Let g be a complex semisimple Lie algebra, Vλ an irreducible g-module with high weight λ, pI a standard parabolic subalgebra of g with Levi factor £I and nil radical nI, and H*(nI, Vλ) the cohomology group of Λn'I ⊗Vλ. We describe the decomposition of H*(nI, Vλ) into irreducible £1-modules.

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The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

A dissertação tem como tema as álgebras de Lie. Especificamente álgebras de Lie semi-simples e suas propriedades . Para encontramos essas propriedades estudamos os conceitos básicos da teoria das álgebras de Lie e suas representações. Então fizemos a classificação dessas álgebras por diagramas de Dynkin explicitando quais os possíveis diagramas que são associados a uma álgebra de Lie semi-simples. Por fim, demonstramos vários resultados concernentes a essa classificação, dentre esses, o principal resultado demonstrado foi: os diagramas de Dynkin são um invariante completo das álgebras de Lie semi-simples / The dissertation has the theme Lie algebras. Specifically semi-simple Lie algebras and its properties. To find these properties we studied the basic concepts of the theory of Lie algebras and their representations. Then we did the classification by Dynkin diagrams of these algebras and explaining the possible diagrams that are associated with a semi-simple Lie algebra. Finally, we demonstrate several results related to this classification, among these, the main result demonstrated was: the Dynkin diagrams are a complete invariant of semi-simple Lie algebras

A dissertação tem como tema as álgebras de Lie. Especificamente álgebras de Lie semi-simples e suas propriedades . Para encontramos essas propriedades estudamos os conceitos básicos da teoria das álgebras de Lie e suas representações. Então fizemos a classificação dessas álgebras por diagramas de Dynkin explicitando quais os possíveis diagramas que são associados a uma álgebra de Lie semi-simples. Por fim, demonstramos vários resultados concernentes a essa classificação, dentre esses, o principal resultado demonstrado foi: os diagramas de Dynkin são um invariante completo das álgebras de Lie semi-simples / The dissertation has the theme Lie algebras. Specifically semi-simple Lie algebras and its properties. To find these properties we studied the basic concepts of the theory of Lie algebras and their representations. Then we did the classification by Dynkin diagrams of these algebras and explaining the possible diagrams that are associated with a semi-simple Lie algebra. Finally, we demonstrate several results related to this classification, among these, the main result demonstrated was: the Dynkin diagrams are a complete invariant of semi-simple Lie algebras

Cette thèse a pour objet l'étude des espaces classifiants pour les actions propres d'un groupe discret. La dimension géométrique propre est la plus petite dimension possible pour un tel espace (qui existe toujours). Nous montrons tout d'abord que pour un réseau dans le groupe d'isométries d'un espace symétrique de type non-compact sans facteur euclidien, la dimension géométrique propre est égale à la dimension cohomologique virtuelle. La preuve utilise le fait que si le rang réel de l'espace est supérieur ou égal à 2 et le réseau est irréductible, alors il est arithmétique. Dans ce cas, nous pouvons calculer explicitement la dimension cohomologique virtuelle à l'aide du rang rationnel. Dans un deuxième temps, nous cherchons à construire concrètement des espaces classifiants pour les actions propres de dimension minimale. Nous essayons d'adapter la construction du "rétract bien équilibré" de Soulé et Ash (pour le cas SL(n,Z)) aux groupes arithmétiques Sp(2n,Z) et Aut(SL(n,Z)). Nous montrons qu'en fait cette construction ne s'étend pas. / In this thesis we study classifying spaces for proper actions of a discrete group. The proper geometric dimension is the smallest dimension of such a space (which always exists). Firstly we prove that for a lattice in the group of isometries of a symmetric space of the non-compact type without euclidean factors, the proper geometric dimension equals the virtual cohomological dimension. The proof relies on the fact that if the space has real rank at least 2 and if the lattice is irreducible, then it is arithmetic. In this case, the virtual cohomological dimension can be explicitly computed with the rational rank. Secondly we want to construct concretely classifying spaces for proper actions of minimal dimension. We try to adapt the construction of the "well-rounded retract" of Soulé and Ash (in the case SL(n,Z)) for the arithmetic groups Sp(2n,Z) and Aut(SL(n,Z)). We show that in fact this construction does not extend.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In representation theory, having a representation of a group G is equivalent to having a kG-module. Since |G-modules which are sums of irreducible kG-modules form a very important class in the theory of modules, to know conditions for a kG-module be irreducible or completely reducible from the particularities of the field k and the group G become a very important issue, whose solution was originally presented by the German mathematician Heinrich Maschke which proved that if the order of G is not a multiple of the characteristic of the field k, then kG is completely reducible (or semisimple). From there, issues unrelated to representation theory, but that concern the semisimplicity of cross products in general are treated as Maschke-type theorem. Our goal in this dissertation is to present some versions of this theorem, starting with classic versions involving cross products for actions of groups on algebras and then versions for Hopf algebras and smash products. / Na teoria de representações de grupos, ter uma representação de um grupo G é equivalente a ter um kG-módulo. Desde que kG-módulos que são somas de kG-módulos irredutíveis formam uma classe bastante importante na teoria de módulos, conhecer condições para que um kG-módulo seja irredutível ou completamente redutível a partir das particularidades do corpo k e do grupo G passou a ser um problema bastante importante. Problema este cuja solução foi originalmente apresentada pelo matemático alemão Heinrich Maschke que provou que se a ordem do grupo G não for múltiplo da característica do corpo k, então kG é completamente redutível (ou semissimples). A partir daí, questões independentes a teoria de representações, mas que dizem respeito a semissimplicidade de produtos cruzados em geral são tratados como Teorema tipo-Maschke. Nosso objetivo neste trabalho é apresentar algumas versões deste teorema. Iniciamos com versões mais clássicas envolvendo produtos cruzados globais e parciais para em seguida estudarmos versões em álgebras de Hopf e produtos smash.

