. Then for all r P Q, qr :� erh. Since q is not a root of unity, for r, s P Q, we haveqr � qs if and only if r � s. We set qi :� qdi , for 0 ¤ i ¤ n.

We use the following notations, for m, r, s P N, r ¥ s and z P C�,

rmsz �zm�z�m

z�z�1 , rmsz! �±mj�1rjsz,

�r

s

�z

� rrsz !rssz !rr�ssz ! .

De�nition 1.1.1. One de�nes the quantum a�ne algebra Uqpgq as the C-algebra generatedby ei, fi, k

�1i , 0 ¤ i ¤ n, together with the following relations, for 0 ¤ i, j ¤ n,

kikj � kjki, rei, fjs � δi,jki � k�1

i

qi � q�1i

, kik�1i � k�1

i ki � 1,

kiejk�1i � q

Ci,ji ej , kifjk

�1i � q

�Ci,ji ej ,

1�Ci,j¸r�0

p�1qrep1�Ci,j�rqi eje

prqi � 0, pi � jq,

1�Ci,j¸r�0

p�1qrfp1�Ci,j�rqi fjf

prqi � 0, pi � jq

where xprqi � xri {rrsqi !, pxi � ei, fiq.

The algebra Uqpgq has another presentation, with the Drinfeld generators ([Dri88],[Bec94])

x�i,rpi P I, r P Zq, φ�i,�mpi P I,m ¥ 0q, k�1i pi P Iq,

and some relations we will not recall here, but which are also q-deformations of the Weyland Serre relations.

Example 1.1.2. For g � sl2, one has the following correspondence

e1 � x�1,0, f1 � x�1,0,

e0 � k�11 x�1,1, f0 � x�1,�1k1.

Let us introduce the generating series, for i P I

φ�i pzq �¸m¥0

φ�i,�mz�m � k�1

i exp

��pqi � q�1

i q¸r¡0

hi,�rz�r

�P Uqphqrz�1s.

Thus, the pφ�i,�mqiPI,m¥0 and the pk�1i , hi,�rqiPI,r¡0 generate the same subalgebra of Uqpgq:

the `-Cartan subalgebra Uqpgq0.

1.1. Background 33

The phi,�rqiPI,r¡0 can be useful because their relations with the Drinfeld generators aresimpler to write. For example, for all i, j P I, r P Zzt0u and m P Z,

rhi,r, x�j,ms � �

rr 9Ci,jsqr

x�j,r�m. (1.1.1.1)

The quantum a�ne algebra has a triangular decomposition, associated to the Drinfeldgenerators: let Uqpgq� be the subalgebra of Uqpgq generated by the px�i,rqiPI,rPZ. Then([Bec94]),

Uqpgq � Uqpgq� b Uqpgq0 b Uqpgq�. (1.1.1.2)

The algebra Uqpgq has a natural Q-grading, with, for all i P I,m P Z, r ¡ 0,

degpx�i,mq � �αi, degphi,rq � degpk�1i q � 0. (1.1.1.3)

The Borel subalgebra

De�nition 1.1.3. The Borel algebra Uqpbq is the subalgebra of Uqpgq generated by theei, k

�1i , for 0 ¤ i ¤ n.

Let Uqpbq� � Uqpgq� X Uqpbq and Uqpbq0 � Uqpgq0 X Uqpbq, then [BCP99]

Uqpbq� � xx�i,myiPI,m¥0, Uqpbq0 � xφ�i,r, k�i yiPI,r¡0.

Remark 1.1.4. In general, such a nice description does not exist for Uqpbq�, except wheng � sl2. In that case, Uqpbq� is isomorphic to the algebra de�ned by the generatorspx�1,mqm¥1, together with the relations ([Jan96, Section 4.21]), for all m, l ¥ 1, i, j P I

x�i,m�1x�j,l � q�

9Ci,jx�j,lx�i,m�1 � q�

9Ci,jx�i,mx�j,l�1 � x�j,l�1x

�i,m. (1.1.1.4)

We also use the subalgebra U¥0q pbq:

U¥0q pbq :� xx�i,m, φ

�i,r, k

�i yiPI,m¥0,r¡0. (1.1.1.5)

The triangular decomposition of (1.1.1.2) carries over ([Bec94]):

Uqpbq � Uqpbq� b Uqpbq0 b Uqpbq�.

From now on, we are going to consider representations of the Borel algebra Uqpbq.Example 1.1.5. The algebra Uqpgq has only one 1-dimensional representation (of type 1),but Uqpbq has an in�nite family of one-dimensional representations, indexed by PQ: for eachω P PQ, rωs denote the one-dimensional representation on which the peiqiPI act trivially

and ki act by multiplication by qωpαiqi .

It may seem that by studying representations of Uqpbq we consider many more repre-sentations, but we will see that for �nite-dimensional representations, the simple modulesare essentially the same.


Hopf algebra structure

The algebra Uqpgq has a Hopf algebra structure, where the coproduct and the antipodeare given by, for i P t0, . . . , nu,

∆peiq � ei b 1� ki b ei,

∆pfiq � fi b k�1i � 1b fi,

∆pkiq � ki b ki, Spkiq � k�1i

Speiq � �k�1i ei, Spfiq � �fiki.

(1.1.1.6)

With these coproducts and antipodes, the Borel algebra Uqpbq is a Hopf subalgebra ofUqpgq.

We have the following result for the coproducts in Uqpbq, where the Q-grading followsfrom the Q-grading on Uqpgq de�ned in (1.1.1.3).

Proposition 1.1.6. [Dam98, Proposition 7.1] For r ¡ 0 and m P Z,

∆phi,rq P hi,r b 1� 1b hi,r � U�q pbq b U�q pbq,

∆px�i,mq P x�i,m b 1� Uqpbq b pUqpbqX�q,

where U�q pbq (resp. U�q pbq) is the subalgebra of Uqpbq consisting of elements of positive(resp. negative) Q-degree, and X� �

°iPI,mPZCx

�i,m � U�q pbq.

These relations of "approximate coproducts" are going to be crucial in the de�nitionof the asymptotical standard modules in Section 1.4.3.

1.1.2 Representations of the Borel algebra

In this Section, we recall some results on the representations of Uqpbq. First of all,we recall the notion of `-weights and highest `-weight modules. These notions are at thecenter of the study of Uqpbq-modules, as are weights and highest weight modules in thestudy of representations of semi-simple Lie algebras. Then, we cite some results on the�nite-dimensional representations of the Borel algebra. And �nally, we recall Hernandez-Jimbo's category O for the representations of Uqpbq.

Highest `-weight modules

Let V be a Uqpbq-module and ω P PQ a weight. One de�nes the weight space of V ofweight ω by

Vω :� tv P V | kiv � qωpα_i qi v, 0 ¤ i ¤ nu.

The vector space V is said to be Cartan diagonalizable if V �À

ωPPQVω.

De�nition 1.1.7. A series ΨΨΨ � pψi,mqiPI,m¥0 of complex numbers, such that ψi,0 P qQi for

all i P I is called an `-weight. The set of `-weights is denoted by P`. One identi�es the`-weight ΨΨΨ to its generating series:

ΨΨΨ � pψipzqqiPI , ψipzq �°m¥0 ψi,mz

m.

1.1. Background 35

The sets PQ and P` have group structures (the elements of P` are invertible formalseries) and one has a surjective group morphism $ : P` Ñ PQ which satis�es ψip0q �

q$pΨΨΨqpα_i qi . Let V be a Uqpbq-module and ΨΨΨ P P` an `-weight. One de�nes the `-weightspace of V of `-weight ΨΨΨ by

VΨΨΨ :� tv P V | Dp ¥ 0,@i P I,@m ¥ 0, pφ�i,m � ψi,mqpv � 0u.

A non-zero vector v P V which belongs to an `-weight space is called an `-weight vector.

Remark 1.1.8. As φ�i,0 � ki, one has VΨΨΨ � V$pΨΨΨq.

De�nition 1.1.9. Let V be a Uqpbq-module. It is said to be of highest `-weight ΨΨΨ P P` ifthere is v P V such that V � Uqpbqv,

eiv � 0,@i P I and φ�i,mv � ψi,mv, @i P I,m ¥ 0.

In that case, the `-weight ΨΨΨ is entirely determined by V , it is called the `-weight of V , andv is the highest `-weight vector of V .

Proposition 1.1.10. [HJ12] For all ΨΨΨ P P` there is, up to isomorphism, a unique simplehighest `-weight module of `-weight ΨΨΨ, denote it by LpΨΨΨq.

Example 1.1.11. For ω P PQ, the one-dimensional representation de�ned in Example 1.1.5

is the simple representation rωs � LpΨΨΨωq, with pΨΨΨωqipzq � qωpα^i qi , for all i P I.

Let us de�ne some particular simple modules.

De�nition 1.1.12. For all i P I and a P C�, de�ne the fundamental representation Vi,a

as the simple module LpYi,aq, where Yi,apzqj �

#qi

1�aq�1i z

1�aqizif j � i

1 if j � i.

If we generalize this de�nition:

De�nition 1.1.13. For i P I, a P C� and k ¥ 0, the Kirillov-Reshetikhin module (or

KR-module) W piqk,a is the simple Uqpbq-module,

Wpiqk,a � LpYi,aYi,aq2i

� � �Yi,aq

2pk�1qi

q.

We are going to see in the next section that these are �nite-dimensional representations.Let us de�ne another family of `-weights. For i P I and a P C�, let ΨΨΨ�1

i,a be the `-weightsatisfying

pψ�1i,a qjpzq �

"p1� azq�1 if i � j,

1 otherwise.

Then,

De�nition 1.1.14. For i P I and a P C�, de�ne

L�i,a :� LpΨΨΨ�1i,a q.

The modules L�i,a (resp. L�i,a) are the positive prefundamental representations (resp. nega-

tive prefundamental representations).

The construction of the pL�i,aqiPI,aPC� is detailed in [HJ12]. It is an asymptotical con-struction, in particular, they are in�nite dimensional representations.

For all `-weight ΨΨΨ, one can consider the normalized `-weight

ΨΨΨ � pΨΨΨp0qq�1ΨΨΨ, (1.1.2.1)

which is an `-weight of weight 0. For example, for i P I, a P C�, Yi,a � ΨΨΨi,aq�1ipΨΨΨi,aqiq

�1.


Finite-dimensional representations

Let C be the category of �nite-dimensional Uqpbq-modules. The prωsqωPPQ and theKirillov-Reshetikhin modules are examples of �nite dimensional simple Uqpbq-modules.

As stated before, these are not so di�erent from the Uqpgq-modules. In particular, onehas

Proposition 1.1.15. [HJ12, References for Proposition 3.5] Let V be a simple �nite-dimensional Uqpgq-module. Then V is simple as a Uqpbq-module.

Using this result and the classi�cation of �nite-dimensional simple module of quantuma�ne algebras in [CP95a], as well as [FH15, remark 3.11], one has

Proposition 1.1.16. Let ΨΨΨ P P`. Then the simple Uqpbq-module LpΨΨΨq is �nite dimensionalif and only if, there exists ω P PQ such that ΨΨΨ1 � ΨΨΨ{ΨΨΨω satis�es: for all i P I, Ψ1

ipzq is ofthe form

Ψ1ipzq � q

degpPiqi

Pipzq�1i q

Pipzqiq,

where Pi are polynomials.Moreover, in that case, the action of Uqpbq can be uniquely extended to an action of

Uqpgq.Remark 1.1.17. Equivalently, LpΨΨΨq is �nite-dimensional if and only if ΨΨΨ is a monomialin the pYi,aqiPI,aPC� . In particular, fundamental representations, and more generally KR-modules, are examples of �nite-dimensional Uqpbq-modules.

Example 1.1.18. For g � sl2, for all a P C�,

Va � Cv�a ` Cv�a ,

with v�a of `-weight Ya and v�a of `-weight Y �1aq2

.This example will be used later.

Category O

The category O of representations of the Borel algebra was �rst de�ned in [HJ12],mimicking the usual de�nition of the category O BGG for Kac-Moody algebras. Here, weuse the de�nition in [HL16b], which is slightly di�erent.

For all λ P PQ, de�ne Dpλq :� tω P PQ | ω ¤ λu.

De�nition 1.1.19. A Uqpbq-module V is in the category O if

1. V is Cartan diagonalizable,

2. For all ω P PQ, one has dimpVωq 8,

3. There is a �nite number of λ1, . . . , λs P PQ such that all the weights that appear inV are in the cone

�sj�1Dpλjq.

The category O is a monoidal category [HJ12].

Example 1.1.20. All �nite dimensional Uqpbq-modules are in the category O. Moreover thepositive and negative prefundamental representations are also in the category O.

Let P r` be the set of `-weights Ψ such that, for all i P I, Ψipzq is rational. We need the

following result.

Theorem 1.1.21. [HJ12] Let Ψ P P`. Simple objects in the category O are highest `-weightmodules. The simple module LpΨq is in the category O if and only if Ψ P P r

` . Moreover, ifV is in the category O and VΨ � 0, then Ψ P P r

` .

1.1. Background 37

1.1.3 A q-character theory

The q-characters of �nite-dimensional representations of quantum a�ne algebras wereintroduced in [FR99] using transfer-matrices. Here, we consider representations in thecategory O, which are not necessarily �nite-dimensional. Hence we use the q-charactermorphism on the Grothendieck ring of the category O de�ned in [HJ12]. More precisely,it is the version of [HL16b] we use here (since we also use the de�nition of the category Ofrom [HL16b]).

After recalling the de�nition of the q-characters, we use it to de�ne some interestingsubcategories of the category O: the categories O�,O� and O�Z ,O

�Z , as in [HL16b].

q-characters for category O

Let E` be the additive group of all maps c : P r` Ñ Z whose support, supppcq � tΨ P

P r` | cpΨq � 0u satis�es: $psupppcqq is contained in a �nite union of sets of the form Dpµq,

and, for all ω P PQ, the set supppcq X$�1pωq is �nite. Similarly, E is the additive groupof maps c : PQ Ñ Z whose support is contained in a �nite union of sets of the form Dpµq.The map $ naturally extends to a surjective morphism $ : E` Ñ E .

For all ΨΨΨ P P r` , let rΨΨΨs � δΨΨΨ P E` (resp. rωs � δω P E , for all ω P PQ).

Remark 1.1.22. One notices that this notation is coherent with the ones from Exam-ple 1.1.11. Indeed, for all ω P PQ, the simple one-dimensional representation rωs � LpΨΨΨωqis identi�ed with the map δω P E .

De�nition 1.1.23. Let V be a module in the category O. The q-character of V is thefollowing element of E`:

χqpV q �¸

ΨΨΨPP r`

dimpVΨΨΨqrΨΨΨs. (1.1.3.1)

The character of V is the following element of E :

χpV q � $pχqpV qq �¸ωPPQ

dimpVωqrωs.

Remark 1.1.24. [HL16b, Section 3.2] In the category O, every object does not neces-sarily have �nite length. But, as for the category O of a classical Kac-Moody algebra(see [Kac90]), the multiplicity of a simple module is well-de�ned. Hence we have itsGrothendieck ring K0pOq. Its elements are formal sums, for each M P O,

rM s �¸

ΨΨΨPP r`

λΨΨΨ,M rLpΨΨΨqs, (1.1.3.2)

where λΨΨΨ,M is the multiplicity of LpΨΨΨq in M . These coe�cients satisfy¸ΨΨΨPP r

` ,ωPPQ

|λΨΨΨ,M |dimppLpΨΨΨqqωqrωs P E .

K0pOq has indeed a ring structure, because of 3, in De�nition 1.1.19.

The q-character morphism is the group morphism,

χq : K0pOq ÞÑ E`

which sends a class rV s of a representation V to χqpV q. It is well de�ned, as χq is compatiblewith exact sequences.


Remark 1.1.25. The de�nition of a q-character makes sense for more general representationsthan that in the category O. For every representation V with �nite-dimensional `-weightspaces, (1.1.3.1) has a sense. However the resulting q-character is not necessarily in the ringE`. Moreover, the module is not necessarily the sum of its `-weight spaces (for example,the Verma modules associated to the `-weights, as in [HJ12, Section 3.1]).

For V a module in the category O having a unique `-weight ΨΨΨ whose weight is maximal,one can consider its normalized q-character χqpV q:

χqpV q :� rΨΨΨ�1s � χqpV q.

For i P I and a P C�, de�ne Ai,a as

Yi,aq�1iYi,aqi

�� ¹tjPI|Cj,i��1u

Yj,a¹

tjPI|Cj,i��2u

Yj,aq�1Yj,aq¹

tjPI|Cj,i��3u

Yj,aq�2Yj,aqYj,aq2

� �1

.

For all i P I, a P C�, $pAi,aq � αi.

Theorem 1.1.26. [FR99, FM01] For V a simple �nite-dimensional Uqpgq-module, onehas

χqpV q P ZrA�1i,a siPI,aPC� .

One has more precise results when V is a fundamental representation, we will use somelater on.

Categories O� and O�

Let us now recall the de�nitions of some subcategories of the category O, introducedin [HL16b]. These categories are interesting to study because the `-weights of the simplemodules have some unique decomposition.

De�nition 1.1.27. An `-weight of P r` is said to be positive (resp. negative) if it is a

monomial in the following `-weights:• the Yi,a � qiΨΨΨ

�1i,aqi

ΨΨΨi,aq�1i, where i P I and a P C�,

• the ΨΨΨi,a (resp. ΨΨΨ�1i,a ), where i P I and a P C�,

• the rωs, where ω P PQ.

Let us denote by P�` (resp. P�

` ) the ring of positive (resp. negative) `-weights.

De�nition 1.1.28. The category O� (resp. O�) is the category of representations in Owhose simple constituents have a positive (resp. negative) highest `-weight, in the sense of(1.1.3.2), that is: for M in O�, one can write,

χqpMq �¸

ΨΨΨPP�`

λΨΨΨ,M rLpΨΨΨqs.

Remark 1.1.29. (i) The category O� (resp. O�) contains C , the category of �nite-dimensional representations, as well as the positive (resp. negative) prefundamentalrepresentations L�i,a (resp. L

�i,a), for all i P I, a P C�.

(ii) The generalized Baxter's relations in [FH15] are satis�ed in the Grothendieck ringsK0pO�q.

(iii) Positive `-weights have a unique factorization into a product of Yi,a and ΨΨΨi,a. Inparticular, for g � sl2, this implies a unique factorization of simple modules intoproducts of prime simple representations in O� (see [HL16b, Theorem 7.9]).

Theorem 1.1.30. [HL16b] The categories O� and O� are monoidal categories.

1.2. Limits of q-characters of standard modules 39

The categories O�Z and O�Z

First, let us recall the in�nite quiver de�ned in [HL16a, Section 2.1.2]. Let V � I � Zand Γ be the quiver with vertex set V whose arrows are given by

ppi, rq Ñ pj, sqq ðñ pCi,j � 0 and s � r � diCi,jq.

Select one of the two connected components of Γ (see [HL16a, Lemma 2.2]) and call itΓ. The vertex set of Γ is denoted by V .

Example 1.1.31. For g � sl2, the in�nite quiver is

Γ :

(1,-2)

(1,0) (1,2) (1,4)

(1,-1) (1,1) (1,3) (1,5)

� � �

� � ��

� � �

In that case, the choice of a connected component is the choice of a parity.

De�nition 1.1.32. [HL16b] De�ne the category O�Z (resp. O�Z ) as the subcategory ofrepresentations of O� (resp. O�) whose simple components have a highest `-weight ΨΨΨsuch that the roots and poles of Ψipzq are of the form qri , with pi, rq P V .

Remark 1.1.33. (i) [HL16b, 4.3] One does not lose any information by only studyingthe subcategories O�Z , instead of O�. Indeed, as in the case of �nite dimensionalrepresentations, each simple object in O� has a decomposition into a tensor productof simple objects which are essentially in O�Z .

(ii) Moreover, these categories are interesting to study by themselves, because they arecategori�cation of some cluster algebras (see [HL16b, Theorem 4.2]).

1.2 Limits of q-characters of standard modules

In this section, we recall the de�nition of standard modules for �nite-dimensional repre-sentations. Then we show that the q-characters of a speci�c sequence of standard modulesconverge to some limit. We conjecture that this limit is the q-character of some Uqpbq-module which respects some of the structure of the �nite-dimensional standard modules.

1.2.1 Standard modules for �nite dimensional representations

For �nite-dimensional representations of a quantum a�ne algebra, one can de�ne thestandard module associated to a given highest `-weight. Let m be a monomial m �Yi1,a1Yi2,a2 � � �YiN ,aN , then the associated standard module is the following tensor productof fundamental representations

Mpmq :� Vi1,a1 b Vi2,a2 b � � � b ViN ,aN , (1.2.1.1)

where the tensor product is written so that:

if k l, then al{ak R qN. (1.2.1.2)

Recall the following result, which is a weaker form of [Cha02, Theorem 5.1 and Corollary5.3]:


Lemma 1.2.1. For pa, bq P pC�q2 such that a{b R qZ, and all i, j P I, then

Vi,a b Vj,b � Vj,b b Vi,a, and this module is irreducible.

The de�nition of the standard module Mpmq by the expression (1.2.1.1) is unambigu-ous, it does not depend on the order of the factors, as long as (1.2.1.2) is satis�ed. Indeed,with Lemma 1.2.1, as long as the condition (1.2.1.2) is satis�ed, two such tensor productsare isomorphic.

Moreover, each simple �nite-dimensional representation has a highest `-weight m thatcan be uniquely written as a monomial in the Yi,a's.

One would want to de�ne analogs of the standard modules for a larger category ofrepresentations, namely the category O.

1.2.2 Limits of q-character

It is known ([Nak03a, Theorem 1.1] for simply laced types and [Her06, Theorem 3.4]in general) that the normalized q-characters of KR-modules, seen as polynomials in A�1

i,a ,have limits as formal power series. In [HJ12, Section 6.1], the normalized q-character ofa negative prefundamental representation is explicitly obtained, as a formal power seriesin ZrrA�1

i,a ss, as a limit of a sequence of normalized q-characters of KR-module (hence�nite-dimensional representations).

The idea here is to consider the limit of a speci�c sequence of normalized q-charactersof �nite-dimensional standard modules. The formal power series we obtain is a conjecturalnormalized q-character for a potential in�nite-dimensional standard module.

For i P I and a P C�, consider the negative `-weight ΨΨΨ � ΨΨΨ�1i,a .

One has pΨΨΨqj � 0 for j � i and pΨΨΨqi pzq �1

1�az . Heuristically, one wants to write

1

1� az�

1� aq�2i z

1� az�

1� aq�4i z

1� aq�2i z

�1� aq�6

i z

1� aq�4i z

�1� aq�8z

1� aq�6z� � � �

��Yi,aq�1

iYi,aq�3

iYi,aq�5

iYi,aq�7

i� � �

ipzq,

with the normalized `-weights de�ned in (1.1.2.1). Thus consider, for N ¥ 1,

mN :�N�1¹k�0

Yi,aq�2k�1i

.

As stated before, from [HJ12, Theorem 6.1], one knows that, as formal power series, thenormalized q-characters satisfy

χqpLpmN qq ÝÝÝÝÝÑNÑ�8

χqpL�i,aq.

For N ¥ 1, one can look at the normalized q-character of the standard module associ-ated to the same `-weight mN . Consider the standard module

SN :� Vi,aq�1ib Vi,aq�3

ib Vi,aq�5

ib � � � b Vi,aq�2N�1

i. (1.2.2.1)

And its normalized q-character

χN :� χqpSN q �N�1¹k�0

χq

�Vi,aq�2k�1

i

�

N�1¹k�0

χq

�LpYi,aq�2k�1

iq.

1.2. Limits of q-characters of standard modules 41

Theorem 1.2.2. For all N ¥ 1, χN P ZrA�1j,aql

sjPI,lPZ.

As a formal power series in pA�1j,aql

qjPI,lPZ, χN has a limit as N Ñ �8,

χN ÝÝÝÝÝÑNÑ�8

χ8i,a P ZrrA�1j,aql

ssjPI,lPZ. (1.2.2.2)

Remark 1.2.3. Here we consider formal sums of monomials in the pA�1j,aql

qjPI,lPZ:

ZrrA�1j,aql

ssjPI,lPZ Q¸

αPNpI�Zq

cαA�1α ,

where each α is a �nitely supported sequence of non-negative integers on I � Z, Aα �±pj,lqPI�ZA

αpj,lq

j,aql. This de�nition is necessary as the result we have to deal with has homo-

geneous parts (for the standard degree degpA�1α q �

°pj,lqPI�Z αpj, lq) with in�nitely many

terms, which was not the case in the limits considered in [HJ12]. For example, here wewant to be able to consider elements such as

°lPZA

�1j,aql

, for all j P I. We get a well-de�nedring of formal power series.

The topology on ZrrA�1j,aql

ssjPI,lPZ is the one of the pointwise convergence: a sequence

of its elements converges only if for each monomial A�1α the corresponding coe�cient

converges, or more precisely is eventually constant, as these coe�cients are integers.

Proof. From Theorem 1.1.26, for all N ¥ 1, χN P ZrA�1j,b sjPI,bPC� . Moreover, by [FM01,

Lemma 6.1], each χqpVi,aq�2k�1i

q is in ZrA�1j,aql

sjPI,lPZ, thus χN too.

By [FM01, proof of Lemma 6.5], for all j P I, b P C�,

χqpVj,bq � 1�¸m, with m ¤ A�1

j,bqj,

for Nakajima's partial order on the monomials in pY �j,bqjPI,bPC� [Nak04]:

m ¤ m1 ô mpm1q�1 �¹�nite

A�1j,b . (1.2.2.3)

Let us �x a monomial m P ZrA�1j,aql

sjPI,lPZ. If the factor χqpVi,aq�2k�1i

q contributes to

the multiplicity of m in χN , then A�1

i,aq�2ki

is a factor of m. Thus, only a �nite number of

factors χqpVi,aq�2k�1i

q can contribute to this multiplicity. In particular, for k large enough,

the factor χqpVi,aq�2k�1i

q does not contribute to the multiplicity of m in χN , for all N ¡ k.Thus the multiplicity of m in χN is stationary, as N Ñ �8.

As this is true for all monomials in ZrA�1j,aql

sjPI,lPZ, the limit of χN as N Ñ �8 is wellde�ned as a formal power series (see Remark 1.2.3).

Example 1.2.4. For g � sl2, one can compute this formula explicitly. Consider the `-weightΨΨΨ � ΨΨΨ�1

1,1. In this case, the normalized q-character of SN is known,

χN �N�1¹k�0

�1�A�1

1,q�2k

.

One can write

χN �N�1

m�0

¸0¤k1 k2 �� km¤N�1

m¹i�1

A�11,q�2ki

.


This formal power series has a limit

χN ÝÝÝÝÝÑNÑ�8

�8

m�0

¸0¤k1 k2 �� km

m¹i�1

A�11,q�2ki

P ZrrA�11,q�2ksskPN. (1.2.2.4)

Remark 1.2.5. (i) Except for the weight 0, all weight spaces are in�nite-dimensional.Thus this formula is not the (normalized) q-character of a representation in the cate-gory O. Nor is the result a formal power series is the "classical" sense. Nonetheless,it can still be a q-character, as stated in Remark 1.1.25.

(ii) We will see later that this q-character also has a decomposition into a sum of q-characters of simple representations, as in (1.1.3.2).

Let us generalize the statement of Theorem 1.2.2. Let ΨΨΨ be a negative `-weight in P�` .

It can be written as a �nite product of pYi,aqiPI,aPC� , pΨΨΨ�1i,a qiPI,aPC� and rωs, with ω P PQ.

Let us write

ΨΨΨ � rωs �r¹

k�1

Yik,ak �s¹l�1

ΨΨΨ�1jl,bl

. (1.2.2.5)

Each factor ΨΨΨ�1i,a can be seen as a limit of

±Yi,aq�2k�1

i. For N ¥ 1, consider the �nite-

dimensional standard module SN , which is the tensor product of the Vik,ak , for 1 ¤ k ¤ r,

and the

ÝÝÑN�1âk�0

Vjl,blq�2k�1jl

, for 1 ¤ l ¤ s, ordered so as to satisfy the condition (1.2.1.2) of

Section 1.2.1 (this is a direct generalization of the SN de�ned in (1.2.2.1)). Then,

Corollary 1.2.6. The sequence of normalized q-characters of SN converges as N Ñ �8,as a formal power series. The limit χ8ΨΨΨ can be written

χ8ΨΨΨ �r¹

k�1

pχqpVik,akqq �s¹l�1

χ8jl,bl P ZrrA�1i,a ssiPI,aPC� .

1.2.3 A conjecture

As explained in the previous section, the formal power series

χ8ΨΨΨ � limNÑ�8

χqpSN q

is a good candidate for the normalized q-character of the asymptotical standard moduleassociated to the negative `-weight ΨΨΨ.

We would like to build, for all negative `-weight ΨΨΨ, a Uqpbq-module MpΨΨΨq with �nite-dimensional `-weight spaces, whose q-character is rΨΨΨs �χ8ΨΨΨ . The following conjecture claimsthat a module with the right q-character and which retains some structure of the �nite-dimensional standard modules exists.

Conjecture 1.2.7. For all negative `-weight ΨΨΨ, there exists a Uqpbq-module MpΨΨΨq with�nite-dimensional `-weight spaces, such that χqpMpΨΨΨqq � rΨΨΨs � χ8ΨΨΨ and the sum of these`-weights spaces is a U¥0

q pbq-module containing a sub-Uqpbq�-module isomorphic to SN forany N ¥ 0.

Remark 1.2.8. Let us note here that the resulting module is not necessarily equal to thesum of its `-weight spaces. We will see later that even for g � sl2, for certain `-weights ΨΨΨ,there is no Uqpbq-module MpΨΨΨq satisfying the properties of the Conjecture, which is thesum of its `-weight spaces.

Section 1.3 will prove this conjecture when g � sl2.

1.3. Asymptotical standard modules 43

1.3 Asymptotical standard modules

From now on, let us assume that g � sl2. We hope in other work to extend theseresults to the type A and then to other types.

For simplicity, we omit the notation for the node from now on: Y1,a � Ya and Ψ1,a � Ψa.The aim of this section is to prove Conjecture 1.2.7 for g � sl2. We build, for every

negative `-weight ΨΨΨ such that LpΨΨΨq is in O�Z , an asymptotical standard Uqpbq-moduleMpΨΨΨq. This module has �nite-dimensional `-weight spaces and its q-character is the formalpower series χ8ΨΨΨ .

The construction will take place in two parts

1. Recall the de�nition of U¥0q pbq in (1.1.1.5). First, we build a U¥0

q pbq-module T , with�nite-dimensional `-weight spaces and the correct q-character.

2. Then, we build by induction from T a Uqpbq-module T c. We show that T c still has�nite-dimensional `-weight spaces and the same q-character.

1.3.1 Construction of the U¥0q pbq-module.

Fundamental example: the case where ΨΨΨ � ΨΨΨ�11

We build a U¥0q pbq-module T with �nite-dimensional `-weight spaces whose q-character

is exactly the limit in (1.2.2.4).For N ¥ 1, consider the standard module SN as de�ned in (1.2.2.1), and the normalized

standard module SN , obtained by normalizing the `-weights in each factor of SN .

SN :� LpYq�1q b LpYq�3q b LpYq�5q b � � � b LpYq�2N�1q. (1.3.1.1)

As in example 1.1.18, each one of the factors of SN is a two dimensional representation

LpYq�2r�1q � Cv�r ` Cv�r ,

where v�r is of `-weight Yq�2r�1 and v�r is of `-weight r�2ω1spYq�2r�1q�1.Let Pf pNq be the set of �nite subsets of N. Let

T :�à

JPPf pNqCvJ . (1.3.1.2)

For J P Pf pNq, the element vJ represents the in�nite simple tensor:

vJ � vε0pJq�1 b v

ε1pJq�3 b v

ε2pJq�5 b v

ε3pJq�7 b � � � ,

where εrpJq �

"� if r P J,� if r R J.

Example 1.3.1. For example,

"vH � v��1 b v��3 b v��5 b v��7 b � � �vt2u � v��1 b v��3 b v��5 b v��7 b � � �

.

Now, we endow the vector space T with a U¥0q pbq-module structure. The key ingre-

dients are the approximate coproducts formulas of Proposition 1.1.6. Let us recall theirexpressions in this context. For r ¡ 0 and m P Z,

∆phrq P hr b 1� 1b hr � U�q pbq b U�q pbq, (1.3.1.3)


∆px�mq P x�m b 1� Uqpbq b pUqpbqX�q, (1.3.1.4)

where U�q pbq (resp. U�q pbq) is the subalgebra of Uqpbq consisting of elements of positive(resp. negative) degree, and X� �

°mPZCx�m � U�q pbq.

Let J P Pf pNq and N ¡ i0 :� maxpJq. Consider the truncated simple tensor vpNqJ ,which is an element of the Uqpbq-module SN de�ned in (1.3.1.1),

vpNqJ :� v

ε0pJq�1 b v

ε1pJq�3 b v

ε2pJq�5 b � � � b v

εN pJq�2N�1 P SN ,

where εrpJq is de�ned as above.

Action of Uqpbq�: The algebra Uqpbq� is generated by the px�mqm¥0.For N ¡ i0, write

vpNqJ � v

pi0qJ b v��2i0�3 b � � � b v��2N�1 � v

pi0qJ b upNq, (1.3.1.5)

Then, using (1.3.1.4), one has, for m ¥ 0,

x�m � vpNqJ �

�x�m � v

pi0qJ

b upNq � 0,

as UqpbqX� acts by 0 on upNq, which is a highest `-weight vector and vpi0qJ P Si0 , which isa Uqpbq-module. That way the action of px�mqm¥0 on vJ is de�ned as

x�m � vJ :��x�m � v

pi0qJ

b u, (1.3.1.6)

where u � v��2i0�3 b � � � b v��2N�1 b � � � . As x�m � vpi0qJ P Si0 , it is a linear combination of

vpi0qK , with maxpKq ¤ i0. Hence, x�m � vJ is the same linear combination, but with the vKinstead of the vpi0qK .

Action of Uqpbq0: The algebra Uqpbq0 is generated by the phr, k�11 qr¥1.

As we normalized the action of k1, for all N ¡ i0, k1 � vpNqI � q�2|J |v

pNqJ . Hence,

naturallyk1 � vJ :� q�2|J |vJ . (1.3.1.7)

For the action of the hr's, let us write, for N ¡ i0, vpNqJ as in (1.3.1.5). Then, using

(1.3.1.3), one has, for r ¥ 1,

hr � vpNqJ �

�hr � v

pi0qJ

b upNq � v

pi0qJ b

�hr � u

pNq� 0,

as U�q pbq sends upNq to a higher weight space, which is t0u.

