Orientadores: Adriano Adrega de Moura, David Hernandez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T00:38:33Z (GMT). No. of bitstreams: 1 Pereira_FernandadeAndrade_D.pdf: 1671660 bytes, checksum: 07b63630df35152e02bbb84739369993 (MD5) Previous issue date: 2014 / Resumo: O conceito de afinização minimal, introduzido por Chari e Pressley, surgiu a partir da impossibilidade de se estender, em geral, uma representação do grupo quântico associado a uma álgebra de Lie simples para o grupo quântico associado à sua álgebra de laços, o que sempre é possível no contexto clássico. Uma classe especial de afinizações minimais é a dos módulos de Kirillov-Reshetikhin, que são afinizações minimais dos módulos irredutíveis quando os pesos máximos são múltiplos dos pesos fundamentais. Esses módulos são objetos centrais no estudo de reticulados integráveis em mecânica estatística. Nas últimas duas décadas, tem sido intensa a investigação científica na direção de se entender as afinizações minimais, devido não só às suas potenciais aplicações em física-matemática, mas também por ser uma teoria muito rica por si só, além de ter forte interação com combinatória. Existe uma classificação quase completa das classes de equivalências de afinizações minimais em termos de polinômios de Drinfeld, devido a Chari e Pressley. A classificação está completa no caso em que o suporte do peso máximo não engloba um subdiagrama de tipo D4, e neste caso existe uma única classe de equivalência. No caso em que o suporte engloba um subdiagrama de tipo D4 a situação depende essencialmente se o suporte contém o vértice trivalente do diagrama ou não. Se ele o contém, a classificação também está completa e existem três classes de equivalências. Caso contrário a classificação não está completa. Neste trabalho apresentamos a classificação das classes de equivalências para álgebras de tipo D. A principal técnica empregada foi a manipulação combinatória de qcaráteres através principalmente de sua descrição via tableaux e, algumas vezes, utilizando-se o algoritmo de Frenkel-Mukhin / Abstract: The concept of minimal affinization, introduced by Chari and Pressley, arose from the impossibility to extend, in general, a representation of the quantum group associated to a simple Lie algebra for the quantum group associated to its loop algebra, which is always possible in the classical context. A special class of minimal affinizations is that of the Kirillov-Reshetikhin modules, which are minimal affinizations of the irreducible modules with highest weight multiple of a fundamental weight. These modules are central objects in the study of integrable lattices in mechanical statistics. In the past two decades it has been intense the scientific research in the direction of understanding the minimal affinizations, not only by their potential applications in mathematical physics, but also for being a very rich theory for itself, in addition to having strong interaction with combinatorics. There exists an almost complete classification of the equivalence classes of the minimal affinizations in terms of Drinfeld polynomials due to Chari and Pressley. The classification is completed in the case where the support of the highest weight does not enclose a subdiagram of type D4, and in this case there is only one equivalence class. In the case where the support encloses a subdiagram of type D4 the situation depends essentially if support contains the trivalent node of the diagram or not. If it contains, the classification is also completed and there are three equivalence classes. Otherwise the classification is not completed. In this work we present the classification of the equivalence classes for algebras of type D. The main technique used was the combinatorial manipulation of qcharacters through mainly its description via tableaux and sometimes using the Frenkel-Mukhin algorithm / Doutorado / Matematica / Doutora em Matemática