Si (π, H) est une représentation unitaire irréductible d'un groupe de Lie G, on sait lui associer son application moment Ψπ. La fermeture de l'image de Ψπ s'appelle l'ensemble moment de π. Généralement, cet ensemble est Conv(Oπ), si Oπ est l'orbite coadjointe associée à π. Mais il ne caractérise pas π : deux orbites distinctes peuvent avoir la même enveloppe convexe fermée. On peut contourner cette non séparation en considérant un surgroupe G+ de G et une application non linéaire ø de g* dans (g+)* telle que, pour les orbites générique, ø(O) est une orbite et Conv (ø(O)) caractérise O. Dans cette thèse, on montre que l'on peut choisir le couple (G+, ø), avec ø de degré ≤ 2 pour tous les groupes nilpotents de dimension ≤ 6, à une exception près, tous les groupes résolubles de dimension ≤ 4, et pour un exemple de groupe de déplacements. Ensuite, on étudie le cas des groupes G = SL(n, R). Pour ces groupes, il existe un tel couple avec ø de degré n, mais il n'en existe pas avec ø de degré 2 si n>2, il n'en existe pas avec ø de degré 3 si n=4. Enfin, on montre que l'application moment Ψπ est celle d'une action fortement hamiltonienne de G sur la variété de Fréchet symplectique PH∞. On construit un foncteur qui associe à tout G un surgroupe de Lie Fréchet G̃, de dimension infinie et, à tout π de G, une action π̃ fortement hamiltonienne, dont l'ensemble moment caractérise π / To a unitary irreducible representation (π,H) of a Lie group G, is associated a moment map Ψπ. The closure of the range of Ψπ is the moment set of π. Generally, this set is Conv(Oπ), if Oπ is the corresponding coadjoint orbit. Unfortunately, it does not characterize π : 2 distincts orbits can have the same closed convex hull. We can overpass this di culty, by considering an overgroup G+ for G and a non linear map ø from g* into (g+)* such that, for generic orbits, ø(O) is an orbit and Conv( ø(O)) characterizes O. In the present thesis, we show that we can choose the pair (G+,ø), with deg ø ≤2 for all the nilpotent groups with dimension ≤6, except one, for all solvable groups with diemnsion ≤4, and for an example of motion group. Then we study the G=SL(n,R) case. For these groups, there exists ø with deg ø =n, if n>2, there is no such ø with deg ø=2, if n=4, there is no such ø with deg ø=3. Finally, we show that the moment map Ψπ is coming from a stronly Hamiltonian G-action on the Frécht symplectic manifold PH∞. We build a functor, which associates to each G an infi nite diemnsional Fréchet-Lie overgroup G̃,and, to each π a strongly Hamiltonian action, whose moment set characterizes π

La position centrale occupée par les Théories Quantiques des Champs Topologiques (TQFTs) dans l’étude de la topologie en basse dimension est due à leur structure extraordinairement riche, qui permet différentes interactions et applications à des questions de nature géométrique. Depuis leur première apparition, un grand effort a été mis dans l’extension des invariants quantiques de 3-variétés en TQFTs et en TQFT Étendues (ETQFTs). Cette thèse s’attaque à ce problème dans deux cadres généraux différents. Le premier est l’étude des invariants quantiques semi-simples de Witten, Reshetikhin et Turaev issus de catégories modulaires. Bien que les ETQFTs correspondantes étaient connues depuis un certain temps, une réalisation explicite basée sur la construction universelle de Blanchet, Habegger, Masbaum et Vogel apparaît ici pour la première fois. L’objectif est de tracer la route à suivre dans la deuxième partie de la thèse, où la même procédure est appliquée à une nouvelle famille d’invariants quantiques non semi-simples due à Costantino, Geer et Patureau. Ces invariants avaient déjà été étendus en TQFTs graduées par Blanchet, Costantino, Geer and Patureau, mais seulement pour une famille explicite d’exemples. Nous posons la première pierre en introduisant la définition de catégorie modulaire relative, un analogue non semi-simple aux catégories modulaires. Ensuite, nous affinons la construction universelle pour obtenir des ETQFTs graduées étendant à la fois les invariants quantiques de Costantino, Geer et Patureau et les TQFTs graduées de Blanchet, Costantino, Geer et Patureau dans ce cadre général / The central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting

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.