The vector upNq is a highest `-weight vector of Sn, whose `-weight is the productYq�2i0�3 Yq�2i0�5 � � � Yq�2N�1 . Hence,

hr � upNq �

q�2i0�2 � q�2N

q � q�1upNq.

Thus it is natural to de�ne, for r ¥ 1, and N ¡ i0

hr � vJ :�

��hr �

q�2i0�2

q � q�1Id

� v

pi0qJ

b u. (1.3.1.8)


As before,�hr �

q�2i0�2

q�q�1 Id� v

pi0qJ P Si0 is a linear combination of vpi0qK , and hr � vJ is the

same linear combination of vK .

Example 1.3.2. As ∆ph1q � h1 b 1� 1b h1 � pq2 � q�2qx�1 b x�0 , then h1 ��v��1 b v��3

��

�q�1pq2 � q�2qv��1 b v��3. Hence,

h1 � vt1u �q�4

q � q�1vt1u � q�1pq2 � q�2qvt0u.

Moreover, h1 � vt0u � p�q � q�2

q�q�1 qvt0u, and h1pvt1u � vt0uq �q�4

q�q�1 pvt1u � vt0uq. We

will see that vt0u is of `-weight pΨΨΨ1q�1A�1

1 and vt1u � vt0u is of `-weight pΨΨΨ1q�1A�1

q�2 .

The combination of the last two paragraphs gives us the following result. As the actionsof the generators of U¥0

q pbq on T de�ned in (1.3.1.6), (1.3.1.7) and (1.3.1.8) are based onactions on �nite-dimensional Uqpbq-modules, they naturally satisfy the relations in U¥0

q pbq.

Proposition 1.3.3. The vector space T has a U¥0q pbq-module structure.

Moreover, the action of Uqpbq0 on the vJ 's is upper-triangular, which allows us toexplicit `-weight vectors.

Partial order on Pf pNq: We de�ne some order on Pf pNq. It has nothing to do withNakajima's partial order on the `-weights recalled in (1.2.2.3).

Remark 1.3.4. For all J P Pf pNq, vJ is a weight vector of weight r|J |ω1s. Moreover, from(1.3.1.8), for all r ¥ 1,

hr � vJ Pà

|K|�|J |,K�ConvpJq

CvK ,

where ConvpJq is the convex hull of J , ConvpJq � tj P N | minpJq ¤ j ¤ maxpJqu.In particular, sets in Pf pNq correspond to `-weights which are generically incomparable

for Nakajima's partial order.

De�nition 1.3.5. For all N P N, let PN pNq :� tJ P Pf pNq | |J | � Nu. The set PN pNq isequipped with the lexicographic order on N -tuples, noted ¨ .

Lemma 1.3.6. For J P Pf pNq and r ¥ 1,

hr � vJ P hr,JvJ �¸

K�ConvpJq,K J

CvK ,

where the hr,J are the coe�cients arising from the action of hr on each component of vJ .

Proof. Using (1.3.1.3) recursively, one has, for N ¥ 1,

∆N phrq PN

k�1

1b � � � b hrloomoonk

b1b � � � b 1� U�q pbq b Uqpbq b � � � b Uqpbq

� 1b U�q pbq b Uqpbq b � � � b Uqpbq � � � � � 1b � � � b 1b U�q pbq b U�q pbq.

Hence, using the previous notations, for N ¡ i0,

hr � vJ �

��hr �

q�2i0�2

q � q�1Id

� v

pi0qJ

b u P hr,JvJ �

¸K�ConvpJq,K J

CvK .


Proposition 1.3.7. The vector space T has a basis of `-weights vectors. More precisely,

T �à

JPPf pNqCwJ , with wJ of `-weight pΨΨΨ1q

�1¹jPJ

A�1q�2j .

As the `-weight spaces are �nite dimensional, one can de�ne a q-character for T ,

χqpT q �¸

JPPf pNqrΨΨΨ�1

1 s¹jPJ

A�1q�2j . (1.3.1.9)

Remark 1.3.8. The normalized q-character of T is exactly the limit χ81,1 of the normalizedq-characters of the sequence of standard modules pSN qN¥1 obtained in (1.2.2.4).

Proof. By Lemma 1.3.6, for all r P N�, J P Pf pNq,

hr � vJ P

��8

m�0

hJ,mr

�vJ �

¸K�ConvpJq,K J

CvK ,

where as before hJ,mr is the coe�cient coming from the action of hr on the m-th componentof vJ .

Hence the action of the phrqrPN� is simultaneously diagonalizable. Let us look at thediagonal terms. For r ¥ 1, J P Pf pNq,

hJ,mr �

#q�p2m�1qr rrsq

r if m R J

�q�p2m�1qr rrsqr if m P J

.

Hence,�8

r�1

��8

m�0

hJ,mr

�zr �

1

q � q�1log

�1

1� z

¹mPJ

1� q�2m�2z

1� q�2m�2z

�.

Thus the vector space T contains a basis of `-weight vectors pwJqJPPf pNq, where for each

J P Pf pNq, wJ is a `-weight pΨΨΨ1q�1

±jPJ A

�1q�2j . Moreover,

wJ P vJ �VectpvK ,K ¨ J,K � Jq.

Remark 1.3.9. However, the action cannot be extended to the full Borel algebra Uqpbq thesame way. For example, for N P N if we consider the truncated pure tensor vector,

vpNqH � v��1 b v��3 b v��5 b � � � v��2N�1 b v��2N�1,

then x�1 acts on vpNqH P LpYq�1q b LpYq�3q b � � � b LpYq�2N�1q, as

x�1 � vpNqH �

N

k�0

q�2k�1vtku.

Which does not have a limit in T as N Ñ �8.

Remark 1.3.10. Hence, if we consider the Conjecture 1.2.7, then necessarily, the sum ofthe `-weight spaces of any potential asymptotical standard module MpΨΨΨ�1

1 q is the in�nitetensor product T (it is the only U¥0

q pbq-module containing a sub-Uqpbq�-module isomorphicto SN for any N ¥ 0). As the action on T cannot be extended, the module MpΨΨΨ�1

1 q mustbe de�ned di�erently.


Generalization of this construction

LetΨΨΨ P P r` be a negative `-weight such that LpΨΨΨq is inO

�Z . In this section, we generalize

Proposition 1.3.7 to this context.As seen in example 1.1.31, the roots and poles of Ψ1pzq all have the same parity. Let

us write ΨΨΨ as a �nite product

ΨΨΨ � rωs �m�

�R2¹r�R1

ΨΨΨ�brq2r

�,

where ω P PQ, m is a monomial in the pYq2l�1qlPZ and br P N.In Section 1.2.2, we have seen that each ΨΨΨ�1

q2rcan be written as an in�nite product of

pYq2k�1qk¤r. Hence, the `-weight ΨΨΨ can be seen as a limit of a series of monomials,

ΨΨΨ � limNÑ�8

rω1sR¹

r��N

Y arq2r�1

�� lim

NÑ�8mN

,

where the sequence parq�8 r¤R is ultimately stationary.

Remark 1.3.11. If ΨΨΨ is only a product of rωs and pYq2l�1qlPZ (if LpΨΨΨq is �nite-dimensional,with Proposition 1.1.16), this sequence is stationary. In that case, the vector space T is�nite-dimensional.

We also know that the sequence of normalized q-characters of the standard modulesassociated to mN converges as N Ñ �8. With notations from Section 1.2.2,

χ8ΨΨΨ � limNÑ�8

pχqpSN qq � limNÑ�8

�R¹

r��N

�1�A�1

q2r

ar�P ZrrA�1

q2rssr¤R. (1.3.1.10)

Theorem 1.3.12. There exists a U¥0q pbq-module TΨΨΨ, with �nite dimensional `-weight

spaces, whose q-character isχqpTΨΨΨq � rΨΨΨs � χ8ΨΨΨ .

Let us introduce a few notations. De�ne the set

J � tpr, kq | r ¤ R, 1 ¤ k ¤ aru.

We consider the following order on Pf pJ q (generalized from the order de�ned in 1.3.5).

De�nition 1.3.13. Let N P N and J,K P Pf pJ q such that |K| � |J | � N .We say that J ¨ K if and only if they are ordered that way for the lexicographical

order on N -tuples, while elements of J are also ordered lexicographically.

Proof. Heuristically, the vector space TΨΨΨ is constructed to be the in�nite tensor product

rω1sb�LpYq2R�1q

baRb�LpYq2R�3q

baR�1

b� � ��LpYq�1q

ba0b� � �

�LpYq�2r�1q

ba�rb� � �

Hence it is generated by the pure tensors:

v�p1q

2R�1 b v�p2q

2R�1 b � � � v�paRq

2R�1 b v�p1q

2R�3 b � � � v�paR�1q

2R�3 b � � � v�p1q

�1 b v�pa0q

�1 b � � �

b v�p1q

�2r�1 b � � � v�pa�rq

�2r�1 b � � � , (1.3.1.11)


with a �nite number of lowest weight components (�).Formally, as in (1.3.1.2), let TΨΨΨ be the vector space

TΨΨΨ :�à

JPPf pJ qCvJ .

As before, as the in�nite pure tensors have a �nite number of lowest weight components,and thanks to the approximate coproduct formulas (1.3.1.3) and (1.3.1.4), the action of

U¥0q pbq on the �nite tensor products SN � rω1s

ÂRr��N

�LpYq2r�1q

barstabilizes as N Ñ

�8 and the limit can be taken as the action of U¥0q pbq on TΨΨΨ.

With the decomposition in the proof of Lemma 1.3.6, one can see that the action ofthe `-Cartan subalgebra Uqpbq0 on the vJ 's is upper triangular, for the order on Pf pJ qde�ned in (1.3.13).

Then, one has, for all J P Pf pJ q, r ¥ 1,

hr.vJ P λr,JvJ �à

K J,K�ConvpJq

CvK ,

where the pλr,Jqr¥1 satisfy ¸r¥1

λr,Jzr �

1

q � q�1logpΦJpzqq,

withΦJpzq �

¸m¥0

φm,Jzm, and pφm,Jqm¥0 � ΨΨΨ

¹pr,kqPJ

A�1q2r.

Hence the vector space TΨΨΨ has a basis of `-weight vectors. Let us write:

TΨΨΨ �à

JPPf pJqCwJ ,

where, for all J P Pf pJq, wJ is an `-weight vector of `-weight ΨΨΨ±

pr,kqPJ A�1q2r

(di�erent wJcan contribute to the same `-weight space).

Thus, TΨΨΨ has �nite dimensional `-weight spaces and its q-character is

χqpTΨΨΨq �¸

JPPf pJqΨΨΨ

¹pr,kqPJ

A�1q2r

� rΨΨΨs � χ8ΨΨΨ .

1.3.2 Construction of induced modules

As stated in Remark 1.3.10, to obtain a Uqpbq-module structure on the in�nite tensorproduct, one needs to extend these modules. This is why our asymptotical standardmodules will be obtained by induction.

Let M be a U¥0q pbq-module. De�ne the Uqpbq-module M c induced from M :

M c � Uqpbq bU¥0q pbqM � Uqpbq� bC M.

This induction preserves the q-character of the module. More precisely, we have Propo-sition 1.3.18, which is obtained from Lemma 1.3.16 bellow.

First of all, we use the following Lemma on the structure of Uqpbq�.


Lemma 1.3.14. [Dam93] The elements�x�m1

x�m2x�m3

� � �x�ms�, where s ¥ 0 and 1 ¤ m1 ¤

m2 ¤ m3 ¤ � � � ¤ ms, form a basis of Uqpbq�.

Remark 1.3.15. This result can also be obtained by seeing the xx�my1¤m¤N � Uqpbq�as successive Ore extensions. Then, with the methods used in [Kas95], we see that theelements px�1 q

i1px�2 qi2 � � � px�N q

iN form a basis of xx�my1¤m¤N .

Let us recall relation (1.1.1.4) and relation (1.1.1.1) in this context:

x�m�1x�l � q�2x�l x

�m�1 � q�2x�mx

�l�1 � x�l�1x

�m, for all m, l ¥ 1, (1.3.2.1)

rhr, x�ms � �

r2rsqr

x�m�r, for all r,m ¥ 1. (1.3.2.2)

Note that the algebra Uqpbq� has a natural N-graduation, which is di�erent from thegraduation coming from the Q-graduation on Uqpgq (1.1.1.3). We note:

degpx�m1x�m2

x�m3� � �x�msq �

s

i�1

mi. (1.3.2.3)

Thanks to the relation (1.3.2.1), this is a well-de�ned graduation on Uqpbq�.

Lemma 1.3.16. Let v P M c be an eigenvector of hr for a certain r ¥ 1. Then v belongsto the subspace 1bM .

Proof. Let us write, with the result of Lemma 1.3.14:

v �¸s¥0

¸1¤m1¤m2¤��¤ms

x�m1x�m2

� � �x�ms b um1,m2,...,ms ,

where all but �nitely many of the um1,m2,...,ms P M are 0. By (1.3.2.2), one has, for alls ¥ 1, and all 1 ¤ m1 ¤ m2 ¤ � � � ¤ ms:

hr�x�m1

x�m2� � �x�ms

�� x�m1

x�m2� � �x�mshr �

r2rsqr

s

j�1

x�m1� � �x�mj�r � � �x

�ms .

Thus, from the de�nition of the induced module,

hr � v � 1b hru1,...,1 �¸s¥1

� ¸1¤m1¤m2¤��¤ms

�x�m1

x�m2� � �x�ms b hrum1,m2,...,ms

�r2rsqr

s

j�1

x�m1� � �x�mj�r � � �x

�ms b um1,m2,...,ms

��.

By hypothesis, there exists λ P C such that hr � v � λv. Thus, from the Q-graduation(1.1.1.3) on Uqpbq�, one has 1b hru1,...,1 � λ1b u1,...,1, and, for all s ¥ 1,¸

1¤m1¤m2¤��¤ms

�x�m1

x�m2� � �x�ms b hrum1,m2,...,ms

�r2rsqr

s

j�1

x�m1� � �x�mj�r � � �x

�ms b um1,m2,...,ms

�� λ

¸1¤m1¤m2¤��¤ms

x�m1x�m2

� � �x�ms b um1,m2,...,ms .


Moreover, from the N-graduation (1.3.2.3) on Uqpbq�, one has, for all s ¥ 1, for all N ¥ 1,¸1¤m1¤m2¤��¤ms°

mi�N�r

x�m1x�m2

� � �x�ms b phr � λqum1,m2,...,ms

�r2rsqr

¸1¤m1¤m2¤��¤ms°

mi�N

s

j�1

x�m1� � �x�mj�r � � �x

�ms b um1,m2,...,ms � 0. (1.3.2.4)

Let us note that the x�m1� � �x�mj�r � � �x

�ms are not necessarily in the basis of Uqpbq� of

Lemma 1.3.14, but from (1.3.2.1) they can be written as a linear combination of these basiselements. Moreover, with Remark 1.3.15, x�m1

� � �x�mj�r � � �x�ms P xx

�my1¤m¤m � Uqpbq�,

where m � maxpmj � r,msq, and can be decomposed in the basis of this subalgebra (notethat mj � r is possibly greater than ms).

Now, suppose that v does not belong to the subspace 1bM . Then, for all s ¥ 1 andall N ¥ 1 such that there exists pm1,m2, . . . ,msq with

°mi � N and um1,m2,...,ms � 0,

consider

ms � max!ms P N | Ds ¥ 1, Dpm1,m2, . . . ,ms�1q,

¸mi � N, um1,...,ms�1,ms � 0

).

(1.3.2.5)Then, we decompose the left-hand term of relation (1.3.2.4) into sums of pure tensors

whose �rst factor is in the basis of Lemma 1.3.14. Then we extract from this sum the termswhose �rst factor is of the form x�m1

� � �x�ms�1x�ms�r. We get a sum equal to 0, from relation

(1.3.2.4). Let us explicit the terms we obtain. As r ¡ 0, the �rst term of the left-hand-side

of (1.3.2.4) does not contribute. The remaining terms have � r2rsqr as a scalar factor. The

terms obtained from j � s are of the form¸1¤m1¤m2¤��¤ms�1¤ms°

mi�N

x�m1� � �x�ms�1

x�ms�r b um1,...,ms�1,ms ,

where the �rst factors of each terms are elements of the considered basis, so no permutationof the x�m is needed.

By rewriting (1.3.2.1),

x�ms�rx�ms

� q�2x�msx�ms�r

� q�2x�ms�r�1x�ms�1 � x�ms�1x

�ms�r�1looooooooooooooooooooooomooooooooooooooooooooooon

does not contribute

,

thus, the terms obtained from j � s� 1 are of the form¸1¤m1¤��¤ms�2¤ms�1�ms°

mi�N

q�2x�m1� � �x�ms�2

x�msx�ms�r

b um1,...,ms�2,ms,ms ,

as only the terms for which ms�1 � ms contribute to the considered sum.Recursively, the terms obtained from j=1 are of the form

q�2ps�1q�x�ms

�s�1x�ms�r b ums,ms,...,ms ,

if N � sms, and 0 otherwise. Thus, the resulting formula is:

0 �¸

1¤m1¤��¤ms�1¤ms°mi�N

Cm1,...,ms�1x�m1� � �x�ms�1

x�ms�r b um1,...,ms�1,ms , (1.3.2.6)


where

Cm1,...,ms�1 �N

k�0

q�2k , with N � 7tmi | mi � msu. (1.3.2.7)

As q is not a root of unity, these Cm1,...,ms�1 are non-zero. This implies that all theum1,...,ms�1,ms are 0, which is a contradiction with the hypothesis taken in the de�nition ofms in (1.3.2.5).

Remark 1.3.17. Recall the de�nition of `-weight vectors from Section 1.1.2: v in the Uqpbq-module V is an `-weight vector if there is an `-weight ΨΨΨ and p P N such that:

@i P I,@m ¥ 0, pφ�i,m � ψi,mqpv � 0.

Proposition 1.3.18. The `-weights vectors of M c are exactly the 1 b u, where u is an`-weight vector of M .

Proof. First of all, if u is an `-weight vector of M , then by construction of M c, 1b u is an`-weight vector of M c.

Conversely, let us write:M c � p1bMq `M¥1, (1.3.2.8)

where M¥1 is generated by the pure tensors for which the �rst factor is of non-zero Q-degree. Notice that from (1.3.2.2), M¥1 is a sub-Uqpbq0-module.

For all `-weight vector v of M c, there exists:

pv � min p P N� | Dψ1 P C, pφ�1 � ψ1q

pv � 0(.

Now suppose there exists non-zero `-weight vectors in M c which is not in 1 b M .Then its projection on M¥1 in the decomposition (1.3.2.8) is a non-zero `-weight vector inM¥1. Consider such a v0 which minimizes pv. Let p0 � pv0 . There is ψ1 P C such thatpφ�1 � ψ1q

p0v0 � 0. By de�nition,

w � pφ�1 � ψ1qp0�1v0 � 0.

But, φ�1 w � ψ1w, as h1 � k�11 φ�1 , and v0 and w are weight vectors, then w is an eigenvector

of h1. By Lemma 1.3.16, w P 1 bM . However, w P M¥1 as M¥1 is stabilized by Uqpbq0.Hence w � 0, which is a contradiction with the de�nition of v0.

Now everything is in place to de�ne the asymptotical standard modules. Let ΨΨΨ be anegative `-weight such that LpΨΨΨq is in O�Z . De�ne the induced Uqpbq-module from theU¥0q pbq-module TΨΨΨ constructed in the previous section:

T cΨΨΨ :� Uqpbq bU¥0q pbq TΨΨΨ.

Theorem 1.3.19. The vector space T cΨΨΨ is a Uqpbq-module, such that its `-weight spacesare �nite-dimensional. As such, T cΨΨΨ satis�es the Conjecture 1.2.7 for ΨΨΨ. One has:

χqpTcΨΨΨq � rΨΨΨs � χ8ΨΨΨ .

Proof. From Proposition 1.3.18 we know that the `-weight vectors of T cΨΨΨ are exactly the1 b u, where u is an `-weight vector of TΨΨΨ. Thus, with the result of Theorem 1.3.12, weknow that T cΨΨΨ has �nite-dimensional `-weight spaces which satisfy:

χqpTcΨΨΨq � χqpTΨΨΨq � rΨΨΨs � χ8ΨΨΨ .


Furthermore, let us look at the way T cΨΨΨ is built to see that it satis�es Conjecture 1.2.7.Recall the standard modules de�ned in Section 1.2.2 and used in Section 1.3.1. For allN ¥ 1, let

SN �

ÝÝÝÑRâ

r��N

�V barq�2r�1

.

Then the sum of `-weights spaces of T cΨΨΨ is the U¥0q pbq-module TΨΨΨ, which contains for all

N ¥ 1 a sub-Uqpbq�-modules isomorphic to SN .

Remark 1.3.20. The space T cΨΨΨ contains submodules without `-weight vectors. For example,for ΨΨΨ � ΨΨΨ�1

1 , the submodule M � x�r b wH � 1b wt0u | r P N� ¡ does not contain any`-weight vectors.

From now on, for all `-weights ΨΨΨ such that LpΨΨΨq is in the category O�Z ,

MpΨΨΨq :� T cΨΨΨ,

will denote the generalized standard module associated to the `-weight ΨΨΨ.

1.4 Decomposition of the q-character of asymptotical stan-

dard modules

As stated in the introduction, for the category C of �nite-dimensional representationsof Uqpbq, it is known ([Nak01a]) that the classes of the standard modules rMpmqs form asecond basis of the Grothendieck ring KpC q (in addition to the classes of simple modules).Moreover, the two bases are triangular with respect to Nakajima's partial ordering ofdominant monomials (see [Nak04])):


m1 m

Pm1,mrLpm1qs, (1.4.0.1)

where the coe�cients Pm1,m P Z are non-negative.The aim of this Section is to show that the q-characters of the modules we have just

built have similar decomposition into sum of q-characters of simple modules.

1.4.1 Partial order on P r`

For a formula of the type (1.4.0.1) to make sense, one needs to de�ne a partial orderon P r

` , which is the index set of both the simple modules and the standard modules in ourcontext.

We draw our inspiration from the partial order de�ned in the proof of [HL16b, Lemma6.4], which is itself a generalization of the order Nakajima used in [Nak04], recalled in(1.2.2.3). The following de�nition is valid for g any untwisted a�ne Kac-Moody algebra.

De�nition 1.4.1. Let ΨΨΨ,ΨΨΨ1 P P r` . we say that ΨΨΨ1 ¤ ΨΨΨ if ΨΨΨ1pΨΨΨq�1 is a monomial in the

A�1i,a , with i P I, a P C�.

Remark 1.4.2. Contrary to the �nite-dimensional case, every `-weight has an in�nite num-ber of lower `-weights.

1.4. Decomposition of the q-character of asymptotical standard modules53

1.4.2 Decomposition for MpΨΨΨ�11 q

Let us recall the q-character of the Uqpbq-module MpΨΨΨ�11 q:

χqpMpΨΨΨ�11 qq �

¸JPPf pNq

rΨΨΨ�11 s

¹jPJ

A�1q�2j .

Theorem 1.4.3. The q-character of the Uqpbq-module MpΨΨΨ�11 q has a decomposition into

a sum of q-characters of simple modules. More precisely,

χqpMpΨΨΨ�11 qq �

�8

m�0

¸1¤r1 r2 �� rmri�1¡ri�1

χqpLpΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm qq.

Remark 1.4.4. This formula is multiplicity-free.

Proof. Let m P N� and pr1, r2, . . . , rmq P pN�qm, satisfying ri�1 ¡ ri � 1 for all 1 ¤ i ¤ m.One has

ΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm � rp�rm � 1qω1s

�Yq�1 � � �Yq�2r1�3

� �Yq�2r1�3 � � �Yq�2r2�3

��

� � ��Yq�2rm�1�3 � � �Yq�2rm�3

ΨΨΨ�1q�2rm�2 ,

The q-sets q�1, q�3, . . . , q�2r1�3

(, q�2ri�3, q�2ri�5, . . . , q�2ri�1�3

(, for 1 ¤ i ¤ m � 1,

and q�2rm�1, q�2rm�3, . . .

(are pairwise in general position. Hence, following [HL16b,

Theorem 7.9], the following tensor product is simple:

LpYq�1 � � �Yq�2r1�3qbLpYq�2r1�3 � � �Yq�2r2�3qb� � �bLpYq�2rm�1�3 � � �Yq�2rm�3qbLpΨΨΨ�1q�2rm�2q

and of highest `-weight ΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm . Thus,

LpΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm q � rp�rm � 1qω1s b LpYq�1 � � �Yq�2r1�3qb

LpYq�2r1�3 � � �Yq�2r2�3q b � � � b LpYq�2rm�1�3 � � �Yq�2rm�3q b LpΨΨΨ�1q�2rm�2q.

One can then compute the q-character of LpΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm q.

rLpΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm qs � rp�rm � 1qω1srLpYq�1 � � �Yq�2r1�3qs

rLpYq�2r1�3 � � �Yq�2r2�3qs � � � rLpYq�2rm�1�3 � � �Yq�2rm�3qsrLpΨΨΨ�1q�2rm�2qs

� rΨΨΨ�11 s

¸JPPf pNqp�q

¹jPJ

A�1q�2j .

where in p�q we consider the �nite subsets of N with m connected components, start-ing with r1, r2, . . . , rm respectively, and with m � 1 connected components, starting with0, r1, r2, . . . , rm respectively.

Thus,

�8

m�0

¸1¤r1 r2 �� rmri�1¡ri�1

rLpΨΨΨ�11 A�1

q�2r1A�1q�2r2

� � �A�1q�2rm qs

� rΨΨΨ�11 s

¸JPPf pNq

¹jPJ

A�1q�2j � χqpMpΨΨΨ�1

1 qq


Remark 1.4.5. One has indeed

χqpMpΨΨΨ�11 qq P

¸ΨΨΨ¤ΨΨΨ�1

1

NrLpΨΨΨqs,

for the partial order on P r` de�ned in Section 1.4.1.

1.4.3 General decomposition

Consider a negative `-weight ΨΨΨ, as in Section 1.3.1, such that LpΨΨΨq is in O�Z . It can bewritten as a �nite product

ΨΨΨ � rωs �m�¹

ΨΨΨ�vrq2r

, (1.4.3.1)

where λ P PQ, m is a monomial in ZrYq2r�1srPZ and vr P N. First we need the followingLemma.

Lemma 1.4.6. For ΨΨΨ1,ΨΨΨ2, negative `-weights as in (1.4.3.1), one has

χqpMpΨΨΨ1ΨΨΨ2qq � χqpMpΨΨΨ1qqχqpMpΨΨΨ2qq. (1.4.3.2)

Proof. If ΨΨΨ1 and ΨΨΨ2 are monomials in ZrYq2r�1srPZ, then the standard modules MpΨΨΨ1qand MpΨΨΨ2q are �nite dimensional and the result is known.

Else, as in (1.3.1.10), their normalized q-characters are limits of normalized q-charactersof �nite dimensional standard modules:

χqpMpΨΨΨiqq � limNÑ�8

χqpMpmiN qq,@i P t1, 2u.

The result is obtained by taking the products of the limits.

Thus, the q-character of the standard module MpΨΨΨq is a product,

χqpMpΨΨΨqq � rωs � χqpmq �¹

χqpΨΨΨaq�va ,

for which each term has a decomposition into a sum of q-characters of simple modules,corresponding to lower highest `-weights.

Finally, using the following lemma, which is straightforward from the de�nition of theorder.

Lemma 1.4.7. The order on the negative `-weights is compatible with the product. Moreprecisely, for ΨΨΨ1,ΨΨΨ2,ΨΨΨ and ΨΨΨ1 some negative `-weights,�

ΨΨΨ ¤ ΨΨΨ1 and ΨΨΨ1 ¤ ΨΨΨΨΨΨ2�ñ ΨΨΨ1 ¤¤¤ ΨΨΨ1ΨΨΨ2.

One can �nally conclude,

Corollary 1.4.8. For every negative `-weight ΨΨΨ such that LpΨΨΨq is in O�Z , one has

χqpMpΨΨΨqq P¸

ΨΨΨ1¤ΨΨΨ

NχqpLpΨΨΨ1qq. (1.4.3.3)

Remark 1.4.9. We showed that the coe�cients in the decomposition (1.4.3.3) are non-negative. It would be interesting to show that these coe�cients can be interpreted asdimensions, which would explain their non-negativity.

Chapter 2Quantum Grothendieck rings as quantumcluster algebras.

This chapter is an adapted version of [Bit19], preprint arXiv:1902.00502.

Abstract.

We de�ne and construct a quantum Grothendieck ring for a certain monoidal sub-

category of the category O of representations of the quantum loop algebra introduced

by Hernandez-Jimbo. We use the cluster algebra structure of the Grothendieck ring

of this category to de�ne the quantum Grothendieck ring as a quantum cluster alge-

bra. When the underlying simple Lie algebra is of type A, we prove that this quantumGrothendieck ring contains the quantum Grothendieck ring of the category of �nite-

dimensional representations of the associated quantum a�ne algebra. In type A1, we

identify remarkable relations in this quantum Grothendieck ring.

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 Category O of representations of quantum loop algebras . . . . . . . . . . . 58

2.2.1 Quantum loop algebra and Borel subalgebra . . . . . . . . . . . . . . 582.2.2 Highest `-weight modules . . . . . . . . . . . . . . . . . . . . . . . . 592.2.3 De�nition of the category O . . . . . . . . . . . . . . . . . . . . . . . 602.2.4 Connection to �nite-dimension UqpLgq-modules . . . . . . . . . . . . 602.2.5 Categories O� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2.6 The category O�Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2.7 The Grothendieck ring K0pOq . . . . . . . . . . . . . . . . . . . . . . 622.2.8 The q-character morphism . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3 Quantum tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3.1 The torus Yt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.3.2 The torus Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 Quantum Grothendieck rings . . . . . . . . . . . . . . . . . . . . . . . . . . 682.4.1 The �nite-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 682.4.2 Compatible pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.4.3 De�nition of KtpO�Z q . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.5 Properties of KtpO�Z q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.1 The bar involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.2 Inclusion of quantum Grothendieck rings . . . . . . . . . . . . . . . . 752.5.3 pq, tq-characters for positive prefundamental representations . . . . . 79

56 Chapter 2. Quantum Grothendieck rings as quantum cluster algebras

2.6 Results in type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.6.1 Proof of the conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 802.6.2 A remarkable subalgebra in type A1 . . . . . . . . . . . . . . . . . . 81

2.1 Introduction

Let g be a simple Lie algebra of Dynkin type A, D or E (also called simply laced types),and let Lg � gbCrt�1s be the loop algebra of g. For q a generic complex number, Drinfeld[Dri88] introduced a q-deformation of the universal enveloping algebra UpLgq of Lg calledthe quantum loop algebra UqpLgq. It is a C-algebra with a Hopf algebra structure, and thecategory C of its �nite-dimensional representations is a monoidal category. The categoryC was studied extensively, in particular to build solutions to the quantum Yang-Baxterequation with spectral parameter (see [Jim89] for a detailed review).

Using the so-called "Drinfeld-Jimbo" presentation of the quantum loop algebra, onecan de�ne a quantum Borel subalgebra Uqpbq, which is a Hopf subalgebra of UqpLgq. Weare here interested in studying a category O of representations of Uqpbq introduced byHernandez-Jimbo [HJ12]. The category O contains all �nite-dimensional Uqpbq-modules,as well as some in�nite-dimensional representations, albeit with �nite-dimensional weightspaces. In particular, this category O contains the prefundamental representations. Theseare a family of in�nite dimensional simple Uqpbq-modules, which �rst appeared in the workof Bazhanov, Lukyanov, Zamolodchikov [BLZ99] for g � sl2 under the name q-oscillatorrepresentations.

These prefundamental representations were also used by Frenkel-Hernandez [FH15]to prove Frenkel-Reshetikhin's conjecture on the spectra of quantum integrable systems[FR99]. More precisely, quantum integrable systems are studied via a partition functionZ, which in turns can be scaled down to the study of the eigenvalues λj of the transfermatrix T . For the 6-vertex (and 8-vertex) models, [Bax72] showed that the eigenvalues ofT have the following remarkable form:

λj � ApzqQjpzq

�2q

Qjpzq�Dpzq

Qjpzq2q

Qjpzq, (2.1.0.1)

where q and z are parameters of the model, the functions Apzq, Dpzq are universal, and Qjis a polynomial. This relation is called the Baxter relation. In the context of representationtheory, relation (2.1.0.1) can be categori�ed as a relation in the Grothendieck ring of thecategory O. For g � sl2, if V is the two-dimensional simple representation of UqpLgq ofhighest loop-weight Yaq�1 , then

rV b L�a s � rω1srL�aq�2s � r�ω1srL

�aq2s, (2.1.0.2)

where r�ω1s are one-dimensional representations of weight�ω1 and L�a denotes the positiveprefundamental representation of quantum parameter a.

Frenkel-Reshetikhin's conjecture stated that for more general quantum integrable sys-tems, constructed via �nite-dimensional representations of the quantum a�ne algebraUqpgq (of which the quantum loop algebra is a quotient), the spectra had a similar form asrelation (2.1.0.1).

Let t be an indeterminate. The Grothendieck ring of the category C has an interestingt-deformation called the quantum Grothendieck ring, which belongs to a non-commutativequantum torus Yt. The quantum Grothendieck ring was �rst studied by Nakajima [Nak04]

2.1. Introduction 57

and Varagnolo-Vasserot [VV03] in relation with quiver varieties. Inside this ring, one cande�ne for all simple modules L classes rLst, called pq, tq-characters. Using these classes,Nakajima was able to compute the characters of the simple modules L, which were com-pletely inaccessible in general, thanks to a Kazhdan-Lusztig type algorithm.

One would want to extend these results to the context of the category O, with theultimate goal of (algorithmically) computing characters of all simple modules in O. Inorder to do that, one �rst needs to build a quantum Grothendieck ring KtpOq inside whichthe classes rLst can be de�ned.

Another interesting approach to this category O is its cluster algebra structure (seebelow). Hernandez-Leclerc [HL13] �rst noticed that the Grothendieck ring of a certainmonoidal subcategory C1 of the category C of �nite-dimensional UqpLgq-modules had thestructure of a cluster algebra. Then, they proved [HL16b] that the Grothendieck ring ofa certain monoidal subcategory O�Z of the category O had a cluster algebra structure, ofin�nite rank, for which one can take as initial seed the classes of the positive prefundamentalrepresentations (the category O�Z contains the �nite-dimensional representations and thepositive prefundamental representations whose spectral parameter satis�es an integralitycondition). Moreover, some exchange relations coming from cluster mutations appearnaturally. For example, the Baxter relation (2.1.0.2) is an exchange relation in this clusteralgebra.

To construct a quantum Grothendieck ring for the category O, the approaches used pre-viously are not applicable anymore. The geometrical approach of Nakajima and Varagnolo-Vasserot (in which the t-graduation naturally comes from the graduation of cohomologicalcomplexes) requires a geometric interpretation of the objects in the category O, which hasnot yet been found. The more algebraic approach consisting of realizing the (quantum)Grothendieck ring as an invariant under a sort of Weyl symmetry, which allowed Hernandezto de�ne a quantum Grothendieck ring of �nite-dimensional representations in non-simplylaced types, is again not relevant for the category O. Only the cluster algebra approachyields results in this context.

In this Chapter, we propose to build the quantum Grothendieck of the category O�Zas a quantum cluster algebra. Quantum cluster algebras are non-commutative versionsof cluster algebras, they live inside a quantum torus, generated by the initial variables,together with t-commuting relations:

Xi �Xj � tΛijXj �Xi. (2.1.0.3)

First of all, one has to build such a quantum torus, and check that it contains the quantumtorus Yt of the quantum Grothendieck ring of the category C . This is proven as the �rstresult of this chapter (Proposition 2.3.3).

Next, one has to show that this quantum torus is compatible with a quantum clus-ter algebra structure based on the same quiver as the cluster algebra structure of theGrothendieck ring K0pO�Z q. In order to do that, we exhibit (Proposition 2.4.10) a compat-ible pair. From then, the quantum Grothendieck ring KtpO�Z q is de�ned as the quantumcluster algebra de�ned from this compatible pair.

We then conjecture (Conjecture 2.5.2) that this quantum Grothendieck ring KtpO�Z qcontains the quantum Grothendieck ring KtpCZq. We propose to demonstrate this conjec-ture by proving that KtpO�Z q contains the pq, tq-characters of the fundamental representa-tions rLpYi,qrqst, as they generate KtpCZq. We state in Conjecture 2.5.7 that these objectscan be obtained in KtpO�Z q as quantum cluster variables, by following the same �nite se-quences of mutations used in the classical cluster algebra K0pO�Z q to obtain the rLpYi,qrqs.


Naturally, Conjecture 2.5.7 implies Conjecture 2.5.2. Finally, we prove Conjecture 2.5.7(and thus Conjecture 2.5.2) in the case where the underlying simple Lie algebra g is oftype A (Theorem 2.6.1). The proof is based on the thinness property of the fundamentalrepresentations in this case. When g � sl2, some explicit computations are possible. Forexample, we give a quantum version of the Baxter relation (2.6.2.1), for all r P Z,

rVq2r�1st � rL�1,q2r

st � t�1{2rω1srL�1,q2r�2st � t1{2r�ω1srL

�1,q2r�2st.

Additionally, we realize a part of the quantum cluster algebra we built as a quotient of thequantum group Uqpsl2q. This is a reminiscence of the result of Qin [Qin16], who constructedUqpgq as a quotient of the Grothendieck ring arising from certain cyclic quiver varieties.

The Chapter is organized as follows. In Section 2.2, we review some results for thecategory O, its subcategories O� and O�Z and their Grothendieck rings. In Section 2.3,after recalling the de�nition of the quantum torus Yt, we de�ne the quantum torus Tt

in which KtpO�Z q will be constructed and we prove the inclusion of the quantum tori. InSection 2.4, we prove that we have all the elements to build a quantum cluster algebra andwe de�ne the quantum Grothendieck ringKtpO�Z q. In Section 2.5, we state some propertiesof the quantum Grothendieck ring. We present the two conjectures regarding the inclusionof the quantum Grothendieck rings in Section 2.5.2. Finally, in Section 2.6 we prove theseconjectures in type A, and we prove �ner properties speci�c to the case when g � sl2.

2.2 Category O of representations of quantum loop algebras

We use the notations from Appendix A. We �rst recall the de�nitions of the quantumloop algebra and its Borel subalgebra, before introducing the Hernandez-Jimbo categoryO of representations, as well as some known results on the subject. We will sporadicallyuse concepts and notations from the two previous sections.

2.2.1 Quantum loop algebra and Borel subalgebra

Fix a nonzero complex number q, which is not a root of unity, and h P C such thatq � eh. Then for all r P Q, qr :� erh is well-de�ned. Since q is not a root of unity, forr, s P Q, we have qr � qs if and only if r � s.

We will use the following standard notations.

rmsz �zm�z�m

z�z�1 , rmsz! �±mj�1rjsz,

�r

s

�z

� rrsz !rssz !rr�ssz !

De�nition 2.2.1. One de�nes the quantum loop algebra UqpLgq as the C-algebra generatedby ei, fi, k

�1i , 0 ¤ i ¤ n, together with the following relations, for 0 ¤ i, j ¤ n:

kikj � kjki, kik�1i � k�1

i ki � 1, ka00 ka11 � � � kann � 1,

rei, fjs � δi,jki � k�1

i

q � q�1,

kiejk�1i � qCijej , kifjk

�1i � q�Cijej ,

1�Cij¸r�0

p�1qrep1�Cij�rqi eje


1�Cij¸r�0

p�1qrfp1�Cij�rqi fjf


(2.2.1.1)

2.2. Category O of representations of quantum loop algebras 59

where xprqi � xri {rrsq!, pxi � ei, fiq.

De�nition 2.2.2. The Borel algebra Uqpbq is the subalgebra of UqpLgq generated by theei, k

�1i , for 0 ¤ i ¤ n.

Both the quantum loop algebra and its Borel subalgebra are Hopf algebras.From now on, except when explicitly stated otherwise, we are going to consider repre-

sentations of the Borel algebra Uqpbq. Particularly, we consider the action of the `-Cartansubalgebra Uqpbq0, a commutative subalgebra of Uqpbq generated by the so-called Drinfeldgenerators:

Uqpbq0 :�Ak�1i , φ�i,r

EiPI,r¡0

.

2.2.2 Highest `-weight modules

Let V be a Uqpbq-module and ω P PQ a weight. One de�nes the weight space of V ofweight ω by

Vω :� tv P V | kiv � qωpα_i qv, 1 ¤ i ¤ nu.

The vector space V is said to be Cartan diagonalizable if V �À

ωPPQVω.

De�nition 2.2.3. A series ΨΨΨ � pψi,mqiPI,m¥0 of complex numbers, such that ψi,m P qQ

for all i P I is called an `-weight. The set of `-weights is denoted by P`. One identi�es the`-weight ΨΨΨ to its generating series :

ΨΨΨ � pψipzqqiPI , ψipzq �°m¥0 ψi,mz

m.

Let us de�ne some particular `-weights which are important in our context.For ω P PQ, let rωs be de�ned as

prωsqipzq � qωpα_i q, 1 ¤ i ¤ n. (2.2.2.1)

For i P I and a P C�, let• ΨΨΨi,a be de�ned as

pΨΨΨi,aqj pzq �

"1� az if j � i1 if j � i

. (2.2.2.2)

• Yi,a be de�ned as

pYi,aqj pzq �

#q 1�aq�1z

1�aqz if j � i

1 if j � i. (2.2.2.3)

The sets PQ and P` have group structures (the elements of P` are invertible formal series)and one has a surjective group morphism $ : P` Ñ PQ which satis�es ψip0q � q$pΨΨΨqpα

_i q,

for all ΨΨΨ P P` and all i P I.Let V be Uqpbq-module and ΨΨΨ P P` an `-weight. One de�nes the `-weight space of V

of `-weight ΨΨΨ by

VΨΨΨ :� tv P V | Dp ¥ 0,@i P I,@m ¥ 0, pφ�i,m � ψi,mqpv � 0u.

Remark 2.2.4. With the usual convention φ�i,0 � ki, one has VΨΨΨ � V$pΨΨΨq.


De�nition 2.2.5. Let V be a Uqpbq-module. It is said to be of highest `-weight ΨΨΨ P P` ifthere is v P V such that V � Uqpbqv,

eiv � 0,@i P I and φ�i,mv � ψi,mv, @i P I,m ¥ 0.

In this case, the `-weight ΨΨΨ is entirely determined by V , it is called the `-weight of V , andv is a highest `-weight vector of V .

Proposition 2.2.6. [HJ12] For all ΨΨΨ P P` there is, up to isomorphism, a unique simplehighest `-weight module of `-weight ΨΨΨ, denoted by LpΨΨΨq.

Example 2.2.7. For ω P PQ, Lprωsq is a one-dimensional representation of weight ω. Wealso denote it by rωs (tensoring by this representation is equivalent to shifting the weightsby ω).

2.2.3 De�nition of the category O

As explained in the Introduction, our focus here is a category O of representations ofthe Borel algebra, which was �rst de�ned in [HJ12], mimicking the usual de�nition of theBGG category O for Kac-Moody algebras. Here, we are going to use the de�nition in[HL16b], which is slightly di�erent.

For all λ P PQ, de�ne Dpλq :� tω P PQ | ω ¤ λu.

De�nition 2.2.8. A Uqpbq-module V is in the category O if :

1. V is Cartan diagonalizable,

2. For all ω P PQ, one has dimpVωq 8,

3. There is a �nite number of λ1, . . . , λs P PQ such that all the weights that appear in Vare in the cone

�sj�1Dpλjq.

The category O is a monoidal category.

Example 2.2.9. All �nite dimensional Uqpbq-modules are in the category O.Let P r

` be the set of `-weights Ψ such that, for all i P I, Ψipzq is rational. We will usethe following result.

Theorem 2.2.10. [HJ12] Let Ψ P P`. Simple objects in the category O are highest `-weightmodules. The simple module LpΨq is in the category O if and only if Ψ P P r

` . Moreover, ifV is in the category O and VΨ � 0, then Ψ P P r

` .

Example 2.2.11. For all i P I and a P C�, de�ne the prefundamental representations L�i,aas

L�i,a :� LpΨΨΨ�i,aq, (2.2.3.1)

for ΨΨΨi,a de�ned in (2.2.2.2). Then from Theorem 2.2.10, the prefundamental representa-tions belong to the category O.

2.2.4 Connection to �nite-dimension UqpLgq-modules

Throughout this chapter, we will use results already known for �nite-dimensional rep-resentations of the quantum loop algebra UqpLgq with the purpose of generalizing some ofthem to the context of the category O of representations of the Borel subalgebra Uqpbq.Let us �rst recognize that this approach is valid.

Let C be the category of all (type 1) �nite-dimensional UqpLgq-modules.


Proposition 2.2.12. [Bow07][CG05, Proposition 2.7] Let V be a simple �nite-dimensionalUqpLgq-module. Then V is simple as a Uqpbq-module.

Using this result and the classi�cation of �nite-dimensional simple module of quantumloop algebras in [CP95a], one has

Proposition 2.2.13. For all i P I, let Pipzq P Crzs be a polynomial with constant term 1.Let ΨΨΨ � pΨiqiPI be the `-weight such that

Ψipzq � qdegpPiqPipzq

�1q

Pipzqq, @i P I. (2.2.4.1)

Then LpΨΨΨq is �nite-dimensional.Moreover the action of Uqpbq can be uniquely extended to an action of UqpLgq, and any

simple object in the category C is of this form.

Hence, the category C is a subcategory of the category O and the inclusion functorpreserves simple objects.

Example 2.2.14. For all i P I and a P C�, consider the simple Uqpbq-module LpΨΨΨq of highest`-weight Yi,a, as in (2.2.2.3), then by Proposition 2.2.13, LpYi,aq is �nite-dimensional. Thismodule is called a fundamental representation and will be denoted by

Vi,a :� LpYi,aq. (2.2.4.2)

In general, simple modules in C are indexed by monomials in the variables pYi,aqiPI,aPC� ,called dominant monomials. Frenkel-Reshetikhin [FR99] de�ned a q-character morphismχq (see Section 2.2.8) on the Grothendieck ring of C . It is an injective ring morphism

χq : K0pC q Ñ Y :� ZrY �1i,a siPI,aPC� . (2.2.4.3)

Example 2.2.15. In the continuity of Example 2.2.14, for g � sl2, one has, for all a P C�,

χqpLpY1,aqq � Y1,a � Y �11,aq2

. (2.2.4.4)

2.2.5 Categories O�

Let us now recall the de�nitions of some subcategories of the category O, introducedin [HL16b]. These categories are interesting to study for di�erent reasons, here we areparticularly interested in the cluster algebra structure of their Grothendieck rings.

De�nition 2.2.16. An `-weight of P r` is said to be positive (resp. negative) if it is a

monomial in the following `-weights :• the Yi,a � qΨΨΨ�1

i,aqΨΨΨi,aq�1 , where i P I and a P C�,• the ΨΨΨi,a (resp. ΨΨΨ�1

i,a ), where i P I and a P C�,• the rωs, where ω P PQ.

De�nition 2.2.17. The category O� (resp. O�) is the category of representations in Owhose simple constituents have a positive (resp. negative) highest `-weight.

The category O� (resp. O�) contains the category of �nite-dimensional representa-tions, as well as the positive (resp. negative) prefundamental representations L�i,a (resp.L�i,a), de�ned in (2.2.11), for all i P I, a P C�.

Theorem 2.2.18. [HL16b] The categories O� and O� are monoidal categories.


2.2.6 The category O�Z

Recall the in�nite quiver Γ from Section A.4 of the Appendix and its set of vertices I.In [HL10], Hernandez and Leclerc de�ned a subcategory CZ of the category C . CZ is

the full subcategory whose objects satisfy: for all composition factor LpΨΨΨq, for all i P I,the roots of the polynomials Pi, as in Proposition 2.2.13 are of the form qr�1, such thatpi, rq P I.

This subcategory is interesting to study because each simple object in C can be writtenas a tensor product of simple objects which are essentially in CZ (see [HL10, Section 3.7]).Thus, the study of simple modules in C is equivalent to the study of simple modules inCZ.

Consider the same type of restriction on the category O.

De�nition 2.2.19. Let OZ be the subcategory of representations of O whose simplecomponents have a highest `-weight ΨΨΨ such that the roots and poles of Ψipzq are of theform qr, such that pi, rq P I.

We also de�ne O�Z as the subcategory of O� whose simple components have a highest`-weight ΨΨΨ such that the roots and poles of Ψipzq are of the form qr, such that pi, rq P I.

2.2.7 The Grothendieck ring K0pOq

Hernandez and Leclerc showed that the Grothendieck rings of the categories O�Z havesome interesting cluster algebra structures.

First of all, de�ne E as the additive group of maps c : PQ Ñ Z whose support iscontained in a �nite union of sets of the form Dpµq. For any ω P PQ, de�ne rωs P E as theδ-function at ω (this is compatible with the notation in Example 2.2.7). The elements ofE can be written as formal sums

c �¸

ωPsupppcq

cpωqrωs. (2.2.7.1)

E can be endowed with a ring structure, where the product is de�ned by

rωs � rω1s � rω � ω1s, @ω, ω1 P PQ.

If pckqkPN is a countable family of elements of E such that for any ω P PQ, ckpωq � 0except for �nitely many k P N, then

°kPN ck is a well-de�ned map from PQ to Z. In that

case, we say that°kPN ck is a countable sum of elements in E .

The Grothendieck ring of the category O can be viewed as a ring extension of E .Similarly to the case of representations of a simple Lie algebra (see [Kac90], Section 9.6),the multiplicity of an irreducible representation in a given representation of the categoryO is well-de�ned. Thus, the Grothendieck ring of the category O is formed of formal sums¸

ΨΨΨPP r`

λΨΨΨrLpΨΨΨqs, (2.2.7.2)

such that the λΨΨΨ P Z satisfy: ¸ΨΨΨPP r

` ,ωPPQ

|λΨΨΨ|dimpLpΨΨΨqωqrωs P E .

In this context, E is identi�ed with the Grothendieck ring of the category of represen-tations of O with constant `-weight.


A notion of countable sum of elements in K0pOq is de�ned exactly as for E .Now consider the cluster algebra ApΓq de�ned by the in�nite quiver Γ of Section A.4

of the Appendix, with in�nite set of coordinates denoted by

zzz �!zi,r | pi, rq P I

). (2.2.7.3)

By the Laurent Phenomenon (see B.4.2 in the Appendix), the cluster algebra ApΓq iscontained in the ring Zrz�1

i,r spi,rqPI . De�ne a E-algebra homomorphism χ : Zrz�1i,r sbZ E Ñ E

by

χpz�1i,r q �

��r

2

ωi

�, ppi, rq P Iq. (2.2.7.4)

The map χ is de�ned on ApΓq bZ E , and for each A P ApΓq bZ E , one can write χpAq �°ωPPQ

Aω b rωs, and |χ|pAq �°ωPPQ|Aω|b rωs.

Consider the completed tensor product

ApΓqbZE , (2.2.7.5)

of countable sums°kPNAk of elements Ak P ApΓq bZ E , such that

°kPN|χ|pAkq is a

countable sum of elements of E , as de�ned above.

Theorem 2.2.20. [HL16b, Theorem 4.2] The category O�Z is monoidal, and the identi�-cation

zi,r b�r

2ωi

�� rL�i,qr s,

�pi, rq P I

, (2.2.7.6)

de�nes an isomorphism of E-algebras

ApΓqbZE � K0pO�Z q. (2.2.7.7)

Example 2.2.21. We mentioned in the introduction that the Baxter relation (2.1.0.2) wasan exchange relation for this cluster algebra structure. Let us detail this. For g � sl2, thequiver Γ is:

...

(1,2)

(1,0)

(1,-2)

...

If we mutate at the node p1, 0q, the new cluster variable obtained is

z11,0 �1

z1,0pz1,2 � z1�2q .

Thus, via the identi�cation (2.2.7.6),

z11,0 �1

rL�1,1s

�r�ω1srL

�1,q2

s � rω1srL�1,q�2s

.

We indeed recognize the Baxter relation.Moreover, the new cluster variable z11,0 identi�es to a fundamental representation:

z11,0 � rLpY1,q�1qs. (2.2.7.8)

Remark 2.2.22. An analog theorem could be written for K0pO�Z q, as these are isomorphicas E-algebras ([HL16b, Theorem 5.17]).


2.2.8 The q-character morphism

Here we detail the notion of q-character on the category O. This notion extends theq-character morphism on the category of �nite-dimensional UqpLgq-modules mentioned inSection 2.2.4.

Similarly to Section 2.2.7, consider E`, the additive group of maps c : P r` Ñ Z such that

the image by $ of its support is contained in a �nite union of sets of the form Dpµq, andfor any ω P PQ, the set supppcq X$�1ptωuq is �nite. The map $ extends naturally to asurjective morphism $ : E` Ñ E . For ΨΨΨ P P r

` , de�ne the delta function rΨΨΨs � δΨΨΨ P E`.The elements of E` can be written as formal sums

c �¸

ΨΨΨPP r`

cpΨΨΨqrΨΨΨs. (2.2.8.1)

Endow E` with a ring structure given by

pc � dqpΨΨΨq �¸

ΨΨΨ1ΨΨΨ2�ΨΨΨ

cpΨΨΨ1qdpΨΨΨ2q, pc, d, P E`,ΨΨΨ P P r` q . (2.2.8.2)

In particular, for ΨΨΨ,ΨΨΨ1 P P r` ,

rΨΨΨs � rΨΨΨ1s � rΨΨΨΨΨΨ1s. (2.2.8.3)

For V a module in the category O, de�ne the q-character of V as in [FR99], [HJ12]:

χqpV q :�¸

ΨΨΨPP r`

dimpVΨΨΨqrΨΨΨs. (2.2.8.4)

By de�nition of the category O, χqpV q is an object of the ring E`.The following result extends the one from [FR99] to the context of the category O.

Proposition 2.2.23 ([HJ12]). The q-character map

χq : K0pOq Ñ E`rV s ÞÑ χqpV q,

(2.2.8.5)

is an injective ring morphism.

Example 2.2.24. For any a P C�, i P I, one has [HJ12, FH15],

χqpL�i,aq � rΨΨΨi,asχi, (2.2.8.6)

where χi � χpL�i,aq P E does not depend on a.For example, if g � sl2,

χ1 � χ �¸r¥0

r�2rω1s. (2.2.8.7)

2.3 Quantum tori

Let t be an indeterminate. The aim of this section is to build a non-commutativequantum torus Tt which will contain the quantum Grothendieck ring for the category O.For the category C of �nite-dimensional UqpLgq-modules, such a quantum torus alreadyexists, denoted by Yt here. Thus one natural condition on Tt is for it to contain Yt. Weshow it is the case in Proposition 2.3.3.

We start this section by recalling the de�nition and some properties of Yt. Here weuse the same quantum torus as in [Her04], which is slightly di�erent from the one used in[Nak04] and [VV03].

2.3. Quantum tori 65

2.3.1 The torus YtIn this section, we consider UqpLgq-modules and no longer Uqpbq-modules. We have

seen in Section 2.2.4 that for �nite-dimension representations, these settings were not toodi�erent.

As seen in (2.2.4.3), the Grothendieck ring of C can be seen as a subring of a ring ofLaurent polynomials

K0pC q � Y � ZrYi,asiPI,aPC�

In order to de�ne a t-deformed non-commutative version of this Grothendieck ring, one�rst needs a non-commutative, t-deformed version of Y, denoted by Yt.

Following [Her04], we de�ne

Y :� ZrY �i,qr | pi, rq P Is, (2.3.1.1)

the Laurent polynomial ring generated by the commuting variables Yi,qr .Let pYt, �q be the Zptq-algebra generated by the pY �

i,qrqpi,rqPI , with the t-commutationsrelations:

Yi,qr � Yj,qs � tNi,jpr�sqYj,qs � Yi,qr , (2.3.1.2)

where Ni,j : ZÑ Z is the antisymmetrical map, de�ned by

Ni,jpmq � Ci,jpm� 1q � Ci,jpm� 1q, @m ¥ 0, (2.3.1.3)

using the notations from Section A.2 of the Appendix.

Example 2.3.1. If we continue Example A.2.3 from the Appendix, for g � sl2, in this case,I � p1, 2Zq, for r P Z, one has

Y1,2r � Y1,2s � t2p�1qs�rY1,2s � Y1,2r, @s ¡ r ¡ 0. (2.3.1.4)

The Zptq-algebra Yt is viewed as a quantum torus of in�nite rank.Let us extend this quantum torus Yt by adjoining a �xed square root t1{2 of t:

Zpt1{2q bZptq Yt. (2.3.1.5)

By abuse of notation, the resulting algebra will still be denoted Yt.For a family of integers with �nitely many non-zero components pui,rqpi,rqPI , de�ne the

commutative monomial±

pi,rqPI Yui,ri,qr as¹

pi,rqPI

Yui,ri,qr :� t

12

°pi,rq pj,sq ui,ruj,sNi,jpr,sqÝÑ� pi,rqPIY

ui,ri,qr , (2.3.1.6)

where on the right-hand side an order on I is chosen so as to give meaning to the sum,and the product � is ordered by it (notice that the result does not depend on the orderchosen).

The commutative monomials form a basis of the Zpt1{2q-vector space Yt.


2.3.2 The torus Tt

We now want to extend the quantum torus Yt to a larger non-commutative algebraTt which would contain at least all the `-weights, and possibly all the candidates for thepq, tq-characters of the modules in the category O�Z .

In particular, Tt contains the ΨΨΨi,qr , for pi, rq P I, and these t-commutes with a relationcompatible with the t-commutation relation between the Yi,qr�1 (2.3.1.2).

Remark 2.3.2. One notices that there is a shift of parity between the powers of q in theY 's and the ΨΨΨ's. From now on, we will consider the ΨΨΨi,qr and the Yi,qr�1 , for pi, rq P I.

We start as in Section 2.3.1. First of all, de�ne

T :� Z�z�i,r | pi, rq P I

�, (2.3.2.1)

the Laurent polynomial ring generated by the commuting variables zi,r.Then, build a t-deformation Tt of T , as the Zrt�1s-algebra generated by the z�i,r, for

pi, rq P I, with a non-commutative product �, and the t-commutations relations

zi,r � zj,s � tFijps�rqzj,s � zi,r,�pi, rq, pj, sq P I

, (2.3.2.2)

where, for all i, j P I, Fij : ZÑ Z is a anti-symmetrical map such that, for all m ¥ 0,

Fijpmq � �¸k¥1

m¥2k�1

Cijpm� 2k � 1q. (2.3.2.3)

Now, letTt :� Zrt�1{2s bZrt�1s Tt. (2.3.2.4)

Similarly, we de�ne the commutative monomials in Tt as,¹pi,rqPI

zvi,ri,qr :� t

12

°pi,rq pj,sq vi,rvj,sFi,jpr,sqÝÑ� pi,rqPIz

vi,ri,qr . (2.3.2.5)

This based quantum torus will be enough to de�ne a structure of quantum clusteralgebra, but for it to contain the quantum Grothendieck ring of the category O�Z , oneneeds to extend it. In order to do that, we draw inspiration from Section 2.2.7. Recall thede�nition of χ from (2.2.7.4). We extend it to the E-algebra morphism χ : Tt bZ E Ñ Ede�ned by imposing χpt�1{2q � 1, as well as

χpz�1i,r q �

��r

2

ωi

�, ppi, rq P Iq.

As before, for z P Tt bZ E , one writes χpzq �°ωPPQ

zωrωs and |χ|pzq �°ωPPQ|zω|rωs.

De�ne the completed tensor product

Tt :� TtbZrt�1{2sE , (2.3.2.6)

of countable sums°kPN zk of elements zk P Tt bZ E , such that

°kPN|χ|pzkq is a countable

sum of E , as in Section 2.2.7.Consistently with the identi�cation (2.2.7.6), and the character of the z�1

i,r , we use the

following notation, for pi, rq P I,

rΨΨΨ�1i,qr s :� z�1

i,r

��r

2ωi

�P Tt. (2.3.2.7)

2.3. Quantum tori 67

Proposition 2.3.3. The identi�cation

J : Yi,qr�1 ÞÑ zi,rz�1i,r�2 � rωisrΨΨΨi,qr srΨΨΨ

�1i,qr�2s, (2.3.2.8)

where the products on the right hand side are commutative, extends to a well-de�ned injec-tive Zptq-algebra morphism J : Yt Ñ Tt.

Proof. One needs to check that the images of the Yi,qr�1 satisfy (2.3.1.2). Thus, we need

to show that, for all pi, rq, pj, sq P I,�zi,rz

�1i,r�2

��zj,sz

�1j,s�2

� tNi,jps�rq

�zj,sz

�1j,s�2

��zi,rz

�1i,r�2

,

which is equivalent to checking that:

2Fi,jps� rq � Fi,jps� r � 2q � Fi,jps� r � 2q � Ni,jps� rq. (2.3.2.9)

Suppose s ¥ r � 2, let m � s� r.

2Fi,jpmq � Fi,jpm� 2q � Fi,jpm� 2q � �¸k¥1

m¥2k�1

Cijpm� 2k � 1q

�¸k¥0

m¥2k�1

Cijpm� 2k � 1q �¸k¥2

m¥2k�1

Cijpm� 2k � 1q

� �Cijpm� 1q � Cijpm� 1q.

Thus 2Fi,jpmq � Fi,jpm� 2q � Fi,jpm� 2q � Ni,jpmq, using (2.3.1.3).If s � r � 1, the left-hand side of (2.3.2.9) is equal to

3Fi,jp1q � Fi,jp3q � Cijp2q � Ni,jp1q.

Example 2.3.4. Let us continue Examples A.2.3 and 2.3.1. For all r P Z. One has

z1,2r � z1,2s � tfps�rqz1,2s � z1,2r ,@r, s P Z, (2.3.2.10)

where f : ZÑ Z is antisymmetric and de�ned by

f|N : m ÞÑp�1qm � 1

2. (2.3.2.11)

And this is compatible with the relations (2.3.1.4).

De�nition 2.3.5. De�ne the evaluation at t � 1 as the E-morphism

π : Tt Ñ E`, (2.3.2.12)

such that

πpzi,rq �

��r

2ωi

�rΨΨΨi,qr s,

πpt�1{2q � 1.

Remark 2.3.6. The identi�cation (2.2.7.6) is between the element zi,r rrωi{2s and the classof the prefundamental representation rL�i,qr s. But this identi�cation is not compatible withthe character χ de�ned in (2.2.7.4), as the character of L�i,qr is χi, as in (2.2.8.6). Here,we choose to identify the variables zi,r with the highest `-weights of the prefundamentalrepresentations (up to a shift of weight), in particular, this identi�cation is compatiblewith the character morphism χ.


2.4 Quantum Grothendieck rings

The aim of this section is to build KtpO�Z q, a t-deformed version of the Grothendieckring of the category O�Z . This ring will be built inside the quantum torus Tt, as a quantumcluster algebra.

Let us summarize the existing objects in this context in a diagram:

CZ � O�Z

K0pCZq � K0pO�Z q � ApΓqbZE

KtpCZq � ???

A natural idea to build a t-deformation of the Grothendieck ring K0pO�Z q is to use itscluster algebra structure and de�ne a t-deformed quantum cluster algebra, as in Section B.3of the Appendix, with the same basis quiver. One has to make sure that the resulting objectis indeed a subalgebra of the quantum torus Tt.

2.4.1 The �nite-dimensional case

We start this section with some reminders regarding the quantum Grothendieck ringof the category of �nite-dimensional UqpLgq-modules.

This object was �rst discussed by Nakajima [Nak04] and Varagnolo-Vasserot [VV03]in the study of perverse sheaves. Then Hernandez gave a more algebraic de�nition, usingt-analogs of screening operators [Her03],[Her04]. This is the version we consider here, withthe restriction to some speci�c tensor subcategory CZ, as in [HL15].

De�nition of the Quantum Grothendieck ring

As in Section 2.2.6, consider CZ the full subcategory of C whose simple componentshave highest `-weights which are monomials in the Yi,qr , with pi, rq P I.

For pi, r � 1q P I, de�ne the commutative monomials

Ai,r :� Yi,qr�1Yi,qr�1

¹j�i

Y �1j,qr P Yt. (2.4.1.1)

For all i P I, let Ki,tpCZq be the Zpt1{2q-subalgebra of Yt generated by the

Yi,qrp1�A�1i,r�1q, Yj,qs

�pi, rq, pj, sq P I , j � i

. (2.4.1.2)

Finally, as in [Her04], de�ne

KtpCZq :�£iPI

Ki,tpCZq. (2.4.1.3)

Remark 2.4.1. Frenkel-Mukhin's algorithm [FM01] allows for the computation of certainq-characters, in particular those of the fundamental representations. In [Her04], Hernandezintroduced a t-deformed version of this algorithm to compute the pq, tq-characters of the

2.4. Quantum Grothendieck rings 69

fundamental representations, and thus to characterized the quantum Grothendieck ring asthe subring of Yt generated for those pq, tq-characters:

KtpCZq �ArLpYi,qrqst | pi, rq P I

E. (2.4.1.4)

pq, tq-characters in KtpCZq

Let us recall some more detailed results about the theory of pq, tq-characters for themodules in the category CZ.

Let M be the set of monomials in the variables pYi,qr�1qpi,rqPI , also called dominantmonomials. From [Her04] we know that for all dominant monomial m, there is a uniqueelement Ftpmq in KtpCZq such that m occurs in Ftpmq with multiplicity 1, and no otherdominant monomial occurs in Ftpmq. These Ftpmq form a Cpt1{2q-basis of KtpCZq.

For all dominant monomial m �±

pi,rqPI Yui,rpmq

i,qr�1 PM, de�ne

rMpmqst :� tαpmqÐÝ�rPZFt

�¹iPI

Yui,rpmq

i,qr�1

�P KtpCZq, (2.4.1.5)

where αpmq P 12Z is �xed such that m appears with coe�cient 1 in the expansion of

rMpmqst on the basis of the commutative monomials. The specialization at t � 1 ofrMpmqst recovers the q-character χqpMpmqq of the standard module Mpmq.

Consider the bar-involution , the anti-automorphism of Yt de�ned by:

t1{2 � t�1{2, Yi,qr�1 � Yi,qr�1 ,�pi, rq P I

. (2.4.1.6)

Remark 2.4.2. The commutative monomials, de�ned in Section 2.3.1, are clearly bar in-variant, as well as the subring KtpCZq.

There is a unique family trLpmqst P KtpCZq | m PMu such that

(i)rLpmqst � rLpmqst, (2.4.1.7)

(ii)rLpmqst P rMpmqst �

¸m1 m

t�1Zrt�1srMpm1qst, (2.4.1.8)

where m1 ¤ m means that mpm1q�1 is a product of Ai,r.

Lastly, we recall this result from Nakajima, proven using the geometry of quiver vari-eties.

Theorem 2.4.3. [Nak04] For all dominant monomial m PM, the specialization at t � 1of rLpmqst is equal to χqpLpmqq.

Moreover, the coe�cients of the expansion of rLpmqst as a linear combination of prod-ucts of Y �1

i,r belong to Nrt�1s.

Thus to all simple modules LpΨΨΨq in CZ is associated an object rLpmqst P KtpCZq, calledthe pq, tq-character. It is compatible with the q-character of the representation.


Remark 2.4.4. With the cluster algebra approach, we shed a new light on this last positivityresult. We interpret the pq, tq-characters of the fundamental modules (and actually allsimple modules which are realized as cluster variables in K0pO�Z q) as quantum clustervariables (Conjecture 2.5.7). Thus using Theorem B.6.1 of the Appendix, we recover thefact that the coe�cients of their expansion on the commutative monomials in the pY �1

i,r q

belong to Nrt�1s.

Remark 2.4.5. In order to fully extended this picture to the context of the category O, andimplement a Kazhdan-Lusztig type algorithm to compute the pq, tq-characters of all simplemodules, one would need an equivalent of the standard modules in this category. Thesedo not exist in general. This question was tackled in Chapter 1, in which equivalents ofstandard modules were de�ned when g � sl2.

2.4.2 Compatible pairs

We now begin the construction of KtpO�Z q.First of all, to de�ne a quantum cluster algebra, one needs a compatible pair, as in

Section B.2 in the Appendix. The basis quiver we consider here is the same quiver Γ asSection A.4 in the Appendix.

Explicitly, the corresponding exchange matrix is the I � I skew-symmetric matrix Bsuch that, for all ppi, rq, pj, sqq P I2,

Bppi,rq,pj,sqq �

$''''&''''%1 if i � j and s � r � 2

or i � j and s � r � 1,�1 if i � j and s � r � 2

or i � j and s � r � 1,0 otherwise.

(2.4.2.1)

Let Λ be the I�I skew-symmetric in�nite matrix encoding the t-commutation relations(2.3.2.2). Precisely, for ppi, rq, pj, sqq P I2 such that s ¡ r,

Λpi,rq,pj,sq � Fi,jps� rq � �¸k¥1

m¥2k�1

Cijpm� 2k � 1q. (2.4.2.2)

Remark 2.4.6. In [HL16b], it is noted that one can use su�ciently large �nite subseed ofΓ instead of an in�nite rank cluster algebra. For our purpose, the same statement staystrue, but one has to check that the subquiver still forms a compatible pair with the torusstructure. Hence, we have to give a more precise framework for the restriction to �nitesubseeds.

For all N P N�, de�ne ΓN , which is a �nite slice of Γ of length 2N � 1, containing anupper and lower row of frozen vertices. More precisely, de�ne IN and IN as

IN :�!pi, rq P I | �2N � 1 ¤ r 2N � 1

), (2.4.2.3)

IN :�!pi, rq P I | �2N � 1 ¤ r 2N � 1

). (2.4.2.4)

Then ΓN is the subquiver of Γ with set of vertices IN , where the vertices is INzIN arefrozen (thus the vertices in IN are the exchangeable vertices).

This way, all cluster variables of ApΓq obtained from the initial seed after a �nitesequence of mutations are cluster variables of the �nite rank cluster algebra ApΓN q, for Nlarge enough. With the same index restrict on B, we will be able to de�ne a size increasingfamily of �nite rank quantum cluster algebras.


Example 2.4.7. Recall from Example A.4.1 of the Appendix the in�nite quiver Γ wheng � sl4. Then the quiver ΓN is the following:

p1, 2Nq p3, 2Nq

p2, 2N � 1q

tt **

p1, 2N � 2q++

OO

p3, 2N � 2qss

OO

p2, 2N � 3qss ++

OO

p1, 2N � 4q++

OO

p3, 2N � 4qss

OO

p2, 2N � 5q

ss ++

OO

� � �

OO

,,

� � �

OO

rr� � �

OO

++ss

p1,�2N � 2q++

OO

p3,�2N � 2qss

OO

p2,�2N � 1q

tt **

OO

p1,�2Nq

OO

p3,�2Nq

OO

p2,�2N � 1q

OO

where the boxed vertices are frozen.

For N P N�, let BN be the corresponding exchange matrix. It is the IN � IN submatrixof B, thus its coe�cients are as in (2.4.2.1).

For all N P N�, let ΛN be the IN � IN submatrix of Λ. It is a �nite pnp2N � 1qq2

skew-symmetric matrix, where n is the rank of the simple Lie algebra g.

Example 2.4.8. For g of type D4, let us explicit a �nite slice of Γ of length 4, containing anupper and lower row of frozen vertices (which is thus not Γ1, of length 3, nor Γ2, of length5):

p1, 2q p3, 2q p4, 2q

p2, 1q

vv (( ,,p1, 0q

))

OO

p3, 0quu

OO

p4, 0q

rr

OO

p2,�1quu ))

OO

,,p1,�2q

))

OO

p3,�2quu

OO

p4,�2q

rr

OO

p2,�3qvv ((

OO

,,p1,�4q

OO

p3,�4q

OO

p4,�4q

OO

p2,�5q

OO

If the set I � tpi, rq P I | i P J1, 4K,�5 ¤ r ¤ 2u is ordered lexicographically by r then i


(reading order), the quiver is represented by the following exchange matrix:

B :�

��

�1 0 0 0 0 0 0 00 �1 0 0 0 0 0 00 0 �1 0 0 0 0 01 1 1 �1 0 0 0 00 0 0 1 �1 0 0 00 0 0 1 0 �1 0 00 0 0 1 0 0 �1 0

�1 �1 �1 0 1 1 1 �11 0 0 �1 0 0 0 10 1 0 �1 0 0 0 10 0 1 �1 0 0 0 10 0 0 1 �1 �1 �1 00 0 0 0 1 0 0 �10 0 0 0 0 1 0 �10 0 0 0 0 0 1 �10 0 0 0 0 0 0 1

��

(2.4.2.5)

The principal part B of B is the square submatrix obtained by omitting the �rst 4 columnsand the last 4 columns. One notices that B is skew-symmetric.

Moreover, using Formula (2.4.2.2), one can compute the corresponding matrix Λ. Weget the following 16� 16 skew-symmetric matrix (with the same order of I as before):��

0 0 0 0 1 0 0 1 1 1 1 2 2 1 1 20 0 0 0 0 1 0 1 1 1 1 2 1 2 1 20 0 0 0 0 0 1 1 1 1 1 2 1 1 2 20 0 0 0 0 0 0 1 1 1 1 3 2 2 2 4

�1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 20 �1 0 0 0 0 0 0 0 1 0 1 1 1 1 20 0 �1 0 0 0 0 0 0 0 1 1 1 1 1 2

�1 �1 �1 �1 0 0 0 0 0 0 0 1 1 1 1 3�1 �1 �1 �1 �1 0 0 0 0 0 0 0 1 0 0 1�1 �1 �1 �1 0 �1 0 0 0 0 0 0 0 1 0 1�1 �1 �1 �1 0 0 �1 0 0 0 0 0 0 0 1 1�2 �2 �2 �3 �1 �1 �1 �1 0 0 0 0 0 0 0 1�2 �1 �1 �2 �1 �1 �1 �1 �1 0 0 0 0 0 0 0�1 �2 �1 �2 �1 �1 �1 �1 0 �1 0 0 0 0 0 0�1 �1 �2 �2 �1 �1 �1 �1 0 0 �1 0 0 0 0 0�2 �2 �2 �4 �2 �2 �2 �3 �1 �1 �1 �1 0 0 0 0

��

(2.4.2.6)

From here, it is easy to check that the product BTΛ is of the form:

BTΛ �

��

0 0 0 0 �2 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 �2 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 �2 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 �2 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 �2 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 �2 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 �2 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 �2 0 0 0 0

�� . (2.4.2.7)


Thus, pΛ, Bq is a compatible pair.

Remark 2.4.9. Notice the coe�cients appearing of the diagonal here are all negative. Fromthe de�nition of compatible pair from [BZ05], for pΛ, Bq to be a compatible pair, theseshould be positive. That is why in De�nition B.2.1 of the Appendix, the coe�cients areof constant sign (the properties of quantum cluster algebras are still satis�ed by takingnegative coe�cients in the compatibility condition). This allows for our t-commutation tobe coherent with that of the quantum torus Yt.

We show that this result is true in general. Furthermore, the speci�c form we obtainin Equation (2.4.2.7) is what we get in general.

Proposition 2.4.10. The pairs pΛ, Bq and�pΛN , BN q

NPN�

are compatible pairs, in the

sense of structure condition for quantum cluster algebras.Moreover,

BTΛ � �2 IdI ,

BTNΛN � �2

�p0q IdIN p0q

�.

Proof. Let ppi, rq, pj, sqq P I2. Let us compute:�BTΛ

pi,rq,pj,sq

�¸

pk,uqPI

bpk,uq,pi,rqλpk,uq,pj,sq. (2.4.2.8)

This is a �nite sum, as each vertex in Γ is adjacent to a �nite number of other vertices.Suppose �rst that r � s. Without loss of generality, we can assume that r s. Then,

using the de�nition of the matrix Λ in (2.4.2.2) and the coe�cients of B in (2.4.2.1), weobtain �

BTΛpi,rq,pj,sq

� �Cijps� r � 1q � Cijps� r � 1q �¸k�i

Ckjps� rq. (2.4.2.9)

Now recall from Lemma A.2.4 of the Appendix, for all pi, jq P I2,

Cijpm� 1q � Cijpm� 1q �¸k�i

Ckjpmq � 0, @m ¥ 1.

Thus, for allpi, jq P I2 and r s, equation (2.4.2.9) gives:�BTΛ

pi,rq,pj,sq

� 0. (2.4.2.10)

Suppose now that r � s. In this case,�BTΛ

pi,rq,pj,rq

� �2Cijp1q � �2δi,j ,

using the other result from Lemma A.2.4. Thus,

BTΛ � �2 IdI . (2.4.2.11)

Now, for all N P N�, let pi, rq P IN and pj, sq P IN . Let us write:�BTΛ

pi,rq,pj,sq

�¸

pk,uqPIN

bpk,uq,pi,rqλpk,uq,pj,sq. (2.4.2.12)

As pi, rq P IN is not a frozen variable, the pj, sq P I such that bpk,uq,pi,rq � 0 are all in IN .Hence, the rest of the reasoning is still valid, and the result follows.


2.4.3 De�nition of KtpO�Z q

Everything is now in place to de�ne KtpO�Z q. Recall the based quantum torus Tt,de�ned in Section 2.3.2. By construction, the associated skew-symmetric bilinear form Λidenti�es with the in�nite skew-symmetric I � I-matrix from the previous section:

Λpepi,rq, epj,sqq � Λpi,rq,pj,sq � Fijps� rq,�pi, rq, pj, sq P I

(2.4.3.1)

where pepi,rqqpi,rqPI is the standard basis of ZpIq.

Let F be the skew-�eld of fractions of Tt. We de�ne the toric frame M : ZpIq Ñ Fzt0uby setting

Mpepi,rqq � zi,r P F , @pi, rq P I . (2.4.3.2)

Then the in�nite rank matrix ΛM satis�es

ΛM � Λ. (2.4.3.3)

From the result of Proposition 2.4.10,

S ��M, B

(2.4.3.4)

is a quantum seed.

De�nition 2.4.11. Let AtpΓq be the quantum cluster algebra associated to the mutation-equivalence class of the quantum seed S.Remark 2.4.12. One could note that this is an in�nite rank quantum cluster algebra,which it not covered by the de�nition given in Appendix B. However, we have a sequenceof quantum cluster algebras pAtpΓN qqNPN� , built on the �nite quivers pΓN qNPN� . As themutation sequences are �nite, one can always assume we are working in the quantumcluster algebra AtpΓN q, with N large enough.

Fix N P N�. Let m � p2N � 1q � n, where n is the rank of the simple Lie algebra g.Consider LN , the sub-lattice of Tt generated by the zi,r, with pi, rq P IN (recall the

de�nition of IN in (2.4.2.4)). LN is of rank m. Consider the toric frame MN which is therestriction of M to LN . In that case,

ΛMN� ΛN , from the previous section.

Thus,

SN :��MN , BN

(2.4.3.5)

is a quantum seed.

De�nition 2.4.13. LetAtpΓN q be the quantum cluster algebra associated to the mutation-equivalence class of the quantum seed SN .Remark 2.4.14. This is an explicit construction of an in�nite rank quantum cluster algebraas a (co)limit of �nite rank quantum cluster algebras. Thus we constructed explicitly theresult of [GG18] in our situation.

De�nition 2.4.15. De�neKtpO�Z q :� AtpΓqbE , (2.4.3.6)

where the tensor product is completed as in (2.3.2.6). The ring KtpO�Z q is a Ert�1{2s-subalgebra of Tt.

For N P N�, with the same completion of the tensor product, de�ne

KtpO�Z,N q :� AtpΓN qbE . (2.4.3.7)

2.5. Properties of KtpO�Z q 75

2.5 Properties of KtpO�Z q

2.5.1 The bar involution

The bar involution de�ned on Yt (see Section 2.4.1) has a counterpart on the largerquantum torus Tt. Besides, as pq, tq-character of simple modules in CZ are bar-invariant byde�nition, it is natural for pq, tq-characters of simple modules inO�Z to also be bar-invariant.

There is unique E-algebra anti-automorphism of Tt such that

t1{2 � t�1{2, zi,r � zi,r, and rωis � rωis, ppi, rq P Iq.

What is crucial to note here is that this de�nition is compatible with the bar-involutionde�ned in general on the quantum torus of any quantum cluster algebra (see [BZ05, Section6]). However, this bar-involution has an important property: all cluster variables areinvariant under the bar involution.

Proposition 2.5.1. All elements of KtpO�Z q of the form χt b 1, where χt P AtpΓq is acluster variable, are invariant under the bar-involution.

2.5.2 Inclusion of quantum Grothendieck rings

As stated earlier, one natural property we would want to be satis�ed by the quantumGrothendieck ringKtpO�Z q is to include the already-existing quantum GrothendieckKtpCZqof the category CZ.

Note that these rings are contained in quantum tori, which are included in one anotherby the injective morphism J from Proposition 2.3.3:

KtpCZq � Yt

KtpO�Z q � Tt.

J

Thus it is natural to formulate the following Conjecture:

Conjecture 2.5.2. The injective morphism J restricts to an inclusion of the quantumGrothendieck rings

J : KtpCZq � KtpO�Z q. (2.5.2.1)

Recall that the quantum Grothendieck ring KtpCZq is generated by the classes of thefundamental representations rLpYi,qr�1qst, for pi, rq P I (see Section 2.4.1). Hence, in orderto prove Conjecture 2.5.2, it is enough to show that the images of these rLpYi,qr�1qst belongto KtpO�Z q.

In Example 2.2.21 we saw how, when the g � sl2, the class of the fundamental repre-sentation rLpY1,q�1qs could be obtained as a cluster variable in ApΓq after one mutation indirection p1, 0q.

This fact is actually true in more generally, as seen in [HL16b], in the proof of Propo-sition 6.1. Let us recall this process precisely.

Fix pi, rq P I. We �rst de�ne a speci�c sequence of vertices in Γ, as in [HL16a]. Recallthe de�nition of the dual Coxeter number h_.

g An Dn E6 E7 E8

h_ n� 1 2n� 2 12 18 30


Let h1 � rh_{2s. Fix an ordering pj1, . . . , jnq of the vertices of the Dynkin diagram of gby taking �rst j1 � i, then all vertices which appear with the same oddity as i in I (the jsuch that pj, rq P I), then the vertices which appear with a di�erent oddity (pj, r� 1q P I).For all k P t2, . . . , h1u, j P t1, . . . , nu, de�ne the sequence Sj,k of k vertices of the columnj of Γ in decreasing order:

Sj,k � pj, r � 2h1 � εq, pj, r � 2h1 � ε� 2q, . . . , pj, r � 2h1 � ε� 2k � 2q, (2.5.2.2)

where ε P t0, 1u, depending of the oddity. Then de�ne

Sk �ÝÑ¤j

Sj,k, (2.5.2.3)

with the order de�ned before. Finally, let:

S � Sh1 � � �S2 pi, r � 2h1q,

by reading left to right and adding one last pi, r � 2h1q at the end.

Example 2.5.3. For g of type D4, and pi, rq � p1, 0q, the sequence S is

S �p1, 6q p1, 4q p1, 2q p3, 6q p3, 4q p3, 2q

p4, 6q p4, 4q p4, 2q p2, 5q p2, 3q p2, 1q

p1, 6q p1, 4q p3, 6q p3, 4q p4, 6q p4, 4q

p2, 5q p2, 3q p1, 6q

Using [HL16a, Theorem 3.1] and elements from the proof of Proposition 6.1 in [HL16b],one gets the following result.

Proposition 2.5.4. Let χi,r be the cluster variable of ApΓq obtained at the vertex pi, r�2h1qafter following the sequence of mutations S, then, via the identi�cation (2.2.7.6)

χi,r � rLpYi,qr�1qs. (2.5.2.4)

To see this result di�erently, if one writes χi,r as a Laurent polynomial in the variablespzj,sq, then χi,r is in the image of J , and

χi,r � J pχqpLpYi,qr�1qq. (2.5.2.5)

Example 2.5.5. Let g � sl3 and pi, rq � p1, 0q. The sequence of vertices S is

S � p1, 4q p1, 2q p2, 3q p2, 1q p1, 4q. (2.5.2.6)

Let us compute the sequence of mutations S:


...

p1, 6q

(1,4)

p1, 2q

p1, 0q

...

...

p2, 5q

p2, 3q

p2, 1q

...

...

p1, 6q

p1, 4q

(1,2)

p1, 0q

...

...

p2, 5q

p2, 3q

p2, 1q

...

...

p1, 6q

p1, 4q

p1, 2q

p1, 0q

...

...

p2, 5q

(2,3)

p2, 1q

...

...

p1, 6q

p1, 4q

p1, 2q

p1, 0q

...

...

p2, 5q

p2, 3q

(2,1)

...

...

p1, 6q

(1,4)

p1, 2q

p1, 0q

...

...

p2, 5q

p2, 3q

p2, 1q

...

...

p1, 6q

p1, 4q

p1, 2q

p1, 0q

...

...

p2, 5q

p2, 3q

p2, 1q

...


The associated cluster variables are:

zp1q1,4 � z1,2z

�11,4z2,5 � z�1

1,4z1,6z2,3,

zp1q1,2 � z1,0z

�11,4z2,5 � z1,0z

�11,2z

�11,4z1,6z2,3 � z�1

1,2z1,6z2,�1,

zp1q2,3 � z2,1z

�12,3 � z1,2z

�11,4z2,5z

�12,3 � z�1

1,4z1,6,

zp2q1,4 � z1,0z

�11,2 � z�1

1,2z1,4z2,1z�12,3 � z�1

2,3z2,5.

Thus, χ1,0 � zp2q1,4 is in the image of J , and

χ1,0 � J pY1,q � Y �11,q3

Y2,q2 � Y �12,q4

q � J pχqpLpY1,qqqq. (2.5.2.7)

Notice also that zp1q2,3 was already in the image of J and that zp1q2,3 � J pχqpLpY2,q2qqq.

Thus, for each pi, rq P I, consider the quantum cluster variables χi,r P KtpO�Z q obtainedfrom the initial quantum seed pzzz,Λq via the sequence of mutations S.

Example 2.5.6. Suppose g � sl2. Consider the quiver Γ1 a well as the skew-symmetricmatrix Λ1,

Γ1 �

p1, 2q

p1, 0q

OO

p1,�2q

OO, Λ1 �

�� 0 �1 01 0 �10 1 0

� .

As seen in Example 2.2.21 (with a shift of quantum parameters), the fundamental repre-sentation rLpY1,q�1qs is obtained in K0pO�Z q after one mutation at p1, 0q (here S � p1, 0q).

The quantum cluster variable obtained after a quantum mutation at p1, 0q, writtenwith commutative monomials, is

χ1,�2 � z1,�2z�11,0 � z1,2z

�11,0 � J pY1,q�1 � Y �1

1,q q � J prLpY1,q�1qstq,

� J�Y1,q�1p1�A�1

1,1q

P J pKtpCZqq,

Thus, we note that in this particular case, the quantum cluster variable χ1,�2 recovers thepq, tq-character rLpY1,q�1qst of the fundamental representation LpY1,q�1q.

In particular, Conjecture 2.5.2 is satis�ed in this case.

This example incites us to formulate another conjecture.

Conjecture 2.5.7. For all pi, rq P I, the quantum cluster variable χi,r recovers, via themorphism J , the pq, tq-character of the fundamental representation LpYi,qr�1q:

χi,r � J�rLpYi,qr�1qst

�. (2.5.2.8)

Remark 2.5.8. Notice that Conjecture 2.5.7 implies Conjecture 2.5.2, and that Conjec-ture 2.5.7 is also satis�ed when g � sl2, from Example 2.5.6.

What can be said, in general, of the quantum cluster variables χi,r ?

Proposition 2.5.9. For all pi, rq P I, the quantum cluster variable χi,r satis�es the fol-lowing properties:


(i) invariant under the bar involution:

χi,r � χi,r. (2.5.2.9)

(ii) the coe�cients of its expansion as a Laurent polynomial in the initial quantum clustervariables tzi,ru are Laurent polynomials in t1{2 with non-negative integers coe�cients:

χi,r Pà

uuu�ui,rPZpIqNrt�1{2szzzuuu. (2.5.2.10)

with zzzuuu �±

pi,rqPI zui,ri,r denoting the commutative monomial.

(iii) its evaluation at t � 1 (as seen in (2.3.2.12)), recovers the q-character of the funda-mental representation LpYi,qr�1q:

πpχi,rq � χqpLpYi,qr�1qq. (2.5.2.11)

Proof. The �rst property is a direct consequence of Proposition 2.5.1 and the second is adirect consequence of the positivity result of Theorem B.6.1 in the Appendix.

For the third property, notice we have used two evaluation maps so far, with the samenotation.• The evaluation map de�ned in (B.5.1) on the bases quantum torus of a quantum clusteralgebra:

π : AtpM, Bq Ñ ZrX�1s,

• The evaluation map de�ned in (2.3.2.12) on Tt:

π : Tt Ñ E`.

These notations are coherent because the map π from (2.3.2.12) is the evaluation mapde�ned on a based quantum torus (of in�nite rank) of a quantum cluster algebra, extended

to a E-morphism on Tt. In this case, the Laurent polynomial ring ZrX�1s is Zrz�1

i,r | pi, rq P

Is, which becomes ErΨΨΨ�1i,r s after extension to a E-morphism and via the identi�cation

(2.2.7.6).Thus we can apply Corollary B.5.3 of the Appendix to this map π. As χi,r is a quantum

cluster variable, its evaluation by π is the cluster variable χi,r, which is obtained from theinitial seed z, via the same sequence of mutations S (the initial seed and quantum seeds are�xed and identi�ed by the evaluation π on the quantum torus Tt). By Proposition 2.5.4,

πpχi,rq � χi,r � χqpLpYi,qr�1qq. (2.5.2.12)

These two properties imply that the χi,r are good candidates for the pq, tq-charactersof the fundamental representations, as stated in Conjecture 2.5.2.

2.5.3 pq, tq-characters for positive prefundamental representations

Recall the q-characters of the positive prefundamental representations in (2.2.8.6), forall i P I, a P C�,

χqpL�i,aq � rΨΨΨi,asχi,

where χi P E is the (classical) character of L�i,a.


De�nition 2.5.10. For pi, rq P I, de�ne

rL�i,qr st :� rΨΨΨi,qr s b χi P KtpO�Z q, (2.5.3.1)

using the notation from (2.3.2.7).

Remark 2.5.11. It is the quantum cluster variable obtained from the initial quantum seed,via the same sequence of mutations used to obtain rL�i,qr s in K0pO�Z q, which in this case,is no mutation at all.

In particular, the evaluation of rL�i,qr st recovers the q-character of rL�i,qr s:

πprL�i,qr stq � rΨΨΨi,qr s b χi � χqpL�i,aq P E`. (2.5.3.2)

2.6 Results in type A

Suppose in this section that the underlying simple Lie algebra g is of type A.

2.6.1 Proof of the conjectures

In this case, the situation of Example 2.5.6 generalizes.

Theorem 2.6.1. Conjecture 2.5.7 is satis�ed in this case.

In this case, the key ingredient is the following well-known result (see for exemple[FR96, Section 11], and references therein).

Theorem 2.6.2. When g is of type A, all `-weight spaces of all fundamental representa-tions LpYi,aq are of dimension 1.

Proof. Fix pi, rq P I. From the second property of Proposition 2.5.9, we know that χi,r canbe written as

χi,r �¸

uuuPZpIqPuuupt

1{2qzzzuuu, (2.6.1.1)

where the Puuupt1{2q are Laurent polynomials with non-negative integer coe�cients. Usingthe third property of Proposition 2.5.9, we deduce the evaluation at t � 1 of equality(2.6.1.1):

χqpLpYi,qr�1qq �¸

uuuPZpIqPuuup1q

¹pi,rqPI

prΨΨΨi,qr sr�rωi{2squi,r P E`. (2.6.1.2)

From the aforementioned theorem, this decomposition is multiplicity-free. Thus, the non-zero coe�cients Puuupt1{2q are of the form tk{2, with k P Z. Finally, as χi,r is bar-invariant,from the �rst property of Proposition 2.5.9, and the zzzuuu are also bar-invariant as com-mutative monomials, we know that the Laurent polynomials Puuupt1{2q are even functions:

Puuup�t1{2q � Puuupt

1{2q. (2.6.1.3)

Thus the variable t1{2 does not explicitly appear in the decomposition (2.6.1.1), and so:

χi,r �¸

uuuPZpIqPuuup1qzzz

uuu,

� J�χqpLpYi,qr�1qq

�.

2.6. Results in type A 81

Moreover, with the same arguments, as rLpYi,qr�1qst is bar-invariant by de�nition,

rLpYi,qr�1qst � χqpLpYi,qr�1qq, (2.6.1.4)

written in the basis of the commutative monomials.Hence, we recover the fact that the quantum cluster variable χi,r is equal, via the

inclusion map J , to the pq, tq-character of LpYi,qr�1q and Conjecture 2.5.7 is satis�ed.

2.6.2 A remarkable subalgebra in type A1

When g � sl2, we can make explicit computations. Recall the formula (2.3.2.10) of thequantum torus from Example 2.3.4:

z1,2r � z1,2s � tfps�rqz1,2s � z1,2r ,@r, s P Z,

where f : ZÑ Z is antisymmetric and de�ned by

f|N : m ÞÑp�1qm � 1

2.

For all r P Z, the pq, tq-character of the prefundamental representation L�1,q2r

de�nedin (2.5.3.1) is

rL�1,q2r

st � rΨΨΨ1q2r sχ1.

Proposition 2.6.3. With these pq, tq-characters, we can write a t-deformed version of theBaxter relation (2.1.0.2), for all r P Z,

rLpY1,q2r�1qst � rL�1,q2r

st � t�1{2rω1srL�1,q2r�2st � t1{2r�ω1srL

�1,q2r�2st. (2.6.2.1)

We call this relation the quantized Baxter relation.

Remark 2.6.4. If we identify the variables Y1,q2r and their images through the injectionJ , this relation is actually the exchange relation related to the quantum mutation inExample 2.5.6 (for a generic quantum parameter q2r).

Now consider the quantum cluster algebra ApΛ1,Γ1q, with notations from Section 2.4.2(Λ1 and Γ1 are given explicitly in Example 2.5.6).

It is a quantum cluster algebra of �nite type (if we remove the frozen vertices fromthe quiver, we get just one vertex, which is a quiver of type A1). It has two quantumclusters, containing the two frozen variables z1,2, z1,�2 and the mutable variables z1,0 and

zp1q1,0 , respectively. Thus, it is generated as a Cpt1{2q-algebra by

E :� rLpY1,q�1qst p� zp1q1,0q, F :� rL�1,1st p� z1,0q,

K :� rω1srL�1,q�2st p� z1,�2q, K 1 :� r�ω1srL

�1,q2

st p� z1,2q.(2.6.2.2)

This algebra is a quotient of a well-known Cpt1{2q-algebra.Let q be a formal parameter. The quantum group Uqpsl2q can be seen as the quotient

Uqpsl2q � D2{@KK 1 � 1

D, (2.6.2.3)

where D2 is the Cpqq-algebra with generators E,F,K,K 1 and relations:

KE � q2EK, K 1E � q�2EK 1

KF � q�2FK, K 1F � q2FK 1

KK 1 � K 1K, and rE,F s � pq � q�1qpK �K 1q.(2.6.2.4)


Remark 2.6.5. • As in [SS19, Remark 3.1], notice that the last relation in (2.6.2.4) is notthe usual relation

re, f s �K �K 1

q � q�1.

But both presentations are equivalent, given the change of variables

E � pq � q�1qe, F � pq � q�1qf.

• D2 is the Drinfeld double [Dri87] of the Borel subalgebra of Uqpsl2q (the subalgebragenerated by K,E).

Proposition 2.6.6. The Cpt1{2q-algebra ApΛ1,Γ1q is isomorphic to the quotient of theDrinfeld double D2 of parameter �t1{2,

ApΛ1,Γ1q�ÝÑ D2{C�t1{2 , (2.6.2.5)

where C�t1{2 is the quantized Casimir element:

C�t1{2 :� EF � t1{2K � t�1{2K 1. (2.6.2.6)

Proof. One has, in ApΛ1,Γ1q,

E � F � t�1{2K � t1{2K 1. (2.6.2.7)

This is the quantized Baxter relation (2.6.2.1). Thus,

rE,F s � p�t1{2 � t�1{2qpK �K 1q.

We check that the other relations in (2.6.2.4) are also satis�ed using the structure of thequantum torus Tt (which is given explicitly in Example 2.3.4).

Hence the map

ApΛ1,Γ1qθÝÑ D2,

sending generators to generators, is well-de�ned and descends onto the quotient

ApΛ1,Γ1q Ñ D2{C�t1{2 .

Moreover, from [CIKLFP13], the cluster monomials in a given cluster in a cluster algebraare linearly independent. In this case, the quantum cluster algebra ApΛ1,Γ1q is of type A1

(without frozen variables), thus of �nite-type. It has two (quantum) clusters: pE,K,K 1qand pF,K,K 1q. Thus, the set of commutative quantum cluster monomials!

EαKβK 1γ | α, β, γ P Z)Y!KβK 1γFα | α, β, γ P Z

), (2.6.2.8)

forms a Cpt1{2q-basis of ApΛ1,Γ1q.Consider the PBW basis of D2:!

EαKβK 1γF δ | α, β, γ, δ P Z). (2.6.2.9)

From the expression of the Casimir element C�t1{2 (2.6.2.6), we deduce a Cpt1{2q-basis ofD2{C�t1{2 , of the same form as (2.6.2.8):!

EαKβK 1γ | α, β, γ P Z)Y!KβK 1γFα | α, β, γ P Z

), . (2.6.2.10)

Hence, the map θ sends a basis to a basis, and thus it is isomorphic.

2.6. Results in type A 83

This result should be compared with the recent work of Schrader and Shapiro [SS19],in which they recognize the same structure of D2 in an algebra built on a quiver, withsome quantum X -cluster algebra structure. In their work, they generalized this result intype A (Theorem 4.4). Ultimately, they obtain an embedding of the whole quantum groupUqpslnq into a quantum cluster algebra. The result of Proposition 2.6.6, together withtheir results, gives hope that one could �nd a realization of the quantum group Uqpgq as aquantum cluster algebra, related to the representation theory of UqpLgq.

Furthermore, de�ne in this case O�1 , the subcategory of O�Z of objects whose image inthe Grothendieck ring K0pO�Z q belongs to the subring generated by rL�

1,q�2s, rL�1,1s, rL

�1,q2

s

and rLpY1,q�1qs. Then O�1 is a monoidal category.From the classi�cation of simple modules when g � sl2 in [HL16b, Section 7], we know

that the only prime simple modules in O�1 are

L�1,q�2 , L

�1,1, L

�1,q2

, LpY1,q�1q. (2.6.2.11)

Moreover, a tensor product of those modules is simple if and only if it does not contain botha factor L�1,1 and a factor LpY1,q�1q (the others are in so-called pairwise general position).Thus, in this situation, the simple modules are in bijection with the cluster monomials:"

simple modulesin O�1

*ÐÑ

"commutative quantum cluster

monomials in ApΛ1,Γ1q

*�L�

1,q�2

bαb�L�1,1

bβb�L�

1,q2

bγÞÑ KαF βK 1γ ,�

L�1,q�2

bα1b�L�

1,q2

bβ1b LpY1,q�1qbγ

1ÞÑ Kα1K 1β1Eγ

1.

Chapter 3Quantum cluster algebra algorithm andpq, tq-characters

Abstract.

We establish a quantum cluster algebra structure on the quantum Grothendieck ring

of a certain monoidal subcategory of the category of �nite-dimensional representations

of a simply-laced quantum a�ne algebra. Moreover, the pq, tq-characters of certain ir-

reducible representations, among which fundamental representations, are obtained as

quantum cluster variables. This approach gives a new algorithm to compute these pq, tq-characters. As an application, we prove that the quantum Grothendieck ring of a larger

category of representations of the Borel subalgebra of the quantum a�ne algebra, de�ned

in a previous work as a quantum cluster algebra, contains indeed the well-known quan-

tum Grothendieck ring of the category of �nite-dimensional representations. Finally, we

display our algorithm on a concrete example.

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.2 Finite-dimensional representations of quantum a�ne algebras . . . . . . . . 88

3.2.1 Quantum a�ne algebra . . . . . . . . . . . . . . . . . . . . . . . . . 883.2.2 Finite-dimensional representations . . . . . . . . . . . . . . . . . . . 893.2.3 q-characters and truncated q-characters . . . . . . . . . . . . . . . . 893.2.4 Cluster algebra structure . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3 Quantum Grothendieck rings . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.1 Quantum torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.2 Commutative monomials . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.3 Quantum Grothendieck ring KtpCZq . . . . . . . . . . . . . . . . . . 933.3.4 pq, tq-characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.5 Truncated pq, tq-characters and quantum Grothendieck ring KtpC

�Z q 95

3.3.6 Quantum T -systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.4 Quantum cluster algebra structure . . . . . . . . . . . . . . . . . . . . . . . 96

3.4.1 A compatible pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.4.2 The quantum cluster algebra At . . . . . . . . . . . . . . . . . . . . 98

3.5 Quantum cluster algebras and quantum Grothendieck ring . . . . . . . . . . 993.5.1 A note on the bar-involution . . . . . . . . . . . . . . . . . . . . . . 1003.5.2 A sequence of vertices . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5.3 Truncated pq, tq-characters as quantum cluster variables . . . . . . . 1013.5.4 Proof of Theorem 3.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 104

86 Chapter 3. Quantum cluster algebra algorithm

3.6 Application to the proof of an inclusion conjecture . . . . . . . . . . . . . . 1073.6.1 The quantum Grothendieck ring KtpO�Z q . . . . . . . . . . . . . . . . 1073.6.2 Intermediate quantum cluster algebras . . . . . . . . . . . . . . . . . 1073.6.3 Inclusion of quantum Grothendieck rings . . . . . . . . . . . . . . . . 111

3.7 Explicit computation in type D4 . . . . . . . . . . . . . . . . . . . . . . . . 115

3.1 Introduction

Finite-dimensional representations of quantum a�ne algebras have been classi�ed byChari-Pressley [CP95b] with a quantum analog of Cartan's highest weight classi�cation of�nite-dimensional representations of simple Lie algebras. Combining this classi�cation withthe notations from Frenkel-Reshetikhin q-character [FR99] theory, one gets the following.

Let g be a �nite-dimensional simple Lie algebra, and Uqpgq be the quantum a�nealgebra. Irreducible �nite-dimensional representations of Uqpgq are indexed by monomialsin the in�nite set of variables tYi,auiPI,aPC� , where I � t1, . . . , nu are the vertices of theDynkin diagram of g. For such a monomial m, the corresponding simple Uqpgq-module isdenoted by Lpmq. If the monomial is just one term m � Yi,a, the corresponding simplemodule LpYi,aq is called a fundamental module. Chari-Pressley classi�cation result alsoimplies that every simple module can be obtained as a subquotient of a tensor product ofsuch fundamental modules.

This classi�cation is a major result in the way of obtaining information of the �nite-dimensional representations of quantum a�ne algebras. However, it gives limited infor-mation of the structure of the modules in themselves. For that purpose, Frenkel andReshetikhin have developed a theory of q-characters, giving the decomposition of the mod-ules into generalized eigenspaces for the action of a large commutative subalgebra of Uqpgq.Frenkel-Mukhin established an algorithm to compute those q-characters [FM01]. Thisalgorithm is guaranteed to work on fundamental modules, but not on all irreducible Uqpgq-modules [NN11].