Orientadores: Adriano Adrega de Moura, Evgeny Mukhin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T22:56:39Z (GMT). No. of bitstreams: 1 Brito_MatheusBatagini_D.pdf: 1928614 bytes, checksum: bd194d09898859744dc51e0bcccd7fa1 (MD5) Previous issue date: 2015 / Resumo: Estudamos representações de dimensão finita para uma álgebra afim quantizada a partir de dois pontos de vista distintos. Na primeira parte deste trabalho estudamos o limite graduado de uma certa subclasse de representações irredutíveis. Seja V uma representação de dimensão finita para uma álgebra do tipo A e suponha que V é isomorfa ao produto tensorial de uma afinização minimal por partes cujo peso máximo é a soma de distintos pesos fundamentais por módulos de Kirillov--Reshetikhin cujos pesos máximos são o dobro de um peso fundamental. Provamos que V admite limite graduado L e que L é isomorfo a um módulo de Demazure de nível dois bem como ao produto de fusão dos limites graduados de cada um dos supramencionados fatores tensoriais de V. Provamos ainda que, se a álgebra for do tipo clássica (resp. G), o limite graduado das afinizações minimais (regulares) (resp. módulos de Kirillov--Reshetikhin) são isomorfos ao módulos CV para alguma R^+ partição descrita explicitamente. Na segunda parte provamos que um módulo para a álgebra afim quantizada do tipo B e posto n é manso se, e somente se, ele é fino. Em outras palavras, os geradores da subálgebra de Cartan afim são diagonalizáveis se, e somente se, os autoespaços generalizados associados têm dimensão um. Classificamos tais módulos e descrevemos seus respectivos q-caracteres. Em alguns casos, o q-caracter é descrito por super standard Young tableaux do tipo (2n|1) / Abstract: We study finite--dimensional representations for a quantum affine algebra from two different points of view. In the first part of this work we study the graded limit of a certain subclass of irreducible representations. Let V be a finite--dimensional representation for a quantum affine algebra of type A and assume that V is isomorphic to the tensor product of a minimal affinization by parts whose highest weight is a sum of distinct fundamental weights by Kirillov-Reshetkhin modules whose highest weights are twice a fundamental weight. We prove that V admits a graded limit L and that L is isomorphic to a level-two Demazure module as well as to the fusion product of the graded limits of each of the aforementioned tensor factors of V. We also prove that if the quantum affine algebra is of classical type (resp. type G), the graded limit of (regular) minimal affinizations (resp. Kirillov--Reshetkin modules) are isomorphic to CV-modules for some R^+ partition explicitly described. In the second part we show that a module for the quantum affine algebra of type B and rank n is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their q-characters. In some cases, the q-characters are described by super standard Young tableaux of type (2n|1) / Doutorado / Matematica / Doutor em Matemática

Given a bialgebra B we present a unifying approach to deformations of associative algebras A with a left B-module algebra structure. Special deformations of the comultiplication of B yield universal deformation formulas, i.e. define deformations of the multiplicative structure for all B-module algebras A. This allows to derive known formulas of Moyal-Vey (1949) and Coll-Gerstenhaber-Giaquinto (1989) from a more general point of view.

In this thesis we explore the geometric representation theory of shifted quantum affine algebras 𝒜^𝜇, using the critical K-theory of certain moduli spaces of infinite flags of quiver representations resembling the moduli of quasimaps to Nakajima quiver varieties. These critical K-theories become 𝒜^𝜇-modules via the so-called critical R-matrix 𝑅, which generalizes the geometric R-matrix of Maulik, Okounkov, and Smirnov. In the asymptotic limit corresponding to taking infinite instead of finite flags, singularities appear in 𝑅 and are responsible for the shift in 𝒜^𝜇. The result is a geometric construction of interesting infinite-dimensional modules in the category 𝒪 of 𝒜^𝜇, including e.g. the pre-fundamental modules previously introduced and studied algebraically by Hernandez and Jimbo. Following Nekrasov, we provide a very natural geometric definition of qq-characters for our asymptotic modules compatible with the pre-existing definition of q-characters. When 𝒜^𝜇 is the shifted quantum toroidal gl₁ algebra, we construct asymptotic modules DT_𝜇 and PT_𝜇 whose combinatorics match those of (1-legged) vertices in Donaldson--Thomas and Pandharipande--Thomas theories. Such vertices control enumerative invariants of curves in toric 3-folds, and finding relations between (equivariant, K-theoretic) DT and PT vertices with descendent insertions is a typical example of a wall-crossing problem. We prove a certain duality between our DT_𝜇 and PT_𝜇 modules which, upon taking q-/qq-characters, provides one such wall-crossing relation.

