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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23681号 / 理博第4771号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 周, 教授 雪江 明彦, 教授 平岡 裕章 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We compute Ext groups between Soergel Bimodules associated to the infinite/finite dihedral group for a realization in characteristic 0 and show that they are free right 𝖱−modules with an explicit basis. We then give a diagrammatic presentation for the corresponding monoidal category of Ext-enhanced Soergel Bimodules. As applications, we compute reduced triply graded link homology 𝐇̅𝐇̅𝐇̅ of the connect sum of two Hopf links as an 𝖱−module and show that the Poincare series for the Hochschild homology of Soergel Bimodules of finite dihedral type categorifies Gomi's trace for finite dihedral groups.

La tesis tiene como objetivo desarrollar y profundizar algunos conceptos del álgebra homológica como los funtores derivados, así como las dimensiones homológicas que son herramientas muy importantes en este área. Después usaremos estos conceptos para demostrar detalladamente el teorema de las Sicigias de Hilbert que permite calcular la dimensión global para el anillo de po-linomios como también para el anillo de series formales bajo cierta condición. Este teorema es de gran importancia ya que actualmente ha generado el desarrollo de una variedad de áreas de estudio e investigación. / Tesis

Les algèbres de Temperley–Lieb de type B, aussi appelées algèbres de Temperley–Lieb à une frontière, sont une famille d’algèbres associatives unitaires de dimension finie généralisant les algèbres de Temperley–Lieb. Elles ont été introduites en 1992 par P.P. Martin et H. Saleur pour la résolution de modèles en mécanique statistique [MS94], mais elles ont rapidement pris de l’importance en théorie de la représentation suite aux travaux de P.P. Martin et D. Woodcock [MW00] [MW03], qui montrent qu’elles s’obtiennent comme quotient d’al- gèbres de Hecke cyclotomiques et qui observent des liens profonds avec la théorie de Lie. Ces quotients sont liés aux algèbres de Khovanov–Lauda–Rouquier (KLR) par les travaux de Brundan et Kleshchev [BK09]; c’est à l’aide des algèbres KLR et de leur formulation diagrammatique que les résultats de ce mémoire seront obtenus. Elles seront maintenant appelées algèbres blob. Ce mémoire porte sur la théorie de la représentation de certains blocs des algèbres blob. Plus précisément, nous trouvons les carquois et relations décrivant les catégories de modules des blocs réguliers en caractéristique nulle. Les résultats sont obtenus par calcul diagram- matique, en utilisant la base cellulaire construite par Plaza–Ryom-Hansen [PRH14] et les idempotents primitifs de Hazi–Martin–Parker [HMP21]. Structure du mémoire: Le premier chapitre rappelle brièvement les notions algébriques qui seront utilisées. Le deuxième chapitre présente les algèbres blob de façon algébrique et diagrammatique, puis plusieurs résultats connus sur celles-ci. Les troisième et quatrième chapitres contiennent tous les résultats originaux, c’est-à-dire le calcul du carquois et relations pour les blocs réguliers. / The Temperley–Lieb algebras of type B, also known as one-boundary Temperley–Lieb al- gebras, are a family of unitary associative algebras of finite dimension that generalize the Temperley–Lieb algebras. They were introduced in 1992 by P.P Martin and H. Saleur for solving models in statistical mechanics [MS94] but they quickly became important in rep- resentation theory following the work of P.P. Martin and D. Woodcock [MW00] [MW03], who showed that they can be realized as quotients of cyclotomic Hecke algebras and observed deep connections with Lie theory. These quotients are related to Khovanov–Lauda–Rouquier (KLR) algebras through the work of Brundan and Kleshchev [BK09]; it is with the help of KLR algebras and their diagrammatic presentation that the results of this thesis will be obtained. They will now be referred to as blob algebras. This thesis focuses on the representation theory of certain blocks of blob algebras. Specif- ically, we find the quivers and relations describing the module categories of regular blocks in characteristic zero. The results are obtained through diagrammatic calculus, using the cellular basis constructed by Plaza–Ryom-Hansen [PRH14] and the primitive idempotents of Hazi–Martin–Parker [HMP21]. Structure: The first chapter briefly recalls the algebraic concepts that will be used. The second chapter presents blob algebras in both algebraic and diagrammatic ways, along with several known results about them. The third and fourth chapters contain all the original results, namely the calculation of quivers and relations for regular blocks.