Then, when g is of simply-laced type, Nakajima [Nak01b] used the input from geometry,and more precisely perverse sheaves on quiver varieties, to construct t-deformations ofthese q-characters, called pq, tq-characters, as elements of a quantum Grothendieck ring. Heintroduced a second base for the Grothendieck ring of the category of �nite-dimensionalrepresentations of Uqpgq, also indexed by the monomials in the variables tYi,auiPI,aPC� ,formed by the standard modules. Geometrically, these standard modules correspond toconstant sheaves, but algebraically, for each monomial m, the standard module Mpmq canbe seen as the tensor product of the fundamental modules corresponding to each of thefactors in m, in a particular order (see also [VV02b]).

He �rst used a t-deformed version of Frenkel-Mukhin's algorithm to compute pq, tq-characters for the fundamental modules, then extended the pq, tq-characters to the standardmodules, denoted rMpmqst. Next, he de�ned pq, tq-characters for the simple modules assome unique family of elements rLpmqst of the quantum Grothendieck ring satisfying someinvariant property, as well as having a decomposition of the form

rLpmqst � rMpmqst �¸

m1 m

Qm1,mptqrMpm1qst, (3.1.0.1)

where is a partial order on the set of Laurent monomials in the variables tYi,auiPI,aPC� ,de�ned by Nakajima, and Qm1,mptq P Zrt�1s is a Laurent polynomial.

3.1. Introduction 87

Nakajima then showed that these pq, tq-characters were indeed t-deformations of theq-characters, in the sense that the evaluation of the pq, tq-characters at t � 1 recoversthe q-characters. Finally, inverting the unitriangular decomposition (3.1.0.1), one getsan algorithm, of the Kazhdan-Lusztig type, to compute the pq, tq-characters, and so theq-characters of all simple �nite-dimensional Uqpgq-modules.

This algorithm is theoretically computable, but as noted in [Nak10], trying to computeit in reality can easily exceed the size of computer memory available. The �rst step of thealgorithm is to compute the pq, tq-characters of the fundamental representations, and forexample, for g of type E8, the 5th fundamental representation requires 120Go of memoryto compute.

In [HL10] Hernandez and Leclerc introduced a new point of view on representations ofquantum a�ne algebras, using the theory of cluster algebras that was developed by Fominand Zelevinsky in the early 2000's [FZ02], [FZ03a], [BFZ05],[FZ07]. In [HL16a] they es-tablished a new algorithm to compute q-characters of a particular class of irreduciblemodules, called Kirillov-Reshetikhin modules, which include the fundamental modules, us-ing the cluster algebra structure of the Grothendieck ring of a subcategory of the categoryof �nite-dimensional Uqpgq-modules. The picture is completed when put into the broadercontext of the category O� of representations of Uqpgq, introduced by Hernandez-Jimboin [HJ12]. In [HL16b], Hernandez and Leclerc showed that the Grothendieck ring of thiscategory, which contains the �nite-dimensional representations, is isomorphic to a clusteralgebra built on an in�nite quiver, while explicitly giving the identi�cation.

In Chapter 2, the author de�ned the quantum Grothendieck ring for this category O�of representations as a quantum cluster algebra, as de�ned by Berenstein and Zelevinsky[BZ05]. However, the question of whether this quantum Grothendieck ring contained thequantum Grothendieck of the category of �nite-dimensional representation, as used byNakajima, was only proven in type A, and remained conjectural for other types.

In this Chapter we propose to show that, when g is of simply-laced type, the quantumGrothendieck ring of a certain monoidal subcategory of the category of �nite-dimensionalUqpgq-modules has a quantum cluster algebra structure (Proposition 3.6.7). The proofrelies heavily on a family of relations satis�ed by the pq, tq-characters of the Kirillov-Reshetikhin modules called quantum T -systems proved in [Nak03a]. These relations aret-deformations of the T -systems relations, �rst stated in [KNS94]. These relations have notbeen generalized to non-simply-laced types, except for type Bn in [HO19]. This is the mainreason why the results of this chapter are limited to ADE types. This quantum clusteralgebra approach gives a new algorithm to compute the pq, tq-characters of the Kirillov-Reshetikhin modules, and in particular of the fundamental modules (see Proposition 3.5.2).This algorithm seems more e�cient, at least in terms of number of steps, than the Frenkel-Mukhin algorithm (see Remark 3.6.10).

For certain subcategories of the category of �nite-dimensional Uqpgq-modules generatedby a �nite number of fundamental modules, Qin obtained in [Qin17] in a di�erent contextresults similar to some results whose direct proofs are given here (see Remark 3.4.7). Inthis present work, we give explicit sequences of mutations to obtain pq, tq-characters offundamental modules.

Next, we use this new result to prove a conjecture that was stated by the author inChapter 2. This previous work dealt with a category O� of representations of the Borelsubalgebra of the quantum a�ne algebra, which contains the �nite-dimensional Uqpgq-modules. The quantum Grothendieck ring of this category was de�ned as a quantumcluster algebra, and it was conjectured that this ring contained the quantum Grothendieck


ring of the category of �nite-dimensional representations. Here, we show that the quantumcluster algebra considered in Chapter 2 can be seen as a twisted version (in the senseof [GL14]) of the quantum cluster algebra occurring in the �nite-dimensional case (seeProposition 3.6.3). As an application, the pq, tq-characters of the fundamental modules areobtained as quantum cluster variables in the quantum Grothendieck ring of the categoryO� (Proposition 3.6.7), and the inclusion of quantum Grothendieck rings conjectured inChapter 2 follows naturally (Theorem 3.6.5 and Corollary 3.6.9).

Note that these results extend the algorithm to compute pq, tq-characters of some simplemodules in the category O�. However, for this category of representations, the question ofde�ning analogs of standard modules remains open. The author tackled this question inChapter 1, and gave a complete answer when the underlying simple Lie algebra is g � sl2.This work is also a partial answer to the �rst point of Nakajima's "to do" list from [Nak11].

The author would also like to note that in type A, parallel results to the ones pre-sented here were proven in [Tur18], via a di�erent approach. In this work, pq, tq-charactersof Kirillov-Reshetikhin modules are also obtained as quantum cluster variables in somequantum cluster algebra, the method uses a generalization of the tableaux-sum notationsintroduced by Nakajima in [Nak03b].

Finally, we use this algorithm to explicitly compute, when g is of type D4, the pq, tq-character of the fundamental representation at the trivalent node.

This chapter is organized as follows. In Section 3.3, we recall results regarding thet-deformation of Grothendieck rings, such as pq, tq-characters and quantum T -systems. InSection 3.4 we prove the existence of a quantum cluster algebra At with t-commutationsrelations coherent with the framework of pq, tq-characters. Then, in Section 3.5 we provethat this quantum cluster algebra is isomorphic to the quantum Grothendieck ring of acertain monoidal subcategory of the category of �nite-dimensional Uqpgq-modules; in thisprocess, we established an algorithm to compute pq, tq-characters of Kirillov-Reshetikhinmodules. Section 3.6 is devoted to the category O�, and to the proof of the inclusionConjecture of Chapter 2. Finally, the explicit computation mentioned just above is donein Section 3.7.

3.2 Finite-dimensional representations of quantum a�ne al-

gebras

We use the notations from Appendix A.In this section, we recall the notations and di�erent results regarding quantum a�ne

algebras and �nite-dimensional representations of quantum a�ne algebras.

3.2.1 Quantum a�ne algebra

Let g be the untwisted a�ne Lie algebra corresponding to g.Fix an nonzero complex number q, which is not a root of unity, and h P C such that

q � eh. Then for all r P Q, qr :� erh. Since q is not a root of unity, for r, s P Q, we haveqr � qs if and only if r � s.

Let Uqpgq be the quantum enveloping algebra of the Lie algebra g (see [CP95a]), it is aC-Hopf algebra.

3.2. Finite-dimensional representations of quantum affine algebras 89

3.2.2 Finite-dimensional representations

Let C be the category of all (type 1) �nite-dimensional Uqpgq-modules. As Uqpgq is aHopf algebra, C is a tensor category. The simple modules in C have been classi�ed byChari-Pressley ([CP95a]), in terms of Drinfeld polynomials.

The simple �nite-dimensional Uqpgq-modules are indexed by the monomials in the in�-nite set of variables pYi,aqiPI,aPC� , called dominant monomials ([FR99]). For such a mono-mial m, let Lpmq denote the corresponding simple Uqpgq-module.

We de�ne the following sets of dominant monomials:

M �

#¹�nite

Yni,ri,qr | pi, rq P I , ni,r P Z¥0, ni,r � 0 except for a �nite number of pi, rq

+,

M� �

#¹�nite

Yni,ri,qr | pi, rq P I�, ni,r P Z¥0, ni,r � 0 except for a �nite number of pi, rq

+.

De�nition 3.2.1. Let CZ be the full subcategory of C of objects whose composition factorsare of the form Lpmq, with m PM.

Let C�Z be the full subcategory of C of objects whose composition factors are of the

form Lpmq, with m PM�.

The category CZ is a tensor category and C�Z is a monoidal category ([HL10, 5.2.4] and

[HL16a, Proposition 3.10]).

Remark 3.2.2. Every simple object in C can be written as a tensor product of simpleobjects which are essentially in CZ (see [HL10, Section 3.7]). Thus, the description of thesimple objects of C reduces to the description of the simple objects of CZ.

Let us introduce some particular irreducible �nite-dimensional representations.

De�nition 3.2.3. For all pi, rq P I, Vi,r :� LpYi,rq is called a fundamental module.For all pi, rq P I , k P Z¡0, let

mpiqk,r :� Yi,rYi,r�2 � � �Yi,r�2k�2, (3.2.2.1)

the corresponding irreducible module Lpmpiqk,rq is called a Kirillov-Reshetikhin module, or

KR-module, and denote byW

piqk,r :� Lpm

piqk,rq. (3.2.2.2)

Note that fundamental module are particular KR-modules, for k � 1,mpiq1,r � Yi,r.

3.2.3 q-characters and truncated q-characters

Frenkel and Reshetikhin introduced in [FR99] an injective ring morphism, called theq-character morphism, on the Grothendieck ring K0pC q of the category C :

χq : K0pC q Ñ Z�Y �1i,a | i P I, a P C�

�. (3.2.3.1)

Moreover, the q-character χqpV q of a Uqpgq-module V gives information about the de-composition into Jordan subspaces for the action of a large commutative subalgebra ofUqpgq.


Here, as we restrict ourselves to the study of the category CZ, the q-character will onlyinvolve variables Y �1

i,qr , for pi, rq P I. Hence, for simplicity of notation we denote them by:Yi,r :� Yi,qr . The q-character we are interested in is the injective ring morphism:

χq : K0pCZq Ñ Y :� Z�Y �1i,r | pi, rq P I

�. (3.2.3.2)

We use the usual notation [FR99],

Ai,r :� Yi,r�1Yi,r�1

�¹j�i

Yj,r,

��1

. (3.2.3.3)

for all pi, rq P J (see (A.4.3) in the Appendix). Note that Ai,r is a Laurent monomial inthe variables Yj,s, with pj, sq P I. The monomials Ai,r are analogs of the simple roots.

Proposition 3.2.4. [FM01, Theorem 4.1] For m a dominant monomial in M, the q-character of the �nite-dimensional irreducible representation Lpmq is of the form

χqpLpmqq � m

�1�

¸p

Mp

�, (3.2.3.4)

where Mp is a monomial in the variables A�1i,r , with pi, rq P J .

Let us recall Nakajima's partial order on monomials. For m and m1 Laurent monomialsin Y,

m ¤ m1 ðñ m1m�1 is a product of Ai,r, with pi, rq P I . (3.2.3.5)

Remark 3.2.5. Note that Proposition 3.2.4 can be translated as follows: for all dominantmonomials m, the monomials occurring in the q-character of the �nite-dimensional irre-ducible representation Lpmq are lower than m, for Nakajima's partial order.

We also recall the truncated q-characters from [HL10]. For m a monomial in M�,the q-character χqpLpmqq may contain Laurent monomials in which variables Yi,r, withpi, rq P IzI� occur. Let χ�q pLpmqq be the Laurent polynomial obtained from χqpLpmqq byremoving any such Laurent monomial. By de�nition

χ�q pLpmqq P Z�Y �1i,r | pi, rq P I�

�. (3.2.3.6)

Example 3.2.6. For g of type A2, one has

χ�q pLpY1,0qq � Y1,0,

χ�q pLpY1,�2qq � Y1,�2 � Y �11,0 Y2,�1,

χ�q pLpY1,�4qq � Y1,�4 � Y �11,�2Y2,�3 � Y �1

2,�1 � χqpLpY1,�4qq.

Proposition 3.2.7. [HL16a, Proposition 3.10] The assignment rLpmqs ÞÑ χ�q pLpmqq ex-tends to an injective ring homomorphism

K0pC�Z q Ñ Z

�Y �1i,r | pi, rq P I�

�. (3.2.3.7)

As such, all simple modules in C�Z are identi�ed with their isoclasses through the

truncated q-character morphism.


3.2.4 Cluster algebra structure

One of the main ingredient we want to use in this work is the cluster algebra structureof the Grothendieck ring of the category C�

Z . See Appendix B for notations and resultsfor cluster algebras and quantum cluster algebras.

Consider the cluster algebra A :� Apuuu,G�q, with initial seed puuu,G�q, where

• uuu are initial cluster variables indexed by I�, uuu �!ui,r | pi, rq P I

�),

• G� is the semi-in�nite quiver with vertex set I� de�ned in the previous section.Consider the identi�cation, for all pi, rq P I�,

ui,r �¹k¥0

r�2k¤0

Yi,r�2k. (3.2.4.1)

This identi�cation makes sense, as these monomials are algebraically independent.From the Laurent phenomenon, we know that all cluster variables of A are Laurent

polynomials in the variables ui,r. Thus, via the identi�cation (3.2.4.1), A is seen as a

subring of Z�Y �1i,r | pi, rq P I�

�.

Theorem 3.2.8. [HL16a, Theorem 5.1] The injective ring homomorphism χ�q is an iso-morphism between the Grothendieck ring of the category C�

Z and the cluster algebra A,after identi�cation (3.2.4.1):

χ�q : K0pC�Z q

�ÝÑ A. (3.2.4.2)

Moreover, truncated q-characters of Kirillov-Reshetikhin modules can be obtained ascluster variables, via the identi�cation of initial seed (3.2.4.1), it is the main result of[HL16a].

3.3 Quantum Grothendieck rings

We will recall in this section the de�nition of the quantum Grothendieck ring of thecategory CZ, introduce that of the category C�

Z , and study those rings.Let t be an indeterminate. The quantum Grothendieck rings of the categories CZ and

C�Z are non-commutative t-deformations of the Grothendieck rings.

3.3.1 Quantum torus

Let Yt be the Zrt�1s-algebra generated by the variables Y �1i,r , for pi, rq P I, and the

t-commutations relations:

Yi,r � Yj,s � tNi,jpr�sqYj,s � Yi,r,

where Ni,j : ZÑ Z is the antisymmetrical map:

Ni,jpmq � Ci,jpm� 1q �Ci,jpm� 1q, @m P Z,

using the notations from Section A.2 of the Appendix.

Remark 3.3.1. Here we work with the quantum torus of [Her04] and [HL15], which isslightly di�erent from the original quantum torus used to de�ne the quantum Grothendieckring in [Nak04] and [VV03].


Example 3.3.2. If we use Example A.2.3 of the Appendix, for g � sl2, in this case, I �p1, 2Zq, for r P Z, one has

Y1,2r � Y1,2s � t2p�1qs�rY1,2s � Y1,2r, @s ¡ r ¡ 0. (3.3.1.1)

The Zptq-algebra Yt is viewed as a quantum torus of in�nite rank.We extend this quantum torus by adjoining a �xed square root t1{2 of t:

Yt :� Zrt1{2s bZrts Yt. (3.3.1.2)

Let Y�t be the quantum torus de�ned exactly the same way, except by only taking asgenerators the Y �1

i,r , for pi, rq P I�.

Let us denote byπ : Yt Ñ Y�t , (3.3.1.3)

the projection of Yt onto Y�t ,

πpYi,rq � 0 if pi, rq P IzI�. (3.3.1.4)

Remark 3.3.3. Even if Y�t is an in�nite rank quantum torus, it can be seen as a limit of�nite rank quantum tori. As �nite rank quantum tori are of polynomial growth, they areOre domains (see [IM94]). Moreover, the Ore condition being local (any pair of elementsof Y�t belongs to some su�ciently larger �nite rank quantum torus), Y�t is an Ore domain.Hence we can consider its skew �eld of fractions Ft.

3.3.2 Commutative monomials

For a family of integers with �nitely many non-zero components pui,rqpi,rqPI , de�ne the

commutative monomial±

pi,rqPI Yui,ri,r as¹

pi,rqPI

Yui,ri,r :� t

12

°pi,rq pj,sq ui,ruj,sNi,jpr,sqÝÑ� pi,rqPIY

ui,ri,r , (3.3.2.1)

where on the right-hand side an order on I is chosen so as to give meaning to the sum,and the product � is ordered by it (notice that the result does not depend on the orderchosen).

The commutative monomials form a basis of the Zrt1{2s-vector space Yt.The non-commutative product of two commutative monomials m1 and m2 in Yt is

given by:m1 �m2 � tDpm1,m2qm2 �m1 � t

12Dpm1,m2qm1m2, (3.3.2.2)

where m1m2 denotes the commutative product of the monomials, and

Dpm1,m2q �¸

pi,rq,pj,sqPI

ui,rpm1quj,spm2qNi,jpr, sq. (3.3.2.3)


3.3.3 Quantum Grothendieck ring KtpCZq

We de�ne the quantum Grothendieck ring KtpCZq of the category CZ as in [HL15,Section 5.4] (see Remark 3.3.1 for original references).

For all pi, rq P J , let Ai,r denote the commutative monomial in Yt de�ned as in (3.2.3.3):

Ai,r :� Yi,r�1Yi,r�1

�¹j�i

Yj,r

��1

.

For all i P I, de�ne Ki,t the subring of Yt generated by the

Yi,r

�1�A�1

i,r�1

, Y �1

j,s

�pi, rq, pj, sq P I , j � i

. (3.3.3.1)

In [Her03], the Ki,t are de�ned as kernels of t-deformed screening operators, motivatedby the results in [FR99]. Let us detail this, as it will be important in the proof of the mainresult. For all i P I, de�ne the free Yt-modules

Y lt,i :�¸

rPZ|pi,rqPI

Yt � Si,r.

Then let Yt,i be the quotient of Y lt,i by the left-Yt-module generated by the elements

Qi,r :� A�1i,r�1Si,r�2 � t2Si,r, @pi, rq P I .

Lemma 3.3.4. For all i P I, the module Yt,i is free.

Proof. The elements Qi,r are linearly independent and for all r0 such that pi, r0q P I �xed,the set tQi,r, Si,r0 | pi, rq P Iu forms a basis of Y lt,i.

Hence Yt,i is a quotient of a free module by a submodule generated by elements of abasis, thus it is free.

From [Her03], for all i P I, there exists a Zrt�1{2s-linear map

Si,t : Yt Ñ Yi,t, (3.3.3.2)

which is a derivation and such that

Ki,t � kerpSi,tq. (3.3.3.3)

Finally, letKtpCZq :�

£iPI

Ki,t. (3.3.3.4)

From [Nak01b] and [Her04] we know that for all dominant monomials m PM, thereexists a unique element Ftpmq P KtpCZq such that m occurs in Ftpmq with multiplicity1 and no other dominant monomial occurs in Ftpmq. Thus, all elements of KtpCZq arecharacterized by the coe�cients of their dominant monomials.

The Ftpmq linearly generate KtpCZq.

Remark 3.3.5. For all pi, rq P I�,

FtpYi,rq � rLpYi,rqst. (3.3.3.5)

The rLpYi,rqst generate KtpC�Z q algebraically.


3.3.4 pq, tq-characters

For a dominant monomial m PM, write it as a commutative monomial in Yt:

m �¹

pi,rqPI

Yni,rpmqi,r P Yt. (3.3.4.1)

De�ne

rMpmqst :� tαpmqÐÝ� rPZF

�¹iPI

Yni,rpmqi,r

�P KtpCZq, (3.3.4.2)

where αpmq P 12Z is �xed such that m occurs with multiplicity one in the expansion of

rMpmqst on the basis of the commutative monomials of Yt, and the product ÐÝ� is takenwith decreasing r P Z.

In particular, from (3.3.3.5), for all pi, rq P I,

rLpYi,rqst � rMpYi,rqst. (3.3.4.3)

One has, for all m PM,

rMpmqstt�1ÝÝÑ χqpMpmqq. (3.3.4.4)

This result is a direct consequence of the de�nition of rMpmqst, as it is satis�ed for thefundamental modules rLpYi,rqst � FtpYi,rq. Thus rMpmqst is called the pq, tq-character ofthe standard module Mpmq.

As in [Nak04], we consider the Z-algebra anti-automorphism of Yt de�ned by:

t1{2 � t�1{2, Yi,r � Yi,r,�pi, rq P I

. (3.3.4.5)

This map is called the bar-involution.

Theorem 3.3.6. [Nak04] There exists a unique family trLpmqstumPM of elements ofKtpCZq such that, for all m PM,• rLpmqst � rLpmqst,• rLpmqst P rMpmqst �

°m1 m t

�1Zrt�1srMpm1qst, where m1 m for Nakajima's partialorder (3.2.3.5).

The following Theorem extends (3.3.4.4), but more importantly gives an algorithm,similar to the Kazhdan-Lusztig algorithm, to compute the pq, tq-characters (and thus theq-characters) of the simple modules.

Theorem 3.3.7. [Nak04, Corollary 3.6] The evaluation at t � 1 of the pq, tq-charactersrecovers the q-characters. For all m PM,

rLpmqstt�1ÝÝÑ χqpLpmqq P Y.

Moreover, the coe�cients of the expansion of rLpmqst as a linear combination of Laurentmonomials in the variables pYi,rqpi,rqPI belong to Nrt�1s.

Note that the positivity result of this Theorem has only been proven for ADE types asyet.


3.3.5 Truncated pq, tq-characters and quantum Grothendieck ring KtpC�Z q

As in Section 3.2.3, one can de�ne truncated versions of the pq, tq-characters.For all dominant monomialsm inM�, let rLpmqs�t be the Laurent polynomial obtained

from rLpmqst by removing any term in which a variable Yi,r, with pi, rq P IzI� occurs:

rLpmqs�t � π prLpmqstq P Y�t , (3.3.5.1)

where π is the projection de�ned in (3.3.1.3).De�ne KtpC

�Z q as the Zrt1{2s-submodule of Y�t generated by the truncated pq, tq-

characters rLpmqs�t of the simple �nite-dimensional modules Lpmq in the category C�Z .

Lemma 3.3.8. The quantum Grothendieck ring KtpC�Z q is actually a subalgebra of Y�t .

Moreover, it is algebraically generated by the truncated pq, tq-characters of the fundamentalmodules:

KtpC�Z q �

ArLpYi,rqs

�t | pi, rq P I

�E. (3.3.5.2)

Proof. For every dominant monomials m1,m2 PM, one can write:

rLpm1qst � rLpm2qst �¸mPM

cmm1,m2pt1{2qrLpmqst. (3.3.5.3)

Hence the image of (3.3.5.3) by the projection π of (3.3.1.3) is:

rLpm1qs�t � rLpm2qs

�t �

¸mPM

cmm1,m2pt1{2qrLpmqs�t .

Thus KtpC�Z q is stable by products.

By de�nition the truncated pq, tq-characters of the fundamental modules LpYi,rq, forpi, rq P I� belong to KtpC

�Z q.

Reciprocally, the pq, tq-characters of the fundamental modules LpYi,rq, for all pi, rq P I,algebraically generate the quantum Grothendieck ring KtpCZq (see remark 3.3.5). Hencethe truncated pq, tq-characters rLpYi,rqs

�t , for all pi, rq P I, algebraically generate KtpC

�Z q.

From Proposition 3.2.4 and Theorem 3.3.6, for all dominant monomials m P M, thepq, tq-character of the simple representation Lpmq is of the form

rLpmqst � m

�1�

¸p

Mp

�,

where Mp is a monomial in the variables pA�1i,r qi,rqPJ , with coe�cients in Zrt�1s. Thus,

π prLpYi,rqstq � 0, if pi, rq P IzI�.

Hence, KtpC�Z q is algebraically generated by the rLpYi,rqs

�t , with pi, rq P I

�.

KtpC�Z q is a t-deformed version of the Grothendieck ring of the category C�

Z , in the

sense that the evaluation rLpmqs�tt�1ÝÝÑ χ�q pLpmqq extends to a ring homomorphism

KtpC�Z q

t�1ÝÝÑ K0pC

�Z q, (3.3.5.4)

where K0pC�Z q is identi�ed with its image under the truncated q-character (3.2.3.7), which

is an injective map.


3.3.6 Quantum T -systems

The pq, tq-characters of the Kirillov-Reshetikhin modules satisfy some algebraic rela-tions called quantum T -systems. Those are t-deformed versions of the T -system relations,which are satis�ed by the q-characters of the KR-modules [KNS94, Nak04, Her06].

Proposition 3.3.9. [Nak03a][HL15, Proposition 5.6] For all pi, rq P I and k P Z¡0, thefollowing relation holds in KtpCZq:

rWpiqk,rst � rW

piqk,r�2st � tαpi,kqrW

piqk�1,r�2st � rW

piqk�1,rst � tγpi,kq �

j�irW

pjqk,r�1st, (3.3.6.1)

where

αpi, kq � �1�1

2

�Ciip2k � 1q � Ciip2k � 1q

, γpi, kq � αpi, kq � 1. (3.3.6.2)

Remark 3.3.10. First of all, one notices that the dominant monomials of W piqk�1,r�2 and

Wpiqk�1,r commute:

mpiqk�1,r�2 �m

piqk�1,r � m

piqk�1,r �m

piqk�1,r�2. (3.3.6.3)

Moreover, the tensor product of the KR-modules W piqk�1,r�2bW

piqk�1,r is irreducible (this re-

sult is proved in [Cha02] and also by explicit computation of its pq, tq-character in [Nak03a]).Thus their respective pq, tq-characters t-commute (see [HL15, Corollary 5.5]). As their dom-inant monomials commute, these pq, tq-characters in fact commute and their product canbe written as a commutative product, as in Section 3.3.2.

By the same arguments, for j � i, the pq, tq-characters rW pjqk,r�1st commute so the order

of the factors in �j�i in (3.3.6.1) does not matter.

By taking the image of (3.3.6.1) through the projection π of (3.3.1.3), one obtains thefollowing relation in KtpC

�Z q. For all pi, rq P I and k P Z¡0,

rWpiqk,rs

�t � rW

piqk,r�2s

�t � tαpi,kqrW

piqk�1,r�2s

�t � rW

piqk�1,rs

�t � tγpi,kq

¹j�i

rWpjqk,r�1s

�t , (3.3.6.4)

where αpi, kq and γpi, kq are de�ned in (3.3.6.2).Note that in (3.3.6.4), the products appearing on the left-hand side are commutative

products, which are well-de�ned from Remark 3.3.10. Hence the change of notations since(3.3.6.1).

3.4 Quantum cluster algebra structure

We de�ne in this section the quantum cluster algebra structure built within the quan-tum torus Y�t .

3.4.1 A compatible pair

For all pi, rq P I�, the variable ui,r, written as in (3.2.4.1), can be seen as commutativemonomial in Y�t . De�ne

Ui,r :�¹k¥0

r�2k¤0

Yi,r�2k P Y�t .

3.4. Quantum cluster algebra structure 97

They satisfy the following t-commutation relations. For all ppi, rq, pj, sqq P pI�q2,

Ui,r � Uj,s � tLppi,rq,pj,sqqUj,s � Ui,r, (3.4.1.1)

whereL ppi, rq, pj, sqq �

¸k¥0

r�2k¤0

¸l¥0

s�2l¤0

Nijps� 2l � r � 2kq. (3.4.1.2)

Let B� be the I� � I�-matrix encoding the quiver G�, for all ppi, rq, pj, sqq P pI�q2:

B� ppi, rq, pj, sqq � |t arrows pi, rq Ñ pj, sq in G�u| � |t arrows pj, sq Ñ pi, rq in G�u|.(3.4.1.3)

Let L be the I� � I� skew-symmetric matrix

L :� pL ppi, rq, pj, sqqqppi,rq,pj,sqqPpI�q2 . (3.4.1.4)

The pair of I�� I�-matrices pL,B�q forms a compatible pair, in the sense of quantumcluster algebras.

More precisely, we prove the following.

Proposition 3.4.1. For all ppi, rq, pj, sqq P pI�q2,

�BT�L

�ppi, rq, pj, sqq �

"�2 if pi, rq � pj, sq0 otherwise.

(3.4.1.5)

Remark 3.4.2. In [BZ05], by de�nition a pair of J � J-matrices pΛ, Bq forms a compatiblepair if TBL is a diagonal matrix with positive integer coe�cients. But as explained inChapter 2, quantum cluster algebras can be built exactly the same way given as data apair pΛ, Bq such that TBL is a diagonal matrix with integer coe�cients with constantsigns.

Proof. Fix ppi, rq, pj, sqq P pI�q2, there are di�erent cases to consider.• If r ¤ �2, one has:�

BT�L

�ppi, rq, pj, sqq � L ppi, r � 2q, pj, sqq � L ppi, r � 2q, pj, sqq

�¸k�i

pL ppk, r � 1q, pj, sqq � L ppk, r � 1q, pj, sqqq .

One has

L ppi, r � 2q, pj, sqq � L ppi, r � 2q, pj, sqq � �Cijps� r � 1q �Cijps� r � 1q

�Cijp�r � 3� ξjq �Cijp�r � 1� ξjq,

where ξ : I Ñ t0, 1u is the height function on the Dynkin diagram of g �xed inSection A.3 of the Appendix.On the other hand, for all k � i, one has

L ppk, r � 1q, pj, sqq � L ppk, r � 1q, pj, sqq � Ckjps� rq �Ckjp�r � 2� ξjq.

Thus, with the reformulation (A.2.6) of Lemma A.2.4 of the Appendix,

�BT�L

�ppi, rq, pj, sqq �

"�2δi,j if s � r,0 otherwise.

(3.4.1.6)


• If r � �1, one has:�BT�L

�ppi,�1q, pj, sqq � L ppi,�3q, pj, sqq

�¸k�i

pL ppk, 0q, pj, sqq � L ppk,�2q, pj, sqqq .

However,

L ppi,�3q, pj, sqq �¸l¥0

s�2l¤0

pNijps� 2l � 3q �Nijps� 2l � 1qq ,

� Cijp4� ξjq �Cijp2� ξjq �Cijpsq �Cijps� 2q.

And, for all k � i,

L ppk, 0q, pj, sqq � L ppk,�2q, pj, sqq � Cijps� 1q �Cijp3� ξjq.

Thus, with relation (A.2.6),

�BT�L

�ppi,�1q, pj, sqq �

"�2δi,j if s � �1,0 otherwise.

(3.4.1.7)

• If r � 0, one has�BT�L

�ppi, 0q, pj, sqq � L ppi,�2q, pj, sqq �

¸k�i

L ppk,�1q, pj, sqq .

However,

L ppi,�2q, pj, sqq � Cijp3� ξjq �Cijp1� ξjq �Cijps� 1q �Cijps� 1q.

And, for all k � i,

L ppk,�4q, pj, sqq � �Cijpsq �Cijp2� ξjq.

Thus, with relation (A.2.6),

�BT�L

�ppi, 0q, pj, sqq �

"�2δi,j if s � 0,0 otherwise.

(3.4.1.8)

The combination of the results (3.4.1.6),(3.4.1.7) and (3.4.1.8) gives the general expression(3.4.1.5).

3.4.2 The quantum cluster algebra At

De�nition 3.4.3. Let T be the based quantum torus with generators t ui,r | pi, rq P I�usatisfying the quasi-commutation relations (3.4.1.1):

ui,r � uj,s � tLppi,rq,pj,sqquj,s � ui,r.

Let F be the skew-�eld of fractions of T .

As pL,B�q forms a compatible pair, it de�nes a quantum seed in F . Let S be themutation equivalence class of the quantum seed pL,B�q.

3.5. Quantum cluster algebras and quantum Grothendieck ring 99

De�nition 3.4.4. Let At be the quantum cluster algebra de�ned by the quantum seed S,as in [BZ05].

By de�nition, At is a Zrt�1{2s-subalgebra of F . However, by the quantum Laurentphenomenon, At is actually a Zrt�1{2s-subalgebra of the quantum torus T .

Lemma 3.4.5. The map

η :T ÝÑ Y�t

ui,r ÞÝÑ Ui,r,(3.4.2.1)

where the variables Ui,r are de�ned in (3.2.4.1), is an isomorphism of quantum tori.

Proof. First of all, this map is well-de�ned because the variables ui,r t-commute exactlyas the variables Ui,r, by de�nition of the matrix L (3.4.1.1).

Secondly, this map is invertible, with inverse:

η�1 :

Y�t ÝÑ T

Yi,r ÞÝÑ

"ui,ru

�1i,r�2 if r � 2 ¤ 0,

ui,r otherwise,.

With this lemma, we know that the quantum cluster algebra At belongs to the quantumtorus Y�t . The following result is the main result of this chapter, it extends Theorem 3.2.8to the quantum setting.

Theorem 3.4.6. The image of the quantum cluster algebra At by the injective ring mor-phism η is the quantum Grothendieck ring of the category C�

Z ,

ηæAt : At�ÝÑ KtpC

�Z q. (3.4.2.2)

Moreover, the truncated pq, tq-characters of the Kirillov-Reshetikhin modules which are inC�Z are obtained as quantum cluster variables.

The proof of this Theorem will be developed in the following section. It is mainly basedon Proposition 3.5.2.

Remark 3.4.7. In [Qin17, Theorem 8.4.3], for certain subcategories of C generated by a�nite number of fundamental modules, Qin proved that there existed an isomorphism be-tween the quantum Grothendieck ring of the category and a quantum cluster algebra, whichidenti�es classes of Kirillov-Reshetikhin modules to cluster variables. It is our understand-ing that this identi�cation coincides with the truncated pq, tq-characters in our work. Here,the isomorphism is given explicitly, and we obtain directly the truncated pq, tq-characters.

3.5 Quantum cluster algebras and quantum Grothendieck

ring

We prove in this section that the quantum Grothendieck ring of the category C�Z is

isomorphic to the quantum cluster algebra we have just de�ned.


3.5.1 A note on the bar-involution

The bar-involution , as de�ned in (3.3.4.5), is a Z-algebra anti-automorphism of thequantum torus Yt. The commutative monomials are invariant under this involution.

On the other hand, the quantum cluster algebra At is also equipped with a Z-linearbar-involution morphism on its quantum torus T (see [BZ05, Section 6]), which satis�es

t1{2 � t�1{2,

ui,r � ui,r.

As noted in Section 2.5.1, these de�nitions are compatible. In our case, they de�neexactly the same involution on Y�t ; the following diagram is commutative:

Y�t //

η�1

��

Y�t

T // T

η

OO. (3.5.1.1)

From [BZ05, Remark 6.4], all cluster variables are invariant under the bar-involution.Thus, the images of the quantum cluster variables in At are bar-invariant elements of Y�t .

We will use the terminology "commutative products", as in Section 3.3.2 for bar-invariant elements of the quantum torus T .

3.5.2 A sequence of vertices

In [HL16a] Hernandez and Leclerc exhibited a particular sequence of mutations in thecluster algebra Apuuu,G�q (see Section 3.2.4) in order to obtain the truncated q-charactersof all the KR-modules, up to a shift of spectral parameter.

The key idea we used was that at each step of this sequence, the exchange relation wasa T -system equation.

We will recall this sequence of mutations and show that, if applied to the quantumcluster algebra At, the quantum exchange relations at each step are in fact quantum T -systems relations, as in (3.3.6).

Recall the height function ξ : I Ñ t0, 1u �xed on the Dynkin diagram of g in Section A.3of the Appendix.

First, �x an order on the columns of G�:

i1, i2, . . . , in, (3.5.2.1)

such that if k ¤ l then ξik ¤ ξil (select �rst the vertices i such that ξi � 0 then the others).Then, the sequence S is de�ned by reading each column, from top to bottom, in this

order.

Example 3.5.1. We follow Examples A.4.1 and A.4.3 of the Appendix, g � sl4 and

I� � p1, 2Z¤0q Y p2, 2Z¤0 � 1q Y p3, 2Z¤0q.

We �x the following order on the columns: 1, 3, 3. Then the sequence S is

S � p1, 0q, p1,�2q, p1,�4q, . . . , p3, 0q, p3,�2q, p3,�4q, . . . , p2,�1q, p2,�3q, p2,�5q, . . . .(3.5.2.2)


3.5.3 Truncated pq, tq-characters as quantum cluster variables

As in [HL16a], let µS be the sequence of quantum cluster mutations in At indexed bythe sequence of vertices S .

For all m ¥ 1, let upmqi,r be the quantum cluster variable obtained at vertex pi, rq afterapplying m times the sequence of mutations µS to the quantum cluster algebra At withinitial seed t ui,r | pi, rq P I�u. By the quantum Laurent phenomenon, the upmqi,r belong tothe quantum torus T . Let

wpmqi,r :� ηpu

pmqi,r q P Y�t , (3.5.3.1)

where η : T Ñ Y�t is the isomorphism de�ned in Lemma 3.4.5.The following result is an extension of Theorem 3.1 from [HL16a] to the quantum

setting.

Proposition 3.5.2. For all pi, rq P I� and m ¥ 0,

wpmqi,r � rW

piqki,r,r�2ms

�t , (3.5.3.2)

where ki,r is de�ned in (A.4.4) in the Appendix.In particular, if 2m ¥ h, this truncated pq, tq-character is equal to its pq, tq-character

andwpmqi,r � rW

piqki,r,r�2mst. (3.5.3.3)

Remark 3.5.3. The sequence of vertices S is in�nite. However, in order to compute one�xed truncated pq, tq-character, one only has to compute a �nite number of mutations inthe in�nite sequence µS . In Section 2.5.2, the exact �nite sequence needed to computethe pq, tq-characters of the fundamental representations Vi,r is given explicitly, we will alsorecall it in Section 3.6.

Proof. We prove this Proposition by induction on m, the number of times the mutationsequence µS is applied on the initial quantum cluster variables t ui,r | pi, rq P I�u.

The base step is given noting, as in [HL16a], that the images by the isomorphismη of the initial quantum cluster variables are indeed truncated pq, tq-characters. For allpi, rq P I�,

wp0qi,r � ηpui,rq � Ui,r �

¹k¥0

r�2k¤0

Yi,r�2k P Y�t .

Thuswp0qi,r � rW

piqki,r,r

s�t . (3.5.3.4)

Let m ¥ 0 and pi, rq P I�. Supposed we have applied m times the mutation sequenceµS , and a pm� 1qth time on all vertices preceding pi, rq in the sequence S , and that allthose previous vertices satisfy (3.5.3.2).

We want to write the quantum exchange relation corresponding to the mutation atvertex pi, rq. From the proof of Theorem 3.1 in [HL16a], we know the shape of the quiverjust before this mutation (At and Apuuu,G�q are de�ned on the same initial quiver and themutation process on the quiver is the same for classical and quantum cluster algebras).

As explained in [HL16a, Section 3.2.3], for a general simply laced Lie algebra g, themutation process takes place at vertices pi, rq having two (or one if r � �ξi) in-goingarrows from pi, r� 2q and outgoing arrows to vertices pj, sq, with j � i. Thus the e�ect of


the mutation sequence µS on two �xed columns of the quiver is the same as the e�ect ofan iteration of the mutation sequence on the corresponding quiver of rank 2.

Let us recall the mutation process on the quiver when g is of type A2.

(1,0)

p1,�2q

p1,�4q

...

p2,�1q

p2,�3q

p2,�5q

...

p1, 0q

(1,-2)

p1,�4q

...

p2,�1q

p2,�3q

p2,�5q

...

p1, 0q

p1,�2q

(1,-4)

...

p2,�1q

p2,�3q

p2,�5q

...

p1, 0q

p1,�2q

p1,�4q

...

p2,�1q

p2,�3q

p2,�5q

...After an in�nite number of mutations (or a su�ciently large one), we start mutating

on the second column.p1, 0q

p1,�2q

p1,�4q

...

(2,-1)

p2,�3q

p2,�5q

...

p1, 0q

p1,�2q

p1,�4q

...

p2,�1q

(2,-3)

p2,�5q

...

p1, 0q

p1,�2q

p1,�4q

...

p2,�1q

p2,�3q

(2,-5)

...

p1, 0q

p1,�2q

p1,�4q

...

p2,�1q

p2,�3q

p2,�5q

...Thus, in general, the quantum exchange relation has the form:

upm�1qi,r � u

pm�1qi,r�2 u

pmqi,r�2

�upmqi,r

�1�

¹j�i0

upm�ξiqj,r�εi

�upmqi,r

�1, (3.5.3.5)

where upmqi,r�2 � 1 if r� 2 ¥ 0 and εi is de�ned in the Appendix in (A.3.2), and both termsare commutative products. This relation can also be written:

upm�1qi,r � u

pmqi,r � tαu

pm�1qi,r�2 u

pmqi,r�2 � tβ

¹j�i0

upm�ξiqj,r�εi

, (3.5.3.6)

where α, β P 12Z.


If we apply η to (3.5.3.6), and use the induction hypothesis, we get the following relationin Y�t :

ηpupm�1qi,r q � rW

piqki,r,r�2ms

�t � tαrW

piqki,r�2,r�2ms

�t rW

piqki,r�2,r�2�2ms

�t

� tβ¹j�i

rWpjqkj,r�εi ,r�1�2ms

�t . (3.5.3.7)

Whereas, the corresponding (truncated) quantum T -system relation (3.3.6.4) is

rWpiqki,r,r�2m�2s

�t � rW

piqki,r,r�2ms

�t � tα

1rW

piqki,r�1,r�2ms

�t rW

piqki,r�1,r�2�2ms

�t

� tβ1¹j�i

rWpjqki,r,r�1�2ms

�t , (3.5.3.8)

where α1, β1 P 12Z are given in (3.3.6.2). Note that one has indeed

ki,r�2 � ki,r � 1, ki,r�2 � ki,r � 1, and kj,r�εi � ki,r, for j � i. (3.5.3.9)

Let k � ki,r and r1 � r � 2m. Let us precise how to obtain the coe�cients α and β.From (3.5.3.5) and (3.5.3.6), α and β are such that the terms

tαrWpiqk�1,r1s

�t rW

piqk�1,r1�2s

�t �

�rW

piqk,r1s

�t

�1, (3.5.3.10)

and

tβ¹j�i

rWpjqk,r1�1s

�t �

�rW

piqk,r1s

�t

�1, (3.5.3.11)

are bar-invariant. Thus, if one takes only the dominant monomials of (3.5.3.10) and(3.5.3.11), they are bar-invariant:

tαmpiqk�1,r1m

piqk�1,r1�2 �

�mpiqk,r1

�1� t�α

�mpiqk,r1

�1�m

piqk�1,r1m

piqk�1,r1�2,

tβ¹j�i

mpjqk,r1�1 �

�mpiqk,r1

�1� t�β

�mpiqk,r1

�1�¹j�i

mpjqk,r1�1.

This enables us to compute α and β.

α �1

2

k�1

l�0

�k�2

p�0

Ni,ip2l � 2pq �k

p�0

Ni,ip2l � 2p� 2q

�

�1

2

k�1

l�0

pCiip2l � 1q �Ciip2l � 2k � 3q �Ciip2l � 3q �Ciip2l � 2k � 1qq

�1

2pCiip2k � 1q �Ciip2k � 1qq �Ciip1q.

Thus α � αpi, kq � �1� 12


. And

β �1

2

¸j�i

k�1

l�0

k�1

p�0

Ni,jp2l � 2p� 1q �1

2

¸j�i

k�1

l�0

pCijp2l � 2q �Cijp2l � 2k � 2qq

�1

2

¸j�i

��Cijp2kq �Cijp0qloomoon�0

� �1

2pCiip2k � 1q �Ciip2k � 1qq , using (A.2.6).


Hence β � γpi, kq � 12


.

Thus α � α1 and β � β1, and


piqk,r1s

�t � rW

piqk,r1�2s

�t � rW

piqk,r1s

�t . (3.5.3.12)

However, rW piqk,r1s

�t is invertible in the skew-�eld of fractions Ft of the quantum torus

Y�t (see Remark 3.3.3). Thus


piqki,r,r�2�2ms

�t . (3.5.3.13)

This concludes the induction.Finally, from [FM01, Corollary 6.14], we know that for all pi, rq P I�, the monomials

m occurring in the q-character χqpWpiqk,rq of the KR-module W piq

k,r are products of Y �1j,r�s,

with 0 ¤ s ¤ 2k � h. Moreover, from Theorem 3.3.6, if one writes the pq, tq-character

rWpiqk,rst of this KR-module as a linear combination of Laurent monomials in the variables

Yi,s, with coe�cients in Zrt�1s, all monomials which occur in this expansion also occur in

its q-character. Thus, if r � 2k � h ¤ 0, the truncated pq, tq-character rW piqk,rs

�t is equal to

the pq, tq-character rW piqk,rst. In particular, for m ¥ h,

wpmqi,r � rW

piqki,r,r�2mst.

3.5.4 Proof of Theorem 3.4.6

We can now prove Theorem 3.4.6. This proof is a quantum analog of the proof of[HL16a, Theorem 5.1]. Naturally, there are technical di�culties brought forth by the non-commutative quantum tori structure. For example, in our situation, the quantum clusteralgebra At is isomorphic to the truncated quantum Grothendieck ring KtpC

�Z q only via

the isomorphism of quantum tori η (3.4.2.1). In particular, we need the following result.

Lemma 3.5.4. The identi�cation

η1 : ui,r ÞÝÑ rWpiqki,r,r

st. (3.5.4.1)

extends to a well-de�ned injective Zrt�1{2s-algebras morphism

η1 : T Ñ Ft,

where Ft is the skew-�eld of fractions of Yt (see Remark 3.3.3).Moreover, the restriction of η1 to the quantum cluster algebra At has its image in the

quantum torus Yt and the Zrt�1{2s-algebra morphisms η, η1 and π satisfy the followingcommutative diagram:

Atη

//

η1!!

Y�t

Ytπ

<<

.

(3.5.4.2)


Proof. From Proposition 3.5.2, for all pi, rq P I�, the full pq, tq-character rW piqki,r,r�2hst is

obtained as the image of the cluster variable sitting at vertex pi, rq after applying h timesthe mutation sequence S (which is locally a �nite sequence of mutations):

ηpuphqi,r q � rW

piqki,r,r�2hst. (3.5.4.3)

In particular, for any two vertices pi, rq, pj, sq, the variables uphqi,r and uphqj,s belong to a

common cluster and t-commute. Thus the pq, tq-characters rW piqki,r,r�2hst t-commute. As

the quantum torus Yt is invariant by shift of quantum parameters (Yj,s ÞÑ Yj,s�2), the pq, tq-

characters rW piqki,r,r

st also t-commute for the product �. Their t-commutation relations are

determined by their dominant monomials, which are ηpui,rq. Thus the rW piqki,r,r

st satisfyexactly the same t-commutations relations are the ui,r. This proves the �rst part of thelemma.

Let X be a cluster variable of At obtained from the initial seed uuu � tui,ru via �nitesequence of mutations σ. We want to show that η1pXq P Yt. As the sequence of mutationsσ is �nite, it will only involve a �nite number of cluster variables. Now apply h times themutation sequence µS to the initial seed so as to replace each cluster variable consideredby uphqi,r (again, we only need a �nite number of mutations). Let us summarize:

η1pui,rq � rWpiqki,r,r

st, ηpuphqi,r q � rW

piqki,r,r�2hst.

Let X 1 be the cluster variable obtained by applying to this new seed the sequence ofmutations σ. By construction, ηpX 1q is equal to η1pXq, up to the downward shift of spectralparameters by 2h: every variable Y �1

j,s is replaced by Y �1j,s�2h. In particular, η1pXq P Yt.

Next, the commutation of diagram (3.5.4.2) is veri�ed as it is satis�ed on the initialseed uuu � tui,ru.

Let Rt be the image of the quantum cluster algebra At

Rt :� ηpAtq P Y�t . (3.5.4.4)

The inclusionKtpC�Z q � Rt is essentially contained in Proposition 3.5.2. For the reverse

inclusion, the main idea is to use the characterization of the quantum Grothendieck ringas the intersection of kernel of operators, called deformed screening operators. We show byinduction on the length of a sequence of mutations that the images of all cluster variablesbelong to those kernels. The images of the initial cluster variables ui,r clearly belong to thequantum Grothendieck ring, and the screening operators being derivations, the exchangerelations force the newly created cluster variables to be in the intersection of the kernelstoo. let us prove this in details.

Proof. Recall from Lemma 3.3.8 that the quantum Grothendieck ring KtpC�Z q is alge-

braically generated by the truncated pq, tq-characters of the fundamental modules:

KtpC�Z q �

ArLpYi,rqs

�t | pi, rq P I

�E.

By Proposition 3.5.2, for all pi, rq P I�,

rLpYi,rqs�t � w

pr�ξi

2q

i,ξi� η

�upr�ξi

2q

i,ξi

P ηpAtq. (3.5.4.5)


Which proves the �rst inclusion:

KtpC�Z q � Rt. (3.5.4.6)

We prove the reverse inclusion as explained just above.As explained in Section 3.3.3, Hernandez proved in [Her03] that for all i P I there

exists operators Si,t : Yt Ñ Yi,t, where Yi,t is a Yt-module, which are Zrt�1s-linear andderivations, such that £

iPI

kerpSi,tq � KtpCZq. (3.5.4.7)

Notice that these operators characterize the quantum Grothendieck ring KtpCZq andnot KtpC

�Z q. Hence the need for Lemma 3.5.4.

Let us prove by induction that all cluster variables Z in At satisfy η1pZq P KtpCZq.Let Z be a quantum cluster variable in At. If Z belongs to the initial cluster variables,

Z � ui,r andη1pZq � η1pui,rq � rWki,r,rst P KtpCZq. (3.5.4.8)

If not, then by induction on the length of the sequence µ, one can assume that Z is obtainedvia a quantum exchange relation

Z � Z1 � tαM1 � tβM2, (3.5.4.9)

where Z1 is a quantum cluster variable of At, M1 and M2 are quantum cluster monomialsof At and

η1pZ1q, η1pM1q, η

1pM2q P KtpC q.

Apply η1 to (3.5.4.9):

η1pZq � η1pZ1q � tαη1pM1q � tβη1pM2q. (3.5.4.10)

For all i P I, apply the derivation Si,t:

Si,t�η1pZq � η1pZ1q

�� Si,t

�η1pZq

�� η1pZ1q � η1pZq � Si,t

�η1pZ1q

�� tαSi,t

�η1pM1q

�� tβSi,t

�η1pM2q

�.

However, by hypothesis,

Si,t�η1pZ1q

�� 0,

Si,t�η1pM1q

�� 0,

Si,t�η1pM1q

�� 0.

Moreover, η1pZ1q � 0 and the images of the screening operator is in a free module over Ytby Lemma 3.3.4. Thus Si,t pη1pZqq � 0, for all i P I. Hence

η1pZq P KtpC�Z q,

which concludes the induction. We have proven

η1pAtq � KtpCZq.

Then, by the commutation of the diagram (3.5.4.2) in Lemma 3.5.4,

Rt � ηpAtq � KtpC�Z q.

Which concludes the proof of the theorem.

3.6. Application to the proof of an inclusion conjecture 107

3.6 Application to the proof of an inclusion conjecture

In this section, we use the quantum cluster algebra structure of the Grothendieck ringC�Z to prove that the quantum Grothendieck ring KtpO�Z q de�ned in Section 2.4.3 contains

KtpC�Z q. In other words, we prove Conjecture 2.5.2 in Chapter 2. The result was already

proven in Chapter 2 in type A, but the core argument used was di�erent.

3.6.1 The quantum Grothendieck ring KtpO�Z q

The quantum Grothendieck ring KtpO�Z q is de�ned as a quantum cluster algebra onthe full in�nite quiver Γ, of which the semi-in�nite quiver G� is a subquiver.

Let us recall some notations. For all i, j P I, Fij : Z Ñ Z is a anti-symmetrical mapsuch that, for all m ¥ 0,

Fijpmq � �¸k¥1

m¥2k�1

Cijpm� 2k � 1q. (3.6.1.1)

Let Tt be the quantum torus de�ned as the Zrt�1{2s-algebra generated by the variables z�i,r,

for pi, rq P I, with a non-commutative product �, and the t-commutations relations

zi,r � zj,s � tFijps�rqzj,s � zi,r,�pi, rq, pj, sq P I

. (3.6.1.2)

Recall also from Proposition 2.3.3 the inclusion of quantum tori J (with a slight shift ofparameters on the zi,r):

J :

"Yt ÝÑ Tt,Yi,r ÞÝÑ zi,r pzi,r�2q

�1 .. (3.6.1.3)

Let Λ be the in�nite skew-symmetric I � I-matrix:

Λpi,rq,pj,sq � Fijps� rq,�pi, rq, pj, sq P I

. (3.6.1.4)

From proposition 2.4.10, the quiver Γ and the skew-symmetric matrix Λ form a compatiblepair. Let AtpΓ,Λq be the associated quantum cluster algebra. Then, as in De�nition 2.4.15,

KtpO�Z q :� AtpΓ,ΛqbE , (3.6.1.5)

where E is a commutative ring and the completion allows for certain countable sums.

3.6.2 Intermediate quantum cluster algebras

The general idea is to see the quantum cluster algebra At as a "sub-quantum clusteralgebra" of AtpΓ,Λq (this term is not well-de�ned). However, as in Section 3.5.4 andcontrary to the aforementioned proof, as we are dealing with quantum cluster algebras inour setting, this is not done trivially. Mainly, one notices that the map

T ÝÑ Ttui,r ÞÝÑ zi,r,

is not a well-de�ned inclusion of quantum tori, as the generators ui,r and zi,r do not satisfythe same t-commutation relations.


First, consider the subquiver Γ� of Γ of index set

I�¤2 :� I X tpi, rq | i P I, r ¤ 2u, (3.6.2.1)

such that the vertices pi, rq, with r ¡ 0 are frozen.To summarize, for ξi � 0 and j � i:

Γ �

...

pi, 2q

pi, 0q

pi,�2q

...

...

pj, 1q

pj,�1q

...

, Γ� �

(i,2)

pi, 0q

pi,�2q

...

(j,1)

pj,�1q

...

, G� �

pi, 0q

pi,�2q

...

pj,�1q

...

.

The quivers Γ and Γ� are only connected by coe�cients (as in [GG18, De�nition 4.1]),thus by [GG18, Theorem 4.5] the inclusion of seeds Γ�,Λ � Γ,Λ induces an inclusion ofthe quantum cluster algebra AtpΓ�,Λq into the quantum cluster algebra of AtpΓ,Λq.

Now we need to link the quantum cluster algebras At and AtpΓ�,Λq.In order to do that, we use a result from [GL14], which deals with graded cluster

algebras.

De�nition 3.6.1. [GL14] A quantum cluster seed pB,Λq is graded if there exists an integercolumn vectorG such that, for all mutable indices k, the kth row of B, Bj satis�es BjG � 0.

Then, the G-degree of the initial variables are set by the vector G: for all clustervariables Xk in the initial cluster X, degGpXq � Gi.

The grading condition is equivalent to the following, for all mutable variables Xk, thesum of the degrees of all variables with arrows to Xk is equal to the sum of the degrees ofall variables with arrows coming from Xk, i.e. exchange relations are homogeneous. Hence,each cluster variable has a well-de�ned degree.

Let us start by considering the quantum cluster algebra AtpΓ�, Lq, built on the quiverΓ�, with coe�cients 1 on the frozen vertices. This quantum cluster algebra is clearlyisomorphic to At, and is a graded quantum cluster algebra of grading G � 0.

Next, for all i P I, we apply the process of [GL14, Theorem 4.6] to add coe�cients fi,while twisting the t-commutation relations.

Letuipj, sq � δi,j ,

�pj, sq P I�¤2

(3.6.2.2)

andtipj, sq � �Fijp2� s� ξiq,

�pj, sq P I�¤2

, (3.6.2.3)


with ξ : I Ñ t0, 1u the height function, �xed in Section A.3 of the Appendix. Then

Lemma 3.6.2. For all i P I, ui and ti are gradings for the ice quiver Γ�.

Proof. For all pj, sq P I�, let Bpj,sq be the pj, sqth "row" of the I�¤2 � I�¤2-skew-symmetricmatrix encoding the adjacency of the ice quiver Γ�.

For all i P I,

Bpj,squi �¸

pk,rqPI�¤2

Bpj,sqpk, rquipk, rq

� uipj, s� 2q � uipj, s� 2q �¸k�j

puipk, s� 1q � uipk, s� 1qq

� δi,j � δi,j �¸k�j

pδi,k � δi,kq � 0.

And

Bpj,sqti �¸

pk,rqPI�¤2

Bpj,sqpk, rqtipk, rq

� tipj, s� 2q � tipj, s� 2q �¸k�j

ptipk, s� 1q � tipk, s� 1qq

� �Fijp�s� ξiq � Fijp4� s� ξiq �¸k�j

p�Fikp3� s� ξiq � Fikp1� s� ξiqq

� �Cijp3� s� ξiq � Cijp1� s� ξiq �¸k�j

Cikp2� s� ξiq � 0,

from Lemma A.2.4 of the Appendix.

Then, by [GL14, Theorem 4.6] (applied n times), the following is a valid set of initialdata for a (multi-)graded quantum cluster algebra Atpuuu, B, L, Gq, where• or all pi, rq P I�¤2,

ui,r �

"ui,rfi if r ¤ 0,fi otherwise

. (3.6.2.4)

and uuu � tui,rupi,rqPI�¤2.

• B � B, the I�¤2 � I�¤2-skew-symmetric adjacency matrix of the quiver Γ�.

• L encodes the t-commutations relations, such that, for all pi, rq P I�, and j P I,

fj � ui,r � ttjpi,rqui,r � fj , (3.6.2.5)

and the fj pairwise commute.• G is a multi-grading, i.e. instead of being an integer column vector, each entry in G isin the lattice ZI . It is de�ned by, for all pi, rq P I�¤2,

Gpui,rq � ei P ZI , (3.6.2.6)

or degi � ui, for all i P I.This is indeed the construction of [GL14], with the initial grading on AtpΓ�, Lq beingG � 0. The new quantum cluster algebra is denoted by Au,tt pΓ�, Lq, to show that it isa twisted version of AtpΓ�, Lq.


Proposition 3.6.3. The quantum cluster algebra Atpuuu, B, Lq is isomorphic to the quantumcluster algebra AtpΓ�,Λq.

Proof. The rest of the data being the same, we only have to check that the I�¤2� I�¤2-skew-

symmetric matrices L and Λ|I�¤2are equal.

From (3.6.1.4), for all�pi, rq, pj, sq P I�¤2

,

Λ ppi, rq, pj, sqq � Fijps� rq.

And L is de�ned asui,r � uj,s � tLppi,rq,pj,sqquj,s � ui,r. (3.6.2.7)

Hence,

L ppi, rq, pj, sqq �

$''''&''''%L ppi, rq, pj, sqq � tipj, sq � tjpi, rq , if pi, rq, pj, sq P I�,

tipj, sq , if

�pi, rq � pi,�ξi � 2q

pj, sq P I�,

0 , if

�pi, rq � pi,�ξi � 2qpj, sq � pj,�ξj � 2q

.

First, notice that, for all i, j P I, m P Z,

Nijpmq � 2Fijpmq � Fijpm� 2q � Fijpm� 2q. (3.6.2.8)

This result is proven in Chapter 2, in the course of the proof of Proposition 2.3.3.Thus, for all pi, rq, pj, sq P I�,

L ppi, rq, pj, sqq �¸k¥0

r�2k¤0

¸l¥0

s�2l¤0

Nijps� 2l � r � 2kq

�¸k¥0

r�2k¤0

pFijps� r � 2kq � Fijps� r � 2k � 2q

�Fijp�ξi � r � 2kq � Fijp�ξj � r � 2k � 2qq

� Fijps� rq � Fijps� ξi � 2q � Fijp�ξj � ξiqlooooooomooooooon�0

�Fijp�ξj � r � 2q

� Fijps� rq � tipj, sq � tjpi, rq.

And of course, for all i P I, pj, sq P I�,

Λ ppi,�ξi � 2q, pj, sqq � Fijps� ξi � 2q � tipj, sq.

Thus, one had indeed,L � Λ|I�¤2

. (3.6.2.9)

From now on, we will use the notations zzz � tzi,r, fjupi,rqPI�,jPI for the initial clusters

variables of both AtpΓ�,Λq and Au,tt pΓ�, Lq.


Remark 3.6.4. • This result is natural, if we look at what the di�erent initial clus-ter variables mean in terms of `-weights. From (3.2.4.1), for all pi, rq P I�, thecluster variable ui,r can be identi�ed with the (commutative) dominant monomialUi,r �

±k¥0

r�2k¤0Yi,r�2k. Whereas, the quantum tori Yt and Tt are compared via the

inclusion J of (3.6.1.3)

J : Yi,r ÞÝÑ zi,r pzi,r�2q�1 , @pi, rq P I .

Thus, the link between the variables ui,r and zi,r is the following:

ui,r � zi,r pfiq�1 , (3.6.2.10)

with the previous convention fi � zi,�ξi�2. More precisely, there are di�erent maps ofquantum tori:

T //

η ��

Tt

Y�t� � // Yt

J

OO, (3.6.2.11)

where identi�cation (3.6.2.10) is the resulting dotted map, which we will denote by ρ:

ρ : T Ñ Tt. (3.6.2.12)

The quantum cluster algebra Au,tt pΓ�, Lq was built with this identi�cation in mind.• This process could also be seen as a quantum version of the multi-grading homogeniza-tion process of the seed puuu,G�q, as in [Gra15, Lemma 7.1], where the multi-grading isde�ned by (3.6.2.6).

3.6.3 Inclusion of quantum Grothendieck rings

In the section, we prove that the quantum Grothendieck ringKtpO�Z q, or more precisely,the quantum cluster algebra AtpΓ,Λq contains the quantum Grothendieck ring KtpCZq,which is the statement of Conjecture 2.5.2 in Chapter 2. Recalled that in Section 2.4.3,the ring KtpO�Z q was de�ned as a completion of the quantum cluster algebra AtpΓ,Λq, butthe aim was to see it as a quantum Grothendieck ring for the category of representationsO�Z from [HJ12] and [HL16b]. As this category contains the category CZ, it was expectedfor the quantum Grothendieck ring KtpO�Z q to contain KtpCZq.

In order to prove this result we will actually prove Conjecture 2 of Chapter 2, which isa stronger result. We state it as follows.

Theorem 3.6.5. The pq, tq-characters of all fundamental representations in CZ are ob-tained as quantum cluster variables in the quantum cluster algebra AtpΓ,Λq.

More precisely, we show that for all i P I, there exists a speci�c �nite sequence ofmutations Si in AtpΓ,Λq such that, if applied to the initial seed tzj,supj,sqPI , the clustervariable sitting at vertex pi,�ξiq is the image by J (of (3.6.1.3)) of the pq, tq-character ofthe fundamental module rLpYi,�ξi�2h1qst, where

h1 � rh

2s, (3.6.3.1)

with h the Coxeter number of the simple Lie algebra g.


Let us de�ne the sequence Si. Let pi1, i2, . . . , jnq be an ordering of the columns of Γ asin (3.5.2.1), such that i1 � i (take �rst all columns j such that ξj � ξi). The sequence Siis a sequence of vertices of Γ, and more precisely of Γ�, de�ned as follows. First read allvertices pi1, rq for �2h1�2 ¤ r ¤ 0, from top to bottom, then all pi2, rq, with �2h1 ¤ r ¤ 0,and so on, then read again all vertices pi1, rq for �2h1 � 4 ¤ r ¤ 0, and continue browsingthe columns successively, until at the last step you only read the vertex pi,�ξiq.

Note that applying this sequence Si of mutations on the quivers Γ� or G� has exactlythe same e�ect on the cluster variable sitting at vertex pi,�ξiq than applying h1 times thein�nite sequence S from Section 3.5.2.

Example 3.6.6. For g of type D4, h � 6, then h1 � 3. Let us give explicitly the sequenceS2 starting on column 2. For simplicity of notations, we assume that the height functionis chosen such that ξ2 � 0. Then the sequence S2 has 15 steps:

S2 � p2, 0q, p2,�2q, p2,�4q, p1,�1q, p1,�3q, p3,�1q, p3,�3q, p4,�1q, p4,�3q,

p2, 0q, p2,�2q, p1,�1qp3,�1qp4,�1qp2, 0q. (3.6.3.2)

Let r0 � �ξi � 2h1. Let χi,r0 P Tt be the quantum cluster variable obtained at ver-tex pi,�ξiq after applying the sequence of mutations Si to the quantum cluster variableAtpΓ�,Λq with initial seed tzj,s, fkupj,sqPI�,kPI .

Proposition 3.6.7. As an element of the quantum torus Tt, χi,r0 belongs to the image ofthe inclusion morphism J .

Moreover,χi,r0 � J prLpYi,r0qstq . (3.6.3.3)

Proof. The cluster variable χi,r0 is a variable of the quantum cluster algebra AtpΓ�,Λq,which is isomorphic to Au,tt pΓ�, Lq from Proposition 3.6.3. By [GL14, Corollary 4.7], thereis a bijection between the quantum cluster variables of Au,tt pΓ�, Lq and those of AtpΓ�, Lq.

With notations from Section 3.5.3, uph1q

i,�ξiis the cluster variable of AtpΓ�, Lq obtained at

vertex pi,�ξiq after applying the mutations of the sequence Si. Also from [GL14, Corollary4.7], we know that there exists integers aj P Z such that

χi,r0 � ρ�uph1qi,�ξi

¹jPI

fajj , (3.6.3.4)

written as a commutative product (both χi,r0 and uph1qi,�ξi

are bar-invariant), with ρ de�ned

in (3.6.2.12). The term uph1qi,�ξi

is a Laurent polynomial in the variables uj,s, which satisfy

ρpuj,sq � zj,spfjq�1 from (3.6.2.10). Thus expression (3.6.3.4) is a way of writing χi,r0 as a

Laurent polynomial in the initial variables tzj,s, fku. However, one can write χi,r0 � N{D,where N is the Laurent polynomial in the cluster variables tzj,s, fku, with coe�cients inZrt�1{2s and not divisible by any of the fk, and D is a monomial in the non-frozen variablestzj,su. Thus

±jPI f

ajj is the smallest monomial such that

ρ�uph1qi,�ξi

¹jPI

fajj

contains only non-negative powers of the frozen variables fk.Moreover, from Proposition 3.5.2,

ηpuph1qi,�ξi

q � wph1qi,�ξi

� rLpYi,r0qst. (3.6.3.5)


However, all Laurent monomials occurring in rLpYi,r0qst already occurred in the q-character χqpLpYi,r0qq, as the pq, tq-character rLpYi,r0qst has positive coe�cients. Indeed,the pq, tq-characters of fundamental modules have been explicitly computed and have beenfound to have non-negative coe�cients (in [Nak03b] for types A and D and [Nak10] fortype E).

From [FM01], all monomials in χqpLpYi,r0qq are products of Y�1j,s , with s ¤ r0 � h, but

by de�nition of r0, r0 � h ¤ 0, and the term with the highest quantum parameter beingthe anti-dominant monomial Y �1

i,r0�h, where : I Ñ I is the involutive map such that

ω0pαjq � �αj , with ω0 the longest element of the Weyl group of g (no relation with thebar-involution of Section 3.5.1).

Consider the change of variables, for pj, sq P I�,

yj,s � η�1pYj,sq �

#uj,suj,s�2

, if s� 2 ¤ 0,

uj,s, otherwise. (3.6.3.6)

Thus

ρpyj,sq �

#zj,szj,s�2

, if s� 2 ¤ 0,zj,sfj, otherwise

. (3.6.3.7)

All monomials occurring in rLpYi,r0qst are commutative monomials in the variables Y �1j,s ,

with s ¤ r0 � h ¤ 0. Moreover, the only monomials in which the variables Y �1j,s , with

s � 2 ¥ 0, occur are the anti-dominant monomial Y �1i,r0�h

, and any possible monomial in

which some variable Yj,r0�h�1 occurs. But for such a monomial m, the variable Y �1j,s with

the highest s in m occurs with a negative power in m (the monomial m is "right-negative",as from [FM01, Lemma 6.5]), thus the variable Yj,r0�h�1 also occurs with a negative power.

The image by ρ � η�1 of any monomial in the variables Y �1j,s , with s � 2 ¤ 0 is a

monomial in the variables tz�1j,s u (without frozen variables). Thus, the image

ρ�uph1qi,�ξi

� ρ � η�1 prLpYi,r0qstq

is a Laurent polynomial with only positive powers of the variables fj .Necessarily,

±jPI f

ajj � 1, and (3.6.3.4) becomes


.

Finally, from the diagram (3.6.2.11),


� J

�η�uph1qi,�ξi

� J prLpYi,r0qstq ,

which concludes the proof.

Remark 3.6.8. At some point in the proof we used the fact that the pq, tq-characters of thefundamental modules had non-negative coe�cients. Note that this part of the proof couldeasily be extended to non-simply laced types, as the pq, tq-characters of their fundamentalrepresentations have also be explicitly computed, and also have non-negative coe�cients(in types B and C, the pq, tq-characters of all fundamental representations are equal totheir respective q-character, and all coe�cients are actually equal to 1 [Her05, Proposition7.2], and see [Her04] for type G2 and [Her05] for type F4).


Corollary 3.6.9. For all pi, rq P I there exists a quantum cluster variable χi,r in thequantum cluster algebra AtpΓ,Λq such that

χi,r � J prLpYi,rqstq .

Proof. For all pi, rq P I, let χi,r be the cluster variable of the quantum cluster algebraAtpΓ,Λq obtained at vertex pi, r � 2h1q after applying the sequence of mutations Si, butstarting at vertex pi, r � 2h1q instead of pi,�ξiq.

Consider the change of variables in AtpΓ,Λq:

s : zj,s ÞÝÑ zj,s�r0�r, @pj, sq P I . (3.6.3.8)

The quantum cluster algebra AtpΓ,Λq is invariant under this shift s, and this changeof variables is clearly invertible (s�1pzj,sq � zj,s�r�r0). One has spzi,rq � zi,r0 , andspzi,r�2h1q � zi,�ξi , thus s pχi,rq � χi,r0 , and from Proposition 3.6.7,

χi,r � s�1 pχi,r0q � s�1 pJ prLpYi,r0qstqq .

However, from the de�nition of the map J in (3.6.1.3), the shift s also acts as a change ofvariables in the quantum torus Yt, s : Yj,u ÞÝÑ Yj,u�r0�r. Hence,

χi,r � J prLpYi,rqstq .

Thus we have proven Theorem 3.6.5.

Remark 3.6.10. • When Conjecture 2.5.7 was �rst formulated in Chapter 2, a recentpositivity result of Davison [Dav18] was mentioned there. This work proves the so-called"positivity conjecture" for quantum cluster algebras, which states that the coe�cientsof the Laurent polynomials into which the cluster variables decompose from the Laurentphenomenon are in fact non-negative. This is an important result, but also a di�cultone, and it is not actually needed in order to obtain our result.• One can note from this proof that we know a close bound on the number of mutationsneeded in order to compute the pq, tq-character of a fundamental module. For g a simple-laced simple Lie algebra of rank n and of (dual) Coxeter number h, if h1 � rh{2s, thenthe number of steps is lower than

nh1ph1 � 1q

2. (3.6.3.9)

We have chosen to go into details on the number of steps required for this processbecause it made sense from an algorithmic point of view to know its complexity.One can compare this algorithm to Frenkel-Mukhin [FM01] to compute q-characters.As explained in [Nak10], when trying to compute q-characters of fundamental rep-resentations of large dimension (for example, in type E8, the q-character of the 5thfundamental representation has approximately 6.4 � 230 monomials), one encountersmemory issues. Indeed, this algorithm has to keep track of all the previously computedterms. This advantage of the cluster algebra approach is that one only had to keep theseed in memory.

3.7. Explicit computation in type D4 115

3.7 Explicit computation in type D4

For g of type D4, with the height function ξ2 � 0, ξ1 � ξ3 � ξ4 � 1, we compute thepq, tq-character of the fundamental representation LpY2,�6q as a quantum cluster variableof the quantum cluster algebra AtpΓ�,Λq, using the algorithm presented in the previousSection.

From Example 3.6.6, the sequence of mutation we have to apply to the initial seed is

S2 � p2, 0q, p2,�2q, p2,�4q, p1,�1q, p1,�3q, p3,�1q, p3,�3q, p4,�1q, p4,�3q,

p2, 0q, p2,�2q, p1,�1qp3,�1qp4,�1qp2, 0q.

One can notice that the required number of steps is indeed lower than 24, which was thebound given in (3.6.3.9).

Let us give explicitly the mutations on the quiver Γ� (which encodes more than G�),as well as the quantum cluster variables obtained at (almost every) step. We give thequantum cluster variables as Laurent polynomials in the variables tzi,r, fjupi,rqPI,jPI , aswell as in the form (3.6.3.4). For completeness, we also compute the multi-degrees of thequantum cluster variables.

This computation was done thanks to the latest version (as of January 2019) of Bern-hard Keller's wonderful quiver mutation applet:

https://webusers.imj-prg.fr/~bernhard.keller/quivermutation/.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q

Mutation at p2, 0q.

deg�zp1q2,0

� e1�e3�e4.

zp1q2,0 � z2,�2 pz2,0q

�1 f1f3f4 � z1,�1z3,�1z4,�1 pz2,0q�1 f2,

�J�Y2,�2 � Y1,�1Y3,�1Y4,�1Y

�12,0

f1f3f4.

https://webusers.imj-prg.fr/~bernhard.keller/quivermutation/


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q

Mutation at p2,�2q.

deg�zp1q2,�2

� e1�e3�e4.

zp1q2,�2 � z2,�4 pz2,0q

�1 f1f3f4 � z1,�1z3,�1z4,�1z2,�4 pz2,�2q�1 pz2,0q

�1 f2

� z1,�3z3,�3z4,�3 pz2,�2q�1 f2,

zp1q2,�2 � J

�Y2,�4Y2,�2 � Y1,�1Y2,�4 pY2,0q

�1 Y3,�1Y4,�1

�Y1,�3Y1,�1 pY2,�2q�1 pY2,0q

�1 Y3,�3Y3,�1Y4,�3Y4,�1

f1f3f4.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q2,�4

� e1�e3�e4.

(3.7.0.1)


zp1q2,�4 � z2,�6 pz2,0q

�1 f1f3f4 � z1,�1z3,�1z4,�1z2,�6 pz2,�2q�1 pz2,0q

�1 f2

� z1,�3z3,�3z4,�3z2,�6 pz2,�4q�1 pz2,�2q

�1 f2 � z1,�5z3,�5z4,�5 pz2,�4q�1 f2,

zp1q2,�4 � J

�Y2,�6Y2,�4Y2,�2 � Y1,�1Y2,�6Y2,�4 pY2,0q

�1 Y3,�1Y4,�1

� Y1,�3Y1,�1Y2,�6 pY2,�2q�1 pY2,0q

�1 Y3,�3Y3,�1Y4,�3Y4,�1

�Y1,�5Y1,�3Y1,�1 pY2,�4q�1 pY2,�2q

�1 Y3,�5Y3,�3Y3,�1Y4,�5Y4,�3Y4,�1

f1f3f4.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q1,�1

� e3 � e4.

zp1q1,�1 � z1,�3 pz1,�1q

�1 f3f4

� pz1,�1q�1 z2,�2 pz2,0q

�1 f1f3f4 � z3,�1z4,�1 pz2,0q�1 f2

zp1q1,�1 � J

�Y1,�3 � pY1,�1q

�1 Y2,�2 � pY2,0q�1 Y3,�1Y4,�1

f3f4.


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q1,�3

� e3 � e4.

zp1q1,�3 � z1,�5 pz1,�1q

�1 f3f4 � z1,�5 pz1,�3q�1 pz1,�1q

�1 z2,�2 pz2,0q�1 f1f3f4

� z1,�5 pz1,�3q�1 pz2,0q

�1 z3,�1z4,�1f2 � pz1,�3q�1 z2,�4 pz2,0q

�1 f1f3f4

� pz1,�3q�1 z1,�1 pz2,�2q

�1 pz2,0q�1 z3,�1z4,�1z2,�4f2 � pz2,�2q

�1 z3,�3z4,�3f2,

zp1q1,�3 � J

�Y1,�5Y1,�3 � Y1,�5 pY1,�1q

�1 Y2,�2 � Y1,�5 pY2,0q�1 Y3,�1Y4,�1

� pY1,�3q�1 pY1,�1q

�1 Y2,�4Y2,0 � pY1,�3q�1 Y2,�4 pY2,0q

�1 Y3,�1Y4,�1

�pY2,�2q�1 pY2,0q

�1 Y3,�3Y3,�1Y4,�3Y4,�1

f3f4.

The next steps of the mutation process are the mutations p3,�1q,p3,�3q and p4,�1q,

p4,�3q. However, the resulting cluster variables are equal zp1q1,�1 zp1q1,�3, up to the exchanges

1 ÐÑ 3 and 1 ÐÑ 4, so we will not give their expressions. We only show the mutation ofthe quiver, and the multi-degrees.


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q3,�1

� e1 � e4.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q3,�3

� e1 � e4.


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q4,�1

� e1 � e3.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


deg�zp1q4,�3

� e1 � e3.


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q

Mutation at p2, 0q.

degpzp2q2,0q � e1 � e3 � e4.

zp2q2,0 � z2,�4 pz2,�2q

�1 f1f3f4

� z1,�3 pz1,�1q�1 pz2,�2q

�1 z2,0z3,�3 pz3,�1q�1 z4,�3 pz4,�1q

�1 f1f3f4

� pz1,�1q�1 z3,�3 pz3,�1q

�1 z4,�3 pz4,�1q�1 f2

1 f3f4

� z1,�3 pz1,�1q�1 pz3,�1q

�1 z4,�3 pz4,�1q�1 f1f

23 f4

� z1,�3 pz1,�1q�1 z3,�3 pz3,�1q

�1 pz4,�1q�1 f1f3f

24

� pz1,�1q�1 z2,�2 pz2,0q

�1 pz3,�1q�1 z4,�3 pz4,�1q

�1 z4,1f21 f

23 f4

� pz1,�1q�1 z2,�2 pz2,0q

�1 z3,�3 pz3,�1q�1 pz4,�1q

�1 f21 f3f

24

� z1,�3 pz1,�1q�1 z2,�2 pz2,0q

�1 pz3,�1q�1 pz4,�1q

�1 f1f23 f

24

� pz2,0q�1 z4,�3f1f2f3 � pz2,0q

�1 z3,�3f1f2f4 � z1,�3 pz2,0q�1 f2f3f4

� pz1,�1q�1 z2

2,�2 pz2,0q�2 pz3,�1q

�1 pz4,�1q�1 f2

1 f23 f

24

��t� t�1

�z2,�2 pz2,0q

�2 f1f2f3f4 � z1,�1 pz2,0q�2 z3,�1z4,�1f

22 ,

zp2q2,0 � J

�Y2,�4 � Y1,�3 pY2,�2q

�1 Y3,�3Y4,�3

� pY1,�1q�1 Y3,�3Y4,�3 � Y1,�3 pY3,�1q

�1 Y4,�3 � Y1,�3Y3,�3 pY4,�1q�1

� pY1,�1q�1 Y2,�2 pY3,�1q

�1 Y4,�3 � pY1,�1q�1 Y2,�2Y3,�3 pY4,�1q

�1

� Y1,�3Y2,�2 pY3,�1q�1 pY4,�1q

�1 � pY2,0q�1 Y4,�3Y4,�1

� pY2,0q�1 Y3,�3Y3,�1 � Y1,�3Y1,�1 pY2,0q

�1

� pY1,�1q�1 pY2,�2q

2 pY3,�1q�1 pY4,�1q

�1 ��t� t�1

�Y2,�2 pY2,0q

�1

�Y1,�1 pY2,0q�2 Y3,�1Y4,�1

f1f3f4.


(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


degpzp2q2,�2q � e1�e3�e4.

Here we do not write explicitly zp2q2,�2, because it has 92 terms.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q


degpzp2q1,�1q � e1.

zp2q1,�1 � z1,�5 pz1,�3q

�1 f1 � pz1,�3q�1 z1,�1z2,�4 pz2,�2q

�1 f1

� pz2,�2q�1 z2,0z3,�3 pz3,�1q

�1 z4,�3 pz4,�1q�1 f1

� z3,�3 pz3,�1q�1 pz4,�1q

�1 f1f4 � pz3,�1q�1 z4,�3 pz4,�1q

�1 f1f3

� z2,�2 pz2,0q�1 pz3,�1q

�1 pz4,�1q�1 f1f3f4 � z1,�1 pz2,0q

�1 f2,


zp2q1,�1 � J

�Y1,�5 � pY1,�3q

�1 Y2,�4 � pY2,�2q�1 Y3,�3Y4,�3 � Y3,�3 pY4,�1q

�1

�pY3,�1q�1 Y4,�3 � Y2,�2 pY3,�1q

�1 pY4,�1q�1 � Y1,�1 pY2,0q

�1f1.

As before, zp2q3,�1 and zp2q4,�1 are similar to zp2q1,�1 so we do not write them.

(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q



(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q




(2,2)

(1,1) (3,1) (4,1)

p2, 0q

p1,�1q p3,�1q p4,�1q

p2,�2q

p1,�3q p3,�3q p4,�3q

p2,�4q

p1,�5q p3,�5q p4,�5q

p2,�6q

Mutation at p2, 0q.

degpzp3q2,0q � 0.

zp3q2,0 � z2,�6 pz2,�4q

�1 � z1,�5 pz1,�3q�1 pz2,�4q

�1 z2,�2z3,�5 pz3,�3q�1 z4,�5 pz4,�3q

�1

� pz1,�3q�1 z1,�1z3,�5 pz3,�3q

�1 z4,�5 pz4,�3q�1

� z1,�5 pz1,�3q�1 pz3,�3q

�1 z3,�1z4,�5 pz4,�3q�1

� z1,�5 pz1,�3q�1 z3,�5 pz3,�3q

�1 pz4,�3q�1 z4,�1

� pz1,�3q�1 z1,�1z2,�4 pz2,�2q

�1 pz3,�3q�1 z3,�1z4,�5 pz4,�3q

�1

� pz1,�3q�1 z1,�1z2,�4 pz2,�2q

�1 z3,�5 pz3,�3q�1 pz4,�3q

�1 z4,�1

� z1,�5 pz1,�3q�1 z2,�4 pz2,�2q

�1 pz3,�3q�1 z3,�1 pz4,�3q

�1 z4,�1

� pz1,�3q�1 z1,�1 pz2,�4q

2 pz2,�2q�2 pz3,�3q

�1 z3,�1 pz4,�3q�1 z4,�1

� pz2,�2q�1 z2,0z4,�5 pz4,�1q

�1 � pz2,�2q�1 z2,0z3,�5 pz3,�1q

�1

� z1,�5 pz1,�1q�1 pz2,�2q

�1 z2,0 ��t� t�1

�z2,�4 pz2,�2q

�2 z2,0

� z1,�3 pz1,�1q�1 pz2,�2q

�2 pz2,0q2 z3,�3 pz3,�1q

�1 z4,�3 pz4,�1q�1

� z4,�5 pz4,�3q�1 pz4,�1q

�1 f4 � z3,�5 pz3,�3q�1 pz3,�1q

�1 f3

� z1,�5 pz1,�3q�1 pz1,�1q

�1 f1 � pz1,�3q�1 z2,�4 pz2,�2q

�1 f1

� z2,�4 pz2,�2q�1 pz3,�3q

�1 f3 � z2,�4 pz2,�2q�1 pz4,�3q

�1 f4

� pz1,�1q�1 pz2,�2q

�1 z2,0z3,�3 pz3,�1q�1 z4,�3 pz4,�1q

�1 f1

� z1,�3 pz1,�1q�1 pz2,�2q

�1 z2,0 pz3,�1q�1 z4,�3 pz4,�1q

�1 f3

� z1,�3 pz1,�1q�1 pz2,�2q

�1 z2,0z3,�3 pz3,�1q�1 pz4,�1q

�1 f4

� pz1,�1q�1 pz3,�1q

�1 z4,�3 pz4,�1q�1 f1f3

� z1,�3 pz1,�1q�1 pz3,�1q

�1 pz4,�1q�1 f3f4

� pz1,�1q�1 z3,�3 pz3,�1q

�1 pz4,�1q�1 f1f4

� pz1,�1q�1 z2,�2 pz2,0q

�1 pz3,�1q�1 pz4,�1q

�1 f1f3f4 � pz2,0q�1 f2.


�J�Y2,�6 � Y1,�5 pY2,�4q

�1 Y3,�5Y4,�5

� pY1,�3q�1 Y3,�5Y4,�5 � Y1,�5 pY3,�3q

�1 Y4,�5 � Y1,�5Y3,�5 pY4,�3q�1

� pY1,�3q�1 Y2,�4 pY3,�3q

�1 Y4,�5 � pY1,�3q�1 Y2,�4Y3,�5 pY4,�3q

�1

� Y1,�5Y2,�4 pY3,�3q�1 pY4,�3q

�1 � pY1,�3q�1 pY2,�4q

2 pY3,�3q�1 pY4,�3q

�1

� pY2,�2q�1 Y4,�5Y4,�3 � pY2,�2q

�1 Y3,�5Y3,�3 � Y1,�5Y1,�3 pY2,�2q�1

��t� t�1

�Y2,�4 pY2,�2q

�1 � Y1,�3 pY2,�2q�2 Y3,�3Y4,�3

� Y4,�5 pY4,�1q�1 � Y3,�5 pY3,�1q

�1 � Y1,�5 pY1,�1q�1

� pY1,�3q�1 pY1,�1q

�1 Y2,�4 � Y2,�4 pY3,�3q�1 pY3,�1q

�1 � Y2,�4 pY4,�3q�1 pY4,�1q

�1

� pY1,�1q�1 pY2,�2q

�1 Y3,�3Y4,�3 � Y1,�3 pY2,�2q�1 pY3,�1q

�1 Y4,�3

� Y1,�3 pY2,�2q�1 Y3,�3 pY4,�1q

�1 � pY1,�1q�1 pY3,�1q

�1 Y4,�3

� Y1,�3 pY3,�1q�1 pY4,�1q

�1 � pY1,�1q�1 Y3,�3 pY4,�1q

�1

�pY1,�1q�1 Y2,�2 pY3,�1q

�1 pY4,�1q�1 � pY2,0q

�1

� J prLpY2,�6qstq .

Remark 3.7.1. Note here that the coe�cient of Y2,�4 pY2,�2q�1 is t� t�1. This coe�cient

is actually the only coe�cient of rLpY2,�6qst to not be equal to 1. The quantum clustervariables being bar-invariant, all coe�cients in the decomposition of the cluster variablesinto sums of commutative polynomials in the variables Y �1

i,r are symmetric polynomials int�1 (P pt�1q � P ptq). Thus this is the only coe�cient that could have been di�erent fromthe corresponding one in rLpY2,�6qst, as it could also have been equal to 2, or any tk� t�k.

Chapter 4Further directions

This chapter contains some leads for future work. We present evidence for furtherdevelopments in the form of partial results.

4.1 Further results in the sl2 case . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.1.1 pq, tq-characters for negative prefundamental representations . . . . . 1274.1.2 Quantum Grothendieck ring for the category O�Z . . . . . . . . . . . 1294.1.3 pq, tq-characters of simple modules . . . . . . . . . . . . . . . . . . . 1304.1.4 Quantum Wronskian relation . . . . . . . . . . . . . . . . . . . . . . 1334.1.5 pq, tq-characters of asymptotical standard modules . . . . . . . . . . 134

4.2 Leads for other types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.2.1 The (q,t)-characters of the negative prefundamental representations . 1364.2.2 Quantum Grothendieck ring for the category O�Z . . . . . . . . . . . 1364.2.3 Extension to non-simply-laced types . . . . . . . . . . . . . . . . . . 137

4.1 Further results in the sl2 case

In this section, we restrict ourselves to the case where g � sl2, and we study represen-tations of the quantum a�ne algebra Uqpsl2q or its Borel subalgebra Uqpbq (see previouschapters for details). The situation of sl2 is particular because we have direct and explicitaccess to a lot of information which is otherwise less accessible.

4.1.1 pq, tq-characters for negative prefundamental representations

In Chapter 2, we de�ned the quantum Grothendieck ring for the category O�Z of repre-sentations of Uqpbq and the pq, tq-characters of the positive prefundamental representations,but the chapter does not deal with the category O�Z nor the pq, tq-characters of the negativeprefundamental representations. The pq, tq-characters of the positive prefundamental rep-resentations are de�ned as elements of a quantum torus Tt, which are both bar-invariantand recover the q-characters when evaluated at t � 1 by the evaluation morphism π.

Recall from Chapter 2 that E is the ring of formal sums of weights, admitting countablesums. Let Ert�1{2s be the ring of Laurent polynomials in the variable t1{2 with coe�cientsin E . Next, recall that Tt is the Zrt�1{2s-algebra generated by the variables z�1

2r , and thet-commutation relations

z2r � z2s � tfps�rqz2s � z2r,

128 Chapter 4. Further directions

where f : Z Ñ Z is the anti-symmetrical map de�ned by fpmq � p�1qm�12 for all m ¥ 0.

We have also de�ne a Ert�1{2s-algebra structure on Tt b E , using the map χ : Tt b E Ñ Ede�ned by χpz2rq � r�rω1s. Then Tt � TtbE , where tensor product is completed to allowsome countable sums.

And, for all r P Z,

rL�q2rst :� rΨΨΨq2r s b χ1 P KtpO�Z q � Tt,

where χ1 P E is the character of L�q2r

, using the notation rΨΨΨq2r s � z2r b rrω1s.When g � sl2, the pq, tq-characters of the negative fundamental representations can be

de�ned naturally, because the q-characters of those representations are explicitly known.Recall that, for all r P Z,

χqpL�q2rq � rΨΨΨ�1

q2rs�

1�A�1q2r

�A�1q2rA�1q2r�2 �A�1

q2rA�1q2r�2A

�1q2r�4 � � � �

P El.

As this element is multiplicity-free, there is only one element rL�2rst in the quantum torusTt with non-negative coe�cients when written as a linear combination of commutativepolynomials, and such that π

�rL�2rst

�� χqpL

�q2rq, it is:

rL�2rst :� rΨΨΨ�1q2rs�

1�A�1q2r

�A�1q2rA�1q2r�2 �A�1

q2rA�1q2r�2A

�1q2r�4 � � � �

, (4.1.1.1)

where rΨΨΨ�1q2rs � rΨΨΨq2r s

�1 � z�12r r�rω1s. Note that we identify the elements of the quantum

torus Yt with their images in Tt by the inclusion map J .This is indeed an element of the quantum torus Tt because

|χ|�rL�2rst

�� 1� |χ|pA�1

q2rq � |χ|pA�1

q2rA�1q2r�2q � |χ|pA�1

q2rA�1q2r�2A

�1q2r�4q � � � �

� 1� r�2ω1s � r�4ω1s � r�6ω1s � . . . � χpL�q2rq � χpL�

q2rq � χ,

which is a countable sum in E .For any simple �nite-dimensional module Lpmq of highest `-weight m, de�ne its nor-

malized pq, tq-character as

χq,tpLpmqq :� m�1 � rLpmqst, (4.1.1.2)

where � is the commutative product between bar-invariant polynomials (rLpmqst is bar-invariant by de�nition).

The prefundamental representations in the category O also have normalized pq, tq-characters:

χq,tpL�q2rq � χ1 �

¸kPN

r�2kω1s,

χq,tpL�q2rq � 1�

�8

m�1

m�1¹k�0

A�1q2r�2k .

We note that the normalized pq, tq-character χq,tpL�q2rq is the limit, as a formal power

series in the variables A�1i,a , of the normalized pq, tq-characters of Kirillov-Reshetikhin mod-

ules, as in [Nak03a]. For all r P Z,

rL�q2rst �

�ΨΨΨq2r

��1lim

NÑ�8χq,t

�WN,q2r�2N�1

�. (4.1.1.3)

4.1. Further results in the sl2 case 129

4.1.2 Quantum Grothendieck ring for the category O�Z

The �nite-dimensional representations of the quantum a�ne algebra Uqpsl2q have beencompletely classi�ed by Chari-Pressley [CP91]. Furthermore, this classi�cation has beenextended to the irreducible representations in the category O� by Hernandez-Leclerc[HL16b]. Let us recall this result. We start with some de�nitions.

De�nition 4.1.1. A q-set is a subset of C� of the form

taq2r1 , aq2r1�2, aq2r1�4, . . . , aq2r2u, (4.1.2.1)

for some a P C�, and �8 ¤ r1 ¤ r2 ¤ �8. The Kirillov-Reshetikhin (KR)-modules Wk,a

and Wl,b are said to be in special position if the union of the q-sets ta, aq2, . . . , aq2k�2u andtb, bq2, . . . , bq2l�2u is a q-set which contains them both properly. The KR-module Wk,a

and the positive prefundamental representation L�b are said to be in special position ifthe union of the q-sets ta, aq2, . . . , aq2k�2u and tbq, bq3, bq5, . . .u is a q-set which containsthem both properly. The KR-module Wk,a and the negative prefundamental representa-tion L�b are said to be in special position if the union of the q-sets ta, aq2, . . . , aq2k�2u andt. . . , bq�5, bq�3, bq�1u is a q-set which contains them both properly. Two positive prefunda-mental representations are never in special position, nor are two negative prefundamentalrepresentations.

Two representations are in general position if they are not in special position.

Theorem 4.1.2. [HL16b, Theorem 7.9] The prime simple objects in the categories O�are the positive (resp. negative) prefundamental representations and the the KR-module(up to invertibles). Any simple representation in the categories O� can be factorized in aunique way as a tensor product of positive (resp. negative) prefundamental representationsand KR-modules (up to permutation of the factors and up to invertibles). Moreover, sucha tensor product is simple if and only if all its factors are pairwise in general position.

From Theorem 4.1.2 we deduce naturally that the Grothendieck ring of the categoryO�Z is algebraically generated by the classes of the KR-modules in O�Z and the negativeprefundamental representations rL�

q2rs, for r P Z. Moreover, the classes of the KR-modules

are algebraically generated by the classes of the fundamental modules. Thus, we de�nethe quantum Grothendieck ring for the category O�Z as follows.

De�nition 4.1.3. Let K�t be the Ert�1{2s-subalgebra of TtbE generated by the elements

rLpYq2r�1qst and rL�q2sst, for k ¥ 0 and r, s P Z. Then

KtpO�Z q :� K�t bZrt�1{2sE , (4.1.2.2)

where the tensor product is completed to allow countable sums, as for Tt.

Inside this quantum Grothendieck ring, one can write a new quantum Baxter relation.

Proposition 4.1.4. For all r P Z, the following relation is satis�ed in KtpO�Z q:

rLpYq2r�1qst � rLq2r st � t1{2rω1srL�q2r�2st � t�1{2r�ω1srL

�q2r�2st. (4.1.2.3)


Proof. For all r P Z,

rLpYq2r�1qst � rL�q2rst �

�Yq2r�1

�1�A�1

q2r�2

��rΨΨΨ�1

q2rs�

1�A�1q2r

�A�1q2rA�1q2r�2 � � � �

� t1{2rω1srΨΨΨ

�1q2r�2s

�1� t�1A�1

q2r

�1�A�1

q2r�2 �A�1q2r�2A

�1q2r�4 � � � �

�A�1

q2r�2 �A�1q2r�2A

�1q2r

�A�1q2r�2A

�1q2rA�1q2r�2 � � � �

� t1{2rω1srL

�q2r�2st � t�1{2r�ω1srL

�q2r�2st.

Remark 4.1.5. The quantum Grothendieck ring KtpO�Z q is de�ned in Chapter 2 as a quan-tum cluster algebra. However, we can see that both de�nitions account to the same ring.Let K�

t be the Ert�1{2s-subalgebra of Tt b E generated by the elements rLpYq2r�1qst andrL�

q2sst, for k ¥ 0 and r, s P Z. From Chapter 2, the classes of the fundamental modules

rLpYq2r�1qst are the cluster variables obtained at vertex p1, 2rq after mutation in directionp1, 2rq, thus

Ert�1{2s�rLpYq2r�1qst, rL

�q2sst | r, s P Z

�(4.1.2.4)

is the lower bound of the cluster algebra AtpΓ,Λq. Besides, this cluster algebra is acyclic,as the quiver Γ is an in�nite line:

� � � // p1,�4q // p1,�2q // p1, 0q // p1, 2q // p1, 4q // � � � .

Thus, by [BZ05, Theorem 7.5], the cluster algebra AtpΓ,Λq is equal to its lower boundK�t , and

KtpO�Z q :� K�t bZrt�1{2sE . (4.1.2.5)

De�nition 4.1.6. Let KtpOZq be the Ert�1{2s-subalgebra of Tt generated by all elementsin KtpO�Z q and KtpO�Z q.

4.1.3 pq, tq-characters of simple modules

Recall the following result, valid for all simple �nite-dimensional Lie algebra g of simply-laced type.

Proposition 4.1.7. Let L and L1 be two simple �nite-dimensional Uqpgq-modules. If thetensor product Lb L1 � L2 is simple then there exists k P Z{2 such that

rLst � rL1st � t2krL1st � rLst � tkrL2st. (4.1.3.1)

When the pq, tq-characters of two representation satisfy an equality such as (4.1.3.1),we say that they t-commute. This result was stated in this form in [HL15, Corollary 5.5],but is actually a direct consequence of the positivity of the structure constants proven in[VV03].

In this section, we will generalize Proposition 4.1.7 to the categories O�Z .First, let us state a lemma, also satis�ed for g of any simply-laced type.

Lemma 4.1.8. For all pi, rq P I, for all pj, sq � pi, rq, rΨΨΨi,rs and A�1j,s commute, and

rΨΨΨi,rs �A�1i,r � t�2A�1

i,r � rΨΨΨi,rs. (4.1.3.2)


Proof. For all pj, sq P I, the image of A�1j,s by the map J is

J pA�1j,s q � J

��Y �1j,s�1Y

�1j,s�1

¹k�j

Yk,s

� � rαisrΨΨΨj,s�2srΨΨΨ�1j,s�2s

¹k�j

rΨΨΨk,s�1srΨΨΨ�1k,s�1s.

Thus, for all pi, rq, pj, sqq P I2, rΨΨΨi,rs �A�1j,s � tαA�1

j,s � rΨΨΨi,rs, with

α � Fijps� r � 2q � Fijps� r � 2q �¸k�j

pFikps� r � 1q � Fikps� r � 1qq ,

� �Cijps� r � 1q �Cijps� r � 1q �¸k�j

Cikps� rq,

� �2δi,j by Lemma pA.2.6q.

We deduce the following.

Lemma 4.1.9. If a KR-module and a positive (resp. negative) prefundamental represen-tation are in general position, then their pq, tq-characters t-commute. Two positive (resp.negative) prefundamental representations always t-commute.

Proof. The pq, tq-characters of positive and negative prefundamental representations beingof very di�erent forms, both cases will be treated separately.• For the category O�: for all r, s P Z, as rLq2r st � rΨΨΨq2ssχ1, the pq, tq-characters rLq2r stand rLq2sst necessarily t-commute.The KR-module Wk,q2r�1 and the positive prefundamental representation L�

q2sare in

special position if and only if r s ¤ r � k. The commutative monomials appearingin rWk,q2r�1st are of the form

Yq2r�1Yq2r�3 � � �Yq2r�2k�1

m�1¹l�0

A�1q2r�2k�2l , (4.1.3.3)

with 0 ¤ m ¤ k.If r � k s or s ¤ r, rΨΨΨq2ss commutes with

±m�1l�0 A�1

q2r�2k�2l for all 0 ¤ m ¤ k by

Lemma 4.1.8. Hence, rWk,q2r�1st and rL�q2sst t-commute.

• For the category O�: The KR-module Wk,q2r�1 and the negative prefundamental rep-resentation L�

q2sare in special position if and only if r ¤ s r � k. The commutative

monomials of rL�q2sst are of the form

rΨΨΨ�1q2ss

q�1¹p�0

A�1q2s�2p , (4.1.3.4)

with 0 ¤ q.If s r, all monomials

±m�1l�0 A�1

q2r�2�2l commute with all monomials±q�1p�0A

�1q2s�2p ,

for 0 ¤ m ¤ k and 0 ¤ q. Moreover, the monomials±m�1l�0 A�1

q2r�2�2l commute with

the highest `-weight rΨΨΨ�1q2ss and all monomials

±q�1p�0A

�1q2s�2p commute with the highest

`-weight of Wk,q2r�1 . Hence, the pq, tq-characters rWk,q2r�1st and rL�q2sst t-commute.


If r�k ¤ s, we use the characterization of the normalized rL�q2sst as a limit of normalized

pq, tq-characters of KR-modules of (4.1.1.3) :

rL�q2sst � rΨΨΨ�1

q2ss limNÑ�8

χq,tpWN,q2s�2N�1q.

For N large enough, the q-set

tq2r�1, q2r�3, . . . , q2r�2k�1u

is included in the q-set

tq2s�2N�1, q2s�2N�3, . . . , q2s�1u.

These q-sets are in general position and thus the tensor product of the KR-modulesWk,q2r�1 and WN,q2s�2N�1 is simple by Theorem 4.1.2, and their pq, tq-characters t-commute by Proposition 4.1.7.Now let α P Z be the t-commutation factor between the highest `-weights of Wk,q2r�1

and L�q2s

mk,q2r�1 � rΨΨΨ�1q2ss � tαrΨΨΨ�1

q2ss �mk,q2r�1 .

And form the di�erence

∆ � rWk,q2r�1st � rL�q2sst � tαrL�

q2sst � rWk,q2r�1st,

� tα{2mk,q2r�1rΨΨΨ�1q2ssA,

where A is a formal power series in the variables A�1q2p

, with coe�cients in Zrt�1{2s. Fixa degree d in those variables and let Ad be the component of A of degree d.Proceed similarly with the pq, tq-characters rWk,q2r�1st and rWN,q2s�2N�1st; let αN bethe t-commutation factor of their highest `-weights, and form the di�erence

∆1 � rWk,q2r�1st � rWN,q2s�2N�1st � tαN rWN,q2s�2N�1st � rWk,q2r�1st,

� tαN {2mk,q2r�1mN,q2s�2N�1B.

For N large enough, the component Bd of degree d of B is equal to A. However, for Nlarge enough, the pq, tq-characters t-commute and ∆1 � 0. Thus, Ad � 0 for all degreed and the pq, tq-characters rWk,q2r�1st and rL

�q2sst t-commute.

Moreover, for all r, s P Z, the pq, tq-characters of the negative prefundamental represen-tations L�

q2rand L�

q2st-commute, and this is shown exactly as above.

For all positive (resp. negative) `-weight ΨΨΨ, from the result of Theorem 4.1.2, writeLpΨΨΨq as a tensor product of KR-modules and positive (resp. negative) `-weights in pairwisegeneral position, with some possible weight:

LpΨΨΨq � rλs bpâ

m�1

Wkm,q2rm�1 bqâl�1

L�q2sl

.

And the pq, tq-character of LpΨΨΨq is de�ned as

rLpΨΨΨqst :� rλs

p¹m�1

rWkm,q2rm�1st

q¹l�1

rL�q2sl

st, (4.1.3.5)


where we take the commutative product, which is well-de�ned because the factors pairwiset-commute from Lemma 4.1.9. Naturally,

rLpΨΨΨqst � rLpΨΨΨqst.

As a direct consequence, one has the following.

Proposition 4.1.10. Let L and L1 be simple modules in the category O� (resp. O�).If their tensor product L b L1 � L2 is simple then their pq, tq-characters rLst and rL1stt-commute.

4.1.4 Quantum Wronskian relation

The quantum Baxter relations take place in the quantum Grothendieck rings KtpO�Z q.However, now that we have de�ned a full quantum Grothendieck ringKtpOZq, we can writerelations which involve both positive and negative prefundamental representations. Amongsuch relations, one can �nd as important examples the quantum Wronskian relations (forg � sl2) or more generally the QQ-systems (for more general g). The quantum Wronskianrelation was described by Bazhanov-Lukyanov-Zamolodchikov in [BLZ99, BLZ03]. It en-codes a famous system of relations called the Bethe Ansatz equations, which are importantin the study of quantum integrable systems (see Introduction). In [KSZ18, Section 3.2],the link between the quantum Wronskian and the Bethe Ansatz was studied in more de-tails. In [FH18], the quantum Wronskian relation was explicitly written as a relation inthe Grothendieck ring K0pOq. For all a P C�, one has

rL�a s � rL�a s � r�α1srL

�aq2s � rL�

aq�2s � χ1. (4.1.4.1)

In [FH18], Frenkel and Hernandez wrote generalizations of the quantum Wronskianrelation, in the form of QQ-relations. The QQ-relations were previously introduced byMasoero, Raimondo, and Valeri in [MRV16, MRV17], in the framework of a�ne opers,which are di�erential operators in one variable associated to the a�ne Kac�Moody algebrathat is Langlands dual to g, and were found to have links with the description of thespectra of quantum g-KdV Hamiltonians. Frenkel and Hernandez proved analogs of thoserelations in the Grothendieck ring K0pOq. For simply-laced types, they can be written asfollows.

For all i P I and a P C�, consider the simple Uqpgq-module

Xi,a :� LpΨΨΨ�1i,a

¹j�i

ΨΨΨj,aqq. (4.1.4.2)

From then, let Q and Q be the following elements of K0pOq, for all i P I and a P C�,

Qi,a � rL�i,as,

Qi,a � rXi,aq�2sχ�1i

��αi

2

�.

Then, from [FH18, Theorem 3.2] in K0pOq,�αi2

�Qi,aq�1Qi,aq �

��αi

2

�Qi,aqQi,aq�1 �

¹j�i

Qj,a. (4.1.4.3)


One note that when g � sl2, Xa � L�a , thus Qa � rL�aq�2sχ

�11

��α1

2

�, and relation (4.1.4.3)

reduces indeed to relation (4.1.4.1) in this case.We have a quantum analog of relation (4.1.4.1) in our quantum Grothendieck ring

KtpOq.

Proposition 4.1.11. For all r P Z,

rL�q2rst � rL

�q2rst � t�1r�α1srL

�q2r�2st � rL

�q2r�2st � χ1. (4.1.4.4)

Proof. For all r P Z,

rL�q2rst � rL

�q2rst �

�rΨΨΨq2r sχ1

��rΨΨΨ�1

q2rsp1�A�1

q2r�A�1

q2rA�1q2r�2 �A�1

q2rA�1q2r�2A

�1q2r�4 � � � � q

� χ1 � t�1A�1

q2r

�1�A�1

q2r�2 �A�1q2r�2A

�1q2r�4 � � � �

χ1,

by Lemma 4.1.8. And

rL�q2r�2st � rL

�q2r�2st �

�rΨΨΨq2r�2sχ1

��rΨΨΨ�1

q2r�2sp1�A�1q2r�2 �A�1

q2r�2A�1q2r�4 � � � � q

� rα1sA

�1q2r

�1�A�1

q2r�2 �A�1q2r�2A

�1q2r�4 � � � �

χ1.

ThusrL�

q2rst � rL

�q2rst � χ1 � t�1r�α1srL

�q2r�2st � rL

�q2r�2st.

4.1.5 pq, tq-characters of asymptotical standard modules

In Chapter 1, we de�ned asymptotical standard modules as Uqpbq-modules. Theseare meant to play the role of the standard modules studied by Nakajima [Nak04] andVaragnolo-Vasserot [VV02b] in the context of the category O�. In [Nak04], Nakajimaused this standard modules to produce an algorithm (a type of "Kazhdan-Lusztig algo-rithm") to compute the pq, tq-characters of all the simple representations of the quantuma�ne algebra.

The aim of the construction of these asymptotical standard modules was to obtaina similar algorithm to compute pq, tq-characters of all irreducible Uqpbq-modules in thecategory O�. Of course, for g � sl2, the pq, tq-characters of the simple modules can beobtained via the classi�cation Theorem 4.1.2 and the knowledge of the pq, tq-characters ofthe KR-modules and the negative prefundamental modules, as we did in (4.1.3.5). Butwith the goal in mind to generalize these results to other types than sl2, it is interestingto complete the picture by producing pq, tq-characters for those asymptotical standardmodules. Moreover, the idea behind the construction of the asymptotical standard moduleswas the limit character formula obtained in Chapter 1, Section 1.2. Thus, it is this formulawe need to t-deform.

Recall that, for all r P Z,

χqpMpΨΨΨ�1q2rqq � rΨΨΨ�1

q2rs

¸JPPf pNq

¹jPJ

A�1q2r�2j , (4.1.5.1)

where Pf pNq denotes the �nite subsets of N.

4.2. Leads for other types 135

This q-character is multiplicity-free, hence there is only one way to de�ne a bar-invariantpq, tq-character with positive coe�cients which specializes at χqpMpΨΨΨ�1

q2rqq at t � 1. The

main issue is to de�ne a quantum torus containing such a formula.Let AE be the non-commutative ring of formal series in variables A�1

q2r, with coe�cients

in Ert�1{2s,AE :� Ert�1{2srrA�1

q2rssrPZ, (4.1.5.2)

where the t-commutation relations are de�ned by:

A�1q2r

�A�1q2s

� tαpr,sqA�1q2s

�A�1q2r, αpr, sq �

$&%�2 if s � r � 1,2 if s � r � 1,0 otherwise.

(4.1.5.3)

The inclusion of quantum tori J satis�es

J pA�1q2rq � z2r�2z

�12r�2. (4.1.5.4)

Thus, the non-commutative ring of polynomials Ert�1{2srA�1q2rsrPZ is included, via the in-

jective map J , in the Ert�1{2s-algebra Tt b E .Finally, let

Tt :� pTt b Eq bErt�1{2srA�1

q2rs AE . (4.1.5.5)

One can de�ne commutative monomials on Tt exactly as in Chapter 2 Section 2.3.For all r P Z, let

rMpΨΨΨ�1q2rqst :� rΨΨΨ�1

q2rs

¸JPPf pNq

¹jPJ

A�1q2r�2j P Tt, (4.1.5.6)

written on the basis of the commutative monomials.More generally, for all `-weights ΨΨΨ such that LpΨΨΨq is in O�Z . Write ΨΨΨ as a �nite product

rλs �m�¹rPZrΨΨΨ�vr

q2rs, (4.1.5.7)

where λ is a weight, m is a monomial in the variables Yq2s , and all but �nitely many of thevr P N are 0, and the number of Yq2r�1 in the monomial m is minimal for such a writing.

Then the pq, tq-character of the standard module MpΨΨΨq, as de�ned in Chapter 1 is

rMpΨΨΨqst :� tαrλsrMpmqst �ÐÝ� rPZrMpΨΨΨ�1

q2rqs�vrt , (4.1.5.8)

where a P 12Z is �xed such that when the right-hand term is written on the basis of the

commutative monomials, the coe�cient of the highest `-weight ΨΨΨ is 1.Note that contrary to (4.1.3.5), the terms in this product do not necessarily t-commute,

so an order has to be chosen, and we �x a decreasing order on r P Z.

4.2 Leads for other types

We have hope to be able to extend some of these results outside of the case whereg � sl2. However, as this an ongoing project, we only present here ideas for these general-izations.

The �rst level of generalization is to g of any simply-laced type.


4.2.1 The (q,t)-characters of the negative prefundamental representa-tions

Here, the simple Lie algebra g is supposed to be of any simply-laced type.Using what we did for sl2 in Section 4.1.1, one could de�ne the normalized pq, tq-

character of the negative prefundamental representation L�i,qr as the limit of normalizedpq, tq-characters of KR-modules:

rL�i,qr st :� pΨΨΨi,qrq�1 lim

NÑ�8χq,t

�W

piq

N,qr�2N�1

P Tt. (4.2.1.1)

Indeed, Nakajima proved in [Nak03a] the existence of this limit. The advantage of thisde�nition is that one can extend properties known in the �nite-dimensional context to thatof the category O�Z .

For example, when g is simply-laced, Nakajima proved in [Nak04] that the coe�cientsof the pq, tq-characters of the simple modules were non-negative. Thus, as each term in thepq, tq-character of rL�i,qr st corresponds to a term in a pq, tq-character of a KR-modules, wededuce that

rL�i,qr st P pΨΨΨi,qrq�1

�1�A�1

i,qrNrt�1{2srrA�1

j,qsss. (4.2.1.2)

From there, one can hope that, similarly as Lemma 4.1.9, the pq, tq-characters of twonegative prefundamental representations are t-commuting elements of the quantum torusTt.

Of course, for other types than sl2, there is no classi�cation theorem as complete asTheorem 4.1.2. However, it is known that a tensor product of two negative prefundamentalrepresentations is simple, so this conjectural result is a requirement for (4.2.1.2) to be anacceptable de�nition of the pq, tq-characters.

4.2.2 Quantum Grothendieck ring for the category O�Z

Our lead for the de�nition of the quantum Grothendieck ring KtpO�Z q is based onour approach for the quantum Grothendieck ring KtpO�Z q in Chapter 2. From [HL16b,Theorem 5.17], we know that the Grothendieck rings K0pO�Z q and K0pO�Z q are isomorphicas E-algebras,

D : K0pO�Z q�ÝÑ K0pO�Z q, (4.2.2.1)

such thatDprL�i,qr sq � rL�

i,q�rs. (4.2.2.2)

Hence, it would make sense for their quantum Grothendieck rings to be isomorphic asErt�1{2s-algebras. Thus,

KtpO�Z q � AtpΓ,Λq, (4.2.2.3)

as Ert�1{2s-algebras. This identi�cation provides the structure of the Ert�1{2s-algebraKtpO�Z q, and the results of Chapter 3 show that

KtpCZq � KtpO�Z q. (4.2.2.4)

However, the identi�cation (4.2.2.3) does not provide information about the pq, tq-characters, as the morphism D is not compatible with the q-character morphism (see[HL16b, Remark 5.15]).

A more interesting angle to study this quantum Grothendieck ring would be to takethe pq, tq-characters of negative prefundamental representations (4.2.1.2) as initial seed of

4.2. Leads for other types 137

a quantum cluster algebra built on the quiver Γ, after showing that these elements of thequantum torus t-commute. Moreover, their t-commutations relations are �xed by theirhighest `-weights, ΨΨΨ�1

i,qr , and they are thus the same than between the z�1i,�r. This would

give a new identi�cationKtpO�Z q � AtpΓ,Λq, (4.2.2.5)

which takes into account pq, tq-characters.

Remark 4.2.1. The di�erence of identi�cations (4.2.2.3) and (4.2.2.5) echoes the two iden-ti�cations of KtpC

�Z q in Chapter 3, by pq, tq-characters and truncated pq, tq-characters.

By performing quantum cluster mutations on the quantum cluster algebraAtpΓ,Λq, oneshould still get the inclusion of quantum Grothendieck rings (4.2.2.4) with this new iden-ti�cation. However, recognizing pq, tq-characters of �nite-dimensional modules formed byusing Laurent series in pq, tq-characters of negative prefundamental representations shouldbe as straightforward as in the positive case, although one may have to use some duality(see [HL16b, Example 5.14]).

Although this would give a new algorithm to obtain the pq, tq-characters of the funda-mental modules, it is not something that makes sense from an algorithmic point of view.Indeed, it requires knowing the pq, tq-characters of the negative prefundamental represen-tations, which are obtained as limits of pq, tq-characters of KR-modules, and are thus moredi�cult to access than the pq, tq-characters of the fundamental modules.

Once both the quantum Grothendieck ringsKtpO�Z q andKtpO�Z q have good realizationsin terms of pq, tq-characters inside the quantum torus Tt, one can de�ne the full quantumGrothendieck ring KtpOZq, as the Ert�1{2s-subalgebra of the quantum torus generated bythe quantum Grothendieck rings KtpO�Z q and KtpO�Z q.

As before, with this full quantum torus, one can write new relations involving both thepositive and the negative prefundamental representations, such as the QQ-systems.

4.2.3 Extension to non-simply-laced types

Most of the work done in this thesis (with the exception of some results in Chapter 1)is restricted to the simple Lie algebras g of simply-laced types, but there is hope we willbe able to extend it in the future.

First of all, one would need a generalization of the t-commutation function F from(2.3.2.2), satisfying a more general form of relation (I.2.2.2), while forming a compatiblepair with the quiver Γ. This would allow for the de�nition of the quantum Grothendieckring of the category O�Z exactly as in De�nition 2.4.15.

Next, we would want to extend Conjecture 2.5.2 and Conjecture 2.5.7 from Chapter 2,proved in Theorem 3.6.5 in Chapter 3. As the results from [HL16b] (there exists �nitesequences of mutations to obtain the classes of the fundamental modules in the clusteralgebra ApΓq) are not restricted to the simply-laced types, these conjectures extend ver-batim.

Now, the proof of Theorem 3.6.5 relies heavily on the quantum T -systems, a family ofrelations in KtpC q, which as of yet have only been stated in types ADE by Nakajima in[Nak03a] (see also [HL15, Proposition 5.6]), and by Hernandez-Oya in [HO19] in type B.Moreover, we have used in a few separate instances the fact that the pq, tq-characters ofsimple modules had non-negative coe�cients. As before, this result is known in simply-laced types from [Nak04] and proven partially in type B in [HO19]. If the approach fromHernandez and Oya is generalized to all non-simply-laced types, then our proof could alsobe extended to all these types.

Appendix

Appendix ACartan data and quantum Cartan data

Let us �x some notations we use in Chapters 2, 3 and 4.

A.1 Root data

Let the g be a simple Lie algebra of rank n and of type A,D or E. Let γ be the Dynkindiagram of g and let I :� t1, . . . , nu be the set of vertices of γ.

The Cartan matrix of g is the n� n matrix C such that

Ci,j �

$&%2 if i � j,

�1 if i � j ( i and j are adjacent vertices of γ ) ,0 otherwise.

Let us denote by pαiqiPI the simple roots of g, pα_i qiPI the simple coroots and pωiqiPIthe fundamental weights. We will use the usual lattices Q �

ÀiPI Zαi, Q� �

ÀiPI Nαi

and P �À

iPI Zωi. Let PQ � P b Q, endowed with the partial ordering : ω ¤ ω1 if andonly if ω1 � ω P Q�.

The Dynkin diagram of g is numbered as in [Kac90], and let a1, a2, . . . , an be the Kaclabels (a0 � 1).

Let h be the (dual) Coxeter number of g:

g An Dn E6 E7 E8

h n� 1 2n� 2 12 18 30(A.1.1)

A.2 Quantum Cartan matrix

Let z be an indeterminate.

De�nition A.2.1. The quantum Cartan matrix of g is the matrix Cpzq with entries,

Cijpzq �

$&%z � z�1 if i � j,�1 if i � j,0 otherwise.

Remark A.2.2. The evaluation Cp1q is the Cartan matrix of g. As detpCq � 0, thendetpCpzqq � 0 and we can de�ne Cpzq, the inverse of the matrix Cpzq. The entries of thematrix Cpzq belong to Qpzq.

142 Appendix A. Cartan data and quantum Cartan data

One can writeCpzq � pz � z�1q Id�A,

where A is the adjacency matrix of γ. Hence,

Cpzq ��8

m�0

pz � z�1q�m�1Am.

Therefore, we can write the entries of Cpzq as power series in z. For all i, j P I,

Cijpzq ��8

m�1

Ci,jpmqzm P Zrrzss. (A.2.1)

Example A.2.3. (i) For g � sl2, one has

C11 ��8

n�0

p�1qnz2n�1 � z � z3 � z5 � z7 � z9 � z11 � � � � (A.2.2)

(ii) For g � sl3, one has

Cii � z � z5 � z7 � z11 � z13 � � � � , 1 ¤ i ¤ 2

Cij � z2 � z4 � z8 � z10 � z14 � � � � , 1 ¤ i � j ¤ 2.

We will need the following lemma :

Lemma A.2.4. For all pi, jq P I2,

Cijpm� 1q � Cijpm� 1q �¸k�j

Cikpmq � 0, @m ¥ 1,

Cijp1q � δi,j .

Proof. By de�nition of C, one has

Cpzq � Cpzq � Id PMnpQpzqq. (A.2.3)

By writing Cpzq as a formal power series, and using the de�nition of Cpzq, we obtain, forall pi, jq P I2,

�8

m�0

��Cijpmqpzm�1 � zm�1q �¸k�j

Cikpmqzm

� � δi,j P Crrzs. (A.2.4)

Which is equivalent to

Cijpm� 1q � Cijpm� 1q �¸k�j

Cikpmq � 0, @m ¥ 1,

Cijp1q �¸k�j

Cijp0q � δi,j ,

Cijp0q � 0.

A.3. Height function 143

Let us extend the functions Cij to symmetrical functions on Z

Ci,jpmq :� Cijpmq � Cijp�mq pm P Zq, (A.2.5)

with the usual convention Cijpmq � 0 if m ¤ 0.Then Lemma A.2.4 translates as:

Cijpm� 1q �Cijpm� 1q �¸k�j

Cikpmq �

"2δi,j if m � 0,0 otherwise.

(A.2.6)

A.3 Height function

As g is simply-laced, its Dynkin diagram γ is a bipartite graph. There is partitionI � I0 \ I1 such that every edge in γ connects a vertex of I0 to a vertex of I1.

De�nition A.3.1. De�ne, for all i P I,

ξi �

"0 if i P I0

1 if i P I1(A.3.1)

The map ξ : I Ñ t0, 1u is called a height function on γ.

Remark A.3.2. In more generality, every function ξ : I Ñ Z satisfying

ξpjq � ξpiq � 1, when j � i

is a height function on γ. It de�nes an orientation of the Dynkin diagram γ:

iÑ j if ξpjq � i� 1.

Our particular choice of height function de�nes a sink-source orientation.

Example A.3.3. If g if of type D5, then γ is

and if we �x ξ1 � 0, then

ξ1 � 1, ξ3 � 1,

ξ2 � 0, ξ4 � 0.

From now on, we �x such a height function ξ.We will also use the notation:

εi :� p�1qξi P t�1u pi P Iq. (A.3.2)

A.4 In�nite quiver

Let us de�ne an in�nite quiver Γ as in [HL16a].First, let

I :�¤iPI

pi, 2Z� ξiq . (A.4.1)

Let Γ be the quiver with vertex set I and arrows

ppi, rq Ñ pj, sqq ðñ pCi,j � 0 and s � r � Ci,jq . (A.4.2)

144 Appendix A. Cartan data and quantum Cartan data

Example A.4.1. For g � sl4, on choice of I is

I � p1, 2Zq Y p2, 2Z� 1q Y p3, 2Zq,

and Γ is the following:...

... p2, 1q

vv ((

OO

...

p1, 0q))

OO

p3, 0qvv

OO

p2,�1qvv ))

OO

p1,�2q))

OO

p3,�2qvv

OO

p2,�3q

vv ((

OO

p1,�4q

OO

...

OO

p3,�4q

OO

...

OO

...

OO

De�nition A.4.2. LetI� :� I X pI � Z¤0q .

And de�ne G� to be the semi-in�nite subquiver of Γ of vertex set I�.Let

J :� pI � Zq zI . (A.4.3)

Example A.4.3. Following Example A.4.1, for g � sl4, with the same choice of heightfunction, one has

I� � p1, 2Z¤0q Y p2, 2Z¤0 � 1q Y p3, 2Z¤0q,

and G� is the following:p1, 0q

))

p3, 0qvv

p2,�1qvv ))

p1,�2q))

OO

p3,�2qvv

OO

p2,�3q

vv ((

OO

p1,�4q

OO

...

OO

p3,�4q

OO

...

OO

...

OO

Finally, we recall a useful notation from [HL16a]. For pi, rq P I�, de�ne

ki,r :��r � ξi

2. (A.4.4)

The vertex pi, rq is the ki,rth vertex in its column in G�, starting at the top.

Appendix BCluster algebras and quantum clusteralgebras

Cluster algebras were de�ned by Fomin and Zelevinsky in the 2000s in a series offundamental papers [FZ02],[FZ03a],[BFZ05] and [FZ07]. They were �rst introduced tostudy total positivity and canonical bases in quantum groups but soon applications tomany di�erent �elds of mathematics were found.

In [BZ05], Berenstein and Zelevinsky introduced natural non-commutative deforma-tions of cluster algebras called quantum cluster algebras.

In this section, we recall the de�nitions of these objects. The interested reader mayrefer fto the aforementioned papers for more details, or to reviews, such as [FZ03b].

B.1 Cluster algebras

Let m ¥ n be two positive integers and let F be the �eld of rational functions over Qin m independent commuting variables. Fix of subset ex � J1,mK of cardinal n.

De�nition B.1.1. A seed in F is a pair pxxx, Bq, where• xxx � tx1, . . . , xmu is an algebraically independent subset of F which generates F .• B � pbi,jq of B is a m � n integer matrix with rows labeled by J1,mK and columnslabeled by ex such that

1. the n� n submatrix B � pbijqi,jPex is skew-symmetrizable.

2. B has full rank n.

The matrix B is called the principal part of B, xxx � txj | j P exu � xxx is the cluster ofthe seed pxxx, Bq, ex are the exchangeable indices, and ccc � xxxrxxx is the set of frozen variables.

For all k P ex, de�ne the seed mutation in direction k as the transformation from pxxx, Bqto µkpxxx, Bq � pxxx1, B1q, with• B1 � µkpBq is the m� n matrix whose entries are given by

b1ij �

#�bij if i � k or j � k,

bij �|bik|bkj�bik|bkj |

2 otherwise.(B.1.1)

This operation is called matrix mutation in direction k. This matrix can also be ob-tained via the operation

B1 � µkpBq � EkBFk, (B.1.2)

146 Appendix B. Cluster algebras and quantum cluster algebras

where Ek is the m�m matrix with entries

pEkqij �

$&%δij if j � k,�1 if j � i � k,

maxp0,�bikq if i � j � k,(B.1.3)

and Fk is the n� n matrix with entries

pFkqij �

$&%δij if i � k,�1 if j � i � k,

maxp0, bkjq if i � k � j.(B.1.4)

• xxx1 � pxxxr txkuq Y tx1ku, where x1k P F is determined by the exchange relation

xkx1k �

¹iPr1,msbik¡0

xbiki �¹

iPr1,msbik 0

x�biki . (B.1.5)

Remark B.1.2. pxxx1, B1q is also a seed in F and the seed mutation operation is involutive:µkpxxx

1, B1q � pxxx, Bq. Thus, we have a equivalence relation: pxxx, Bq is mutation-equivalentto pxxx1, B1q, denoted by pxxx, Bq � pxxx1, B1q, if pxxx1, B1q can be obtained from pxxx, Bq by a �nitesequence of seed mutations.

Graphically, if the matrix B is skew-symmetric, it can be represented by a quiver andthe matrix mutation by a simple operation on the quiver. Fix B a skew-symmetric matrix.De�ne the quiver Q whose set of vertices is J1,mK, where the vertices corresponding to cccare usually denoted by a square 2 and called frozen vertices. For all i P J1,mK, j P ex, bijis the number of arrows from i to j (can be negative if the arrows are from j to i).

i

j

bij

In this context, the operation of matrix mutation can be translated naturally to an op-eration on the quiver Q. For k P ex, the quiver Q1 � µkpQq is obtained from Q by thefollowing operations:• For each pair of arrows iÑ k Ñ j in Q, create an arrow from i to j.• Invert all arrows adjacent to k.• Remove all 2-cycles that were possibly created.

De�nition B.1.3. Let S be a mutation-equivalence class of seeds in F . The cluster algebraApSq associated to S is the Zrccc�s-subalgebra of F generated by all the clusters of all theseeds in S.

B.2 Compatible pairs

A quantum cluster algebra is a non-commutative version of a cluster algebra. Clustervariables will not commute anymore, but, if they are in the same cluster, commute upto some power of an indeterminate t. These powers can be encoded in a skew-symmetricmatrix Λ. In order for the quantum cluster algebra to be well-de�ned, one needs to checkthat these t-commutation relations behave well with the exchange relations. This is madeexplicit via the notion of compatible pairs.

B.3. Definition of quantum cluster algebras 147

De�nition B.2.1. Let B be a m � n integer matrix, with rows labeled by J1,mK andcolumns labeled by ex. Let Λ � pλijq1¤i,j¤m be a skew-symmetric m�m integer matrix.We say that pΛ, Bq forms a compatible pair if, for all i P ex and 1 ¤ j ¤ m, we have

m

k�1

bkiλkj � δi,jdi, (B.2.1)

with pdiqiPex some integers, all positive or negative. Relation (B.2.1) is equivalent to sayingthat, up to reordering, if ex � J1, nK, the matrix BTΛ consists of two blocks, a diagonaln� n block, and a n� pm� nq zero block:��

d1

d2

. . . p0qdn

�� Fix a compatible pair pΛ, Bq and �x k P ex. De�ne, in a similar way as in (B.1.2),

Λ1 � µkpΛq :� ETk ΛEk, (B.2.2)

with Ek from (B.1.3).

Proposition B.2.2 ([BZ05]). The pair pΛ1, B1q is compatible.

We say that pΛ1, B1q is the mutation in direction k of the pair pΛ, Bq, and we will usethe notation:

µkpΛ, Bq :��µkpΛq, µkpBq

� pΛ1, B1q. (B.2.3)

Proposition B.2.3 ([BZ05]). The mutation of a compatible pair is involutive. For anycompatible pair pΛ, Bq and any mutation direction k P ex, µkpµkpΛ, Bqq � pΛ, Bq.

B.3 De�nition of quantum cluster algebras

We now introduce the last notions we need in order to de�ne quantum cluster algebras.Let t be a formal variable. Consider Zrt�1{2s, the ring of Laurent polynomials in the

variable t1{2.Recall that any skew-symmetric integer matrix Λ of size m � m determines a skew-

symmetric Z-bilinear form on Zm, which will also be denoted by Λ:

Λpei, ejq :� λi,j , @i, j P J1,mK, (B.3.1)

where tei | 1 ¤ i ¤ mu is the standard basis of Zm.

De�nition B.3.1. The quantum torus T � pT pΛq, �q associated with the skew-symmetricbilinear form Λ is the Zrt�1{2s-algebra generated by the tXe | e P Zmu, together with thet-commuting relations:

Xe �Xf � tΛpe,fq{2Xe�f � tΛpe,fqXf �Xe, @e, f P Zm. (B.3.2)

The quantum torus T pΛq is an Ore domain (see details in [BZ05]), thus it is containedin its skew-�eld a fractions F � pF , �q. The �eld F is a Qpt1{2q-algebra.


De�nition B.3.2. A toric frame in F is a map M : Zm Ñ Fzt0u of the form

Mpcq � φpXηpcqq, @c P Zm, (B.3.3)

where φ : F Ñ F is a Qpt1{2q-algebra automorphism and η : Zm Ñ Zm is an isomorphismof Z-modules.

For any toric frame M , de�ne ΛM : Zm�Zm Ñ Z, a skew-symmetric bilinear form, by

ΛM pe, fq � Λpηpeq, ηpfqq, @e, f P Zm. (B.3.4)

Then,Mpeq �Mpfq � tΛM pe,fq{2Mpe� fq � tΛM pe,fqMpfq �Mpeq. (B.3.5)

De�nition B.3.3. A quantum seed in F is a pair pM, Bq, where• M is a toric frame in F ,• B is an m� ex integer matrix,• the pair pΛM , Bq is compatible, as in De�nition B.2.1.

Next, we need to de�ne mutations of quantum seeds. Let pM, Bq be a quantum seed,and �x k P ex. De�ne M 1 : Zm Ñ Fzt0u by setting

M 1pfq �

# °kp�0

�fkp

�tdk{2

MpEkf� pbkq if fk ¥ 0,

M 1p�fq�1 otherwise,

where Ek is the matrix from (B.1.3), bk P Zm is the kth column of B and the t-binomialcoe�cient is de�ned by�

r

p

t

:�ptr � t�rqptr�1 � t�r�1q � � � ptr�p�1q � t�r�p�1

ptp � t�pqptp�1 � t�p�1q � � � pt� t�1q, @0 ¤ p ¤ r. (B.3.6)

Recall the de�nition of the mutated matrix B1 � µkpBq from Section B.1. Then themutation in direction k of the quantum seed pM, Bq is the pair µkpM, Bq � pM 1, B1q

Proposition B.3.4 ([BZ05]). (1) The pair pM 1, B1q is a quantum seed.

(2) The mutation in direction k of the compatible pair pΛM , Bq is the pair pΛM 1 , B1q.

For a quantum seed pM, Bq, let X � tX1, . . . , Xmu be the free generating set of F ,given by Xi :� Mpeiq. Let X � tXi | i P exu, we call it the cluster of the quantum seedpM, Bq, and let C � XzX.

For all k P ex, if pM 1, B1q � µkpM, Bq, then the X 1i �M 1peiq are obtained by:

X 1i �

#Xi if i � k,

M��ek �

°bik¡0 bikei

�M

��ek �

°bik 0 bikei

if i � k.

(B.3.7)

The mutation of quantum seeds, as the mutation of compatible pairs, is an involutiveprocess: µkpM

1, B1q � pM, Bq. Thus, as before, we have an equivalence relation: twoquantum seeds pM1, B1q and pM2, B2q are mutation equivalent if pM2, B2q can be obtainedfrom pM1, B1q by a sequence of quantum seed mutations. From (B.3.7), the set C onlydepends on the mutation equivalence class of the quantum seed. The variables in C,pXiqiRex, are called the frozen variable of the mutation equivalence class.

De�nition B.3.5. Let S be a mutation equivalence class of quantum seeds in F and Cthe set of its frozen variables. The quantum cluster algebra ApSq associated with S is theZrt1{2s-subalgebra of the skew-�eld F generated by the union of all clusters in all seeds inS, together with the elements of C and their inverses.

B.4. Laurent phenomenon and quantum Laurent phenomenon 149

B.4 Laurent phenomenon and quantum Laurent phenomenon

One of the main properties of cluster algebras is the so-called Laurent Phenomenonwhich was formulated in [BFZ05]. Quantum cluster algebras present a counterpart to thisresult called the Quantum Laurent Phenomenon.

Here, we follow [BZ05, Section 5]. In order to state this result, one needs the notion ofupper cluster algebras.

Fix pM, Bq a quantum seed, and X � tX1, . . . , Xmu given by Xk � Mpekq. LetZPrX�1s denote the based quantum torus generated by the pXkq1¤k¤m; it is a Zrt�1{2s-subalgebra of F with basis tMpcq | c P Zmu, such that the ground ring ZP is the ring ofinteger Laurent polynomials in the variables t1{2 and pXjqjRex. For k P ex, let pMk, Bkqbe the quantum seed obtained from pM, Bq by mutation in direction k, and let Xk denoteits cluster, thus:

Xk � pXztXkuq Y tX 1ku.

De�ne the quantum upper cluster algebra as the ZP-subalgebra of F given by

UpM, Bq :� ZPrX�1s X£kPex

ZPrX�1k s. (B.4.1)

Theorem B.4.1. [BZ05, Theorem 5.1] The quantum upper algebra UpM, Bq depends onlyon the mutation-equivalence class of the quantum seed pM, Bq.

Thus we use the notation: UpM, Bq � UpSq, where S is the mutation-equivalence classof pM, Bq, one has:

UpSq �£

pM,BqPS

ZPrX�1s. (B.4.2)

Theorem B.4.1 has the following important corollary, which we refer to as the quantumLaurent phenomenon.

Corollary B.4.2. [BZ05, Corollary 5.2] The cluster algebra ApSq is contained in UpSq.Equivalently, ApSq is contained in the quantum torus ZPrX�1s for every quantum seedpM, Bq P S of cluster X.

B.5 Specializations of quantum cluster algebras

Fix a quantum seed pM, Bq and X its cluster. The based quantum torus ZPrX�1sspecializes naturally at t � 1, via the ring morphism:

π : ZPrX�1s Ñ ZrX�1s, (B.5.1)

such thatπpXkq � Xk, p1 ¤ k ¤ mq

πpt�1{2q � 1.

If we restrict this morphism to the quantum cluster algebra ApSq, it is not clear thatwe recover the (classical) cluster algebra ApBq. This question was tackled in a recent paperby Geiss, Leclerc and Schröer [GLS18].

Remark B.5.1. From a combinatorial point of view, the cluster algebras ApSq and ApBqare constructed on the same quiver B, and the mutations have the same e�ect on thequiver. Assume the initial seeds are �xed and identi�ed, via the morphism (B.5.1). Then,each quantum cluster variable in ApSq is identi�ed to a cluster variable in ApBq.


Proposition B.5.2. [GLS18, Lemma 3.3] The restriction of π to ApSq is surjective onApBq, and quantum cluster variables are sent to the corresponding cluster variables.

They also conjectured that the specialization at t � 1 of the quantum cluster algebrais isomorphic to the classical cluster algebra, and gave a proof under some assumptions onthe initial seed.

Nevertheless, by applying Proposition B.5.2 to di�erent seeds (while keeping the iden-ti�cation (B.5.1) of the initial seeds), one gets:

Corollary B.5.3. The evaluation morphism π sends all quantum cluster monomials tothe corresponding cluster monomials.

B.6 Positivity

Let us state a last general result on quantum cluster algebras: Davison's positivitytheorem [Dav18].

We have recalled in Section B.4 that each (quantum) cluster variable can be written as aLaurent polynomial in the initial (quantum) cluster variables (and t1{2). For classical clus-ter algebras, Fomin-Zelevinski conjectured that these Laurent polynomials have positivecoe�cients. The so-called positivity conjecture was proven by Lee-Schi�er in [LS15].

For quantum cluster algebras, the result is the following.

Theorem B.6.1. [Dav18, Theorem 2.4] Let A be a quantum cluster algebra de�ned by acompatible pair pΛ, Bq. For a mutated toric frame M 1 and a quantum cluster monomialY , let us write:

Y �¸

ePZmaept

1{2qM 1peq, (B.6.1)

with aept1{2q P Zrt�1{2s. Then the coe�cients aept1{2q have positive coe�cients.Moreover, they can be written with in the form

aept1{2q � t� degpbeptqq{2beptq, (B.6.2)

where beptq P Nrqs, i.e. each polynomial aept1{2q contains only even or odd powers of t1{2.

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