We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀_∞ ⊃ G_∞ whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G_∞ is given. All Poisson-Lie structures of coboundary type on the group G₀_∞ are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G_∞ of G_∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go_∞ of Go_∞ which is the Witt algebra. A large class of Poisson structures on the space V_λ of λ-densities on the real line is found such that V_λ becomes a homogeneous Poisson space under the action of the Poisson-Lie group G_∞. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G_∞ and G₀_∞. / Ph. D.

Ce texte étudie le groupe quantique Uξ sl(2|1) associé à la superalgèbre de Lie sl(2|1) et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de 3-variétés en utilisant la notion de trace modifiée. D'abord nous prouvons que la H catégorie CH des modules de poids nilpotents sur Uξ sl(2|1) est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus CH possède une structure relativement G-prémodulaire ce qui est une condition suffisante pour construire un invariant de 3-variétés à la Costantino-Geer-Patureau. Cet invariant est le cœur d'une 1+1+1-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d’invariants qui dispense de considérer la catégorie de H l l ses représentations. Nous montrons que le groupe quantique déroulé Uξ sl(2|1)/(e1 , f1 ) possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de 3-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de G-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la G -intégrale. / This text studies the quantum group Uξ sl(2|1) associated with the Lie superalgebra sl(2|1) and a category of finite dimensional representations. The aim is to construct the topological invariants of 3-manifolds using the notion of modified trace. We first prove that the category CH of the nilpotent weight modules over Uξ sl(2|1) is ribbon and that there exists a modified trace on its ideal of projective modules. Furthermore, CH possesses a relative G-premodular structure which is a sufficient condition to construct an invariant of 3-manifolds of Costantino-Geer-Patureau type. This invariant is the heart of a 1+1+1-TQFT (Topological Quantum Field Theory). Next Hennings proposed from a finite dimensional Hopf algebra, a construction of invariants which does not require to consider the category of its representations. We show that the unrolled H l l quantum group Uξ sl(2|1)/(e1 , f1 ) has a completion which is a topological ribbon Hopf algebra. We construct an invariant of 3-manifolds of Hennings type using this algebraic structure, a discrete Fourier transform, and the notion of G-integrals. The integral in a Hopf algebra is central in the construction of Hennings. The notion of modified trace in a category has recently been revealed to be a generalization of the integrals in a finite dimensional Hopf algebra. In a more general context of infinite dimensional Hopf algebras we prove the relation formulated between the modified trace and the G-integral.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by [Eta](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.

No description available.

Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study a subalgebra J_C of J corresponding to the subregular cell C of W . We prove a factorization theorem that allows us to compute products in J_C without inputs from Kazhdan-Lusztig theory, then discuss two applications of this result. First, we describe J_C in terms of the Coxeter diagram of (W, S) in the case (W, S) is simply- laced, and deduce more connections between the diagram and J_C in some other cases. Second, we prove that for certain specific Coxeter systems, some subalgebras of J_C are free fusion rings, thereby connecting the algebras to compact quantum groups arising in operator algebra theory.

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-18T00:34:15Z (GMT). No. of bitstreams: 1 Brito_MatheusBatagini_M.pdf: 1577922 bytes, checksum: 6e151b2dd08dad43a9a5c9840dd36514 (MD5) Previous issue date: 2011 / Resumo: Na presente dissertação estudamos dois exemplos de relacionamento da teoria de álgebras de cluster com teoria de representações. A saber, estudamos os principais resultados dos artigos [5, 26]. O primeiro é uma relação entre álgebras de cluster e representações de certos quivers com relações que também estão relacionadas com triangulações de polígonos regulares. O segundo exemplo trata de um modelo de categorificação monoidal de certas álgebras de cluster via teoria de representações de dimensão finita do grupo quântico associado a uma álgebra de Kac-Moody afim de tipo A / Abstract: In this dissertation we study two examples of interplay between the theory of cluster algebras from one side and representation theory on the other. Namely, we study the main results of the articles [5, 26]. The first one is a relation between cluster algebras of type A and representations of certain quiver with relations which are also related to triangulations of regular polygons. The second example concerns a model of monoidal categorification of certain cluster algebras via finite dimensional representation theory of the quantum group associated to an affine Kac- Moody algebra of type A / Mestrado / Algebra / Mestre em Matemática