<p dir="ltr">On this thesis, we study the validity of the Blackadar-Kirchberg conjecture for C*-<br>algebras that arise as extensions of separable, nuclear, quasidiagonal C*-algebras that satisfy<br>the Universal Coefficient Theorem. More specifically, we show that the conjecture for the<br>C*-algebra in the middle has an affirmative answer if the ideal lies in a class of C*-algebras<br>that is closed under local approximations and contains all separable ASH-algebras, as well<br>as certain classes of simple, unital C*-algebras and crossed products of unital C*-algebras<br>with Z. We also investigate when discrete, amenable groups have C*-algebras of real rank<br>zero. While it is known that this happens when the group is locally finite, the converse in<br>an open problem. We show that if C*(G) has real rank zero, then all normal subgroups of<br>G that are elementary amenable and have finite Hirsch length must be locally finite.<br><br></p>

A complexidade de um módulo M, sobre uma álgebra de dimensão finita R, é a medida do crescimento da dimensão de suas sizigias. No nosso trabalho, estudamos esse conceito, nos concentrando muito mais no caso das álgebras autoinjetiva. Relacionamos esse crescimento com o comportamento da componente do carcás de Auslander-Reiten, a qual o módulo M pertence. Em particular, estudamos, com bastante cuidado, o caso em que a complexidade é 1, o que significa que a dimensão das sizigias são eventualmente constante. Surpreendentemente, o comportamento de todos os módulos numa mesma componente é muito parecido. / The complexity of a module M under a finite dimensional algebra R is the measure of the growth of its syzygies\' dimension. In our work, we study this concept concentrating on the case of the selfinjective algebras. We relate this growth with the behavior of the Auslander-Reiten component containing this module. In particular, we study, carefully, the case in which the complexity is 1. Surprisingly, the behavior of every module in the same component as M is very similar.

Submitted by Alison Vanceto (alison-vanceto@hotmail.com) on 2016-09-21T11:54:01Z No. of bitstreams: 1 DissGEMG.pdf: 900713 bytes, checksum: 6350d5da67ccdaab208cf7961f1161f6 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-21T12:02:05Z (GMT) No. of bitstreams: 1 DissGEMG.pdf: 900713 bytes, checksum: 6350d5da67ccdaab208cf7961f1161f6 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-21T12:02:16Z (GMT) No. of bitstreams: 1 DissGEMG.pdf: 900713 bytes, checksum: 6350d5da67ccdaab208cf7961f1161f6 (MD5) / Made available in DSpace on 2016-09-21T12:12:18Z (GMT). No. of bitstreams: 1 DissGEMG.pdf: 900713 bytes, checksum: 6350d5da67ccdaab208cf7961f1161f6 (MD5) Previous issue date: 2015-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The objective of this dissertation is descrive formally the irreducible representations of finitedimensional semisimple Lie algebras g over a field F algebraically closed with characteristic zero, as also get some multiplicity formulas that allow compute the dimension of the weight space in the representation and also the quantity of weight. In this regard, the newness of this work is the study of a new characters formula, recently published by Schützer [Sch12], and this based in one combinatory given only in terms of not simple positive roots of the Lie algebra. The main results of this dissertation are reviewed and clarified. / O objetivo deste trabalho é descrever formalmente as representações irredutíveis das álgebras de Lie semissimples g de dimensão finita sobre um corpo algebricamente fechado de característica zero, como também obter algumas fórmulas de multiplicidades que permitem calcular a dimensão dos espaços de peso da representação e também a quantidade de pesos. Nesse sentido, a novidade deste trabalho é o estudo de uma nova fórmula de Caracteres, recentemente encontrada por Schützer [Sch12], e que se baseia em uma combinatória dada apenas em termos das raízes positivas não simples da álgebra de Lie. Os principais resultados desse artigo são revistos e clarificados.