This work is an examination of various aspects of twisted vertex operator representations of Kac-Moody algebras. It starts with an introduction to Kac-Moody algebras and string theