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Hall Algebras and Localization of Categories Mikhail Gorsky

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Hall Algebras and Localization of Categories Mikhail Gorsky Hall algebras and localization of categories Mikhail Gorsky To cite this version: Mikhail Gorsky. Hall algebras and localization of categories. General Mathematics [math.GM]. Uni- versité de Paris, 2019. English. NNT : 2019UNIP7170. tel-03148279 HAL Id: tel-03148279 https://tel.archives-ouvertes.fr/tel-03148279 Submitted on 22 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université de Paris École doctorale 386 - Sciences Mathématiques de Paris centre Institut de mathématiques de Jussieu-Paris Rive gauche Algèbres de Hall et localisation des catégories Par Mikhail GORSKY THÈSE DE DOCTORAT DE MATHÉMATIQUES Dirigée par Bernhard KELLER Présentée et soutenue publiquement le 19 décembre 2019 Devant un jury composé de : M. Bernard LECLERC Université de Caen examinateur M. Patrick LE MEUR Institut de mathématiques de Jussieu-PRG examinateur M. Bernhard KELLER Institut de mathématiques de Jussieu-PRG directeur M. Olivier SCHIFFMANN Université Paris-Sud examinateur Mme Michela VARAGNOLO Université Cergy-Pontoise examinatrice 2 Institut de mathématiques de Jussieu-Paris Rive gauche. UMR 7586. Boîte courrier 247 4 place Jussieu 75 252 Paris Cedex 05 Acknowledgements I am indebted to my advisor Bernhard Keller for suggesting the topic of this thesis and many fruitful discussions and detailed explanations ever since I was a master student. I cannot overstate my deep gratitude for his support and patience. I am very grateful to my wife Olga for her constant support. When I was going through difficult times while doing the research presented in this thesis, many people helped me by saying kind words in right times. I would like to specially thank here Gustavo Jasso, Lara Bossinger and Giovanni Cerulli Irelli. For various discussions that helped me in my research, I am grateful to Olivier Schiffmann, Matthias Künzer, Hiroyuki Nakaoka, Pierre-Guy Plamondon, Ben Davison, Sven Meinhardt and many others. 3 4 Résumé Cette thèse concerne les algèbres de Hall. Nous définissons des algèbres de Hall étendues tordues de catégories triangulées et montrons que, dans certains cas, elles sont bien définies même lorsque leurs contreparties non étendues ne le sont pas. Nous montrons que chaque catégorie exacte aux équivalences faibles avec une structure supplémentaire appropriée donne naturellement lieu à une algèbre de Hall étendue tordue de sa catégorie homotopique. Nous montrons que cette construction récupère la catégorification par Bridgeland des groupes quantiques via les algèbres de Hall de complexes et sa généralisation par Lu et Peng. Nous montrons que les algèbres ainsi définies sont fonctorielles par rapport aux foncteurs exacts respectant les équivalences faibles. Cela nous permet de prouver l’invariance par basculement des algèbres de Bridgeland et de catégorifier les symétries de Lusztig des groupes quantiques. Sous des conditions de finitude appropriées, pour deux structures exactes différentes sur la même catégorie additive, l’une ayant strictement moins de conflations que l’autre, nous définissons une filtration sur l’algèbre de Hall de cette dernière dont le gradué associé est l’algèbre de Hall de la première. Cette construction généralise les filtrations de type PBW quantiques. Mots-clés Algèbres de Hall, localisation des catégories, catégorification, groupes quantiques, dégéneres- cences de PBW 5 6 Hall algebras and localization of categories Abstract This thesis concerns Hall algebras. We define twisted extended Hall algebras of triangulated categories and show that in some cases they are well-defined even when their non-extended coun- terparts are not. We show that each exact category with weak equivalences with an appropriate extra structure naturally gives rise to a twisted extended Hall algebra of its homotopy category. We prove that this construction recovers Bridgeland’s categorification of quantum groups via Hall algebras of complexes and its generalization due to Lu and Peng. We prove that the algebras thus defined are functorial under exact functors respecting weak equivalences. This allows us to prove the tilting invariance of Bridgeland’s algebras and to categorify Lusztig’s symmetries of quantum groups. Under suitable finiteness conditions, for two different exact structures on the same additive category with one having strictly less conflations than the other, we define a filtration on the Hall algebra of the latter whose associate graded is the Hall algebra of the former. This construction generalizes quantum PBW-type filtrations. Keywords Hall algebras, localization of categories, categorification, quantum groups, PBW degenerations. Table des matières Introduction 11 0.1 Contexte historique . 11 0.2 Algèbres de Hall étendues et localisation des catégories . 13 0.3 Propriété du coeur et son substitut . 16 0.4 Réduction des structures exactes et dégénérescence des algèbres de Hall . 18 0.5 Remarques supplémentaires . 19 Introduction 21 0.6 Historical context . 21 0.7 Extended Hall algebras and localization of categories . 23 0.8 Heart property and its substitute . 25 0.9 Reduction of exact structures and degeneration of Hall algebras . 27 0.10 Further remarks . 28 1 Preliminaries 31 1.1 Exact categories . 31 1.2 Exact categories with weak equivalences . 34 1.3 Frobenius and stable categories . 35 1.4 Hereditary exact model categories and Hovey triples . 35 1.5 Frobenius pairs with larger exact structure . 37 1.6 Hall algebras and Euler forms . 39 1.7 Derived Hall algebras and Euler forms . 40 1.8 Flat deformations . 41 2 Semi-derived Hall algebras of Frobenius categories 43 2.1 Semi-derived Hall algebras for stable categories . 43 2.2 Semi-derived vs. derived Hall algebras . 48 2.3 Examples . 49 3 Twisted extended Hall algebras of triangulated categories 53 3.1 Twisted extended Hall algebras of triangulated categories . 53 7 8 TABLE DES MATIÈRES 3.2 Idempotented forms . 59 3.3 Hall algebras of exact categories with weak equivalences . 61 3.3.1 Relative Grothendieck monoids and groups . 61 3.3.2 Euler forms and numerical quotients . 64 3.3.3 Middle terms . 65 3.3.4 Hall algebras H(E; W) and Hnum(E; W) . 68 3.3.5 Replacements and relation to H(E) .................... 69 3.3.6 Hereditary exact model categories . 71 3.3.7 Functoriality and heart property . 71 4 Bounded and periodic complexes 73 4.1 Periodic complexes . 73 4.1.1 Z=2−graded complexes . 74 4.1.2 Grothendieck groups of Z=2−graded complexes . 78 4.2 Semi-derived Hall algebras . 81 4.2.1 Finiteness conditions . 81 4.2.2 Euler form and quantum tori . 84 4.2.3 Module structure over the quantum torus . 86 4.2.4 Freeness over the quantum torus . 87 4.2.5 Multiplication . 88 4.2.6 Associativity . 94 4.2.7 Derived invariance . 96 4.2.8 Case with enough projectives . 100 4.2.9 Hereditary case . 102 4.2.10 Example : quiver representations . 106 4.2.11 Tilting objects and derived equivalences . 107 4.3 Z=2-graded version of SDH(E) . 108 4.3.1 Choice of a subcategory . 108 4.3.2 Grothendieck monoids and groups . 112 4.3.3 Euler forms . 114 4.3.4 Construction of the algebra SDHZ=2(E) . 115 4.3.5 Associativity and derived invariance . 118 4.3.6 Hereditary case : Bridgeland’s construction, Drinfeld doubles and quan- tum groups . 120 p 4.3.7 Reflection functors and the braid group action on U q(g) . 122 4.4 Twisted extended Hall algebras of derived categories . 124 5 Reduction of exact structures and degeneration of Hall algebras 127 5.1 Exact structures and degenerations of Hall algebras . 127 5.1.1 General result . 127 TABLE DES MATIÈRES 9 5.1.2 Categories of finite type . 128 5.2 2-periodic complexes and generalized quantum doubles . 131 10 TABLE DES MATIÈRES Introduction 0.1 Contexte historique Les algèbres de Hall fournissent l’un des premiers exemples connus de catégorification addi- tive. Elles sont apparues pour la première fois dans les travaux de Steinitz [74] et Hall [27] sur des p−groupes finis commutatifs. Plus tard, elles sont réapparues dans les travaux de Ringel [64] sur les groupes quantiques. Il a introduit la notion de l’algèbre de Hall d’une catégorie abélienne avec des espaces de dimension finie Hom - et Ext1. En tant qu’espace vectoriel, elle a une base paramétrisée par les classes d’isomorphisme des objets de la catégorie. La multiplication capte des informations sur les extensions entre les objets. On peut considérer cela comme une algèbre de fonctions constructibles sur le champ d’objets de la catégorie, avec le produit de convolution donné par les correspondances de Hecke. Ringel a construit un isomorphisme entre l’algèbre de Hall torduée de la catégorie de repré- sentations d’un carquois de Dynkin simplement lacé Q sur le corps fini Fq et la partie nilpotente du groupe quantique correspondant, spécialisé à la racine carré de q : Up (n ) !H∼ (rep (Q)): q + tw Fq Plus tard, Green [26] a généralisé ce résultat à un carquois valu Q arbitraire en montrant un iso- morphisme entre la partie nilpotente de l’algèbre enveloppante universelle quantifiée de l’algèbre de Kac-Moody correspondante et la “sous-algèbre de composition” dans H (rep (Q)) engen- tw Fq drée par les classes des objets simples. À l’aide du groupe de Grothendieck de la catégorie des représentations de carquois, il introduit une version étendue de l’algèbre de Hall qui récupère la partie de Borel du groupe quantique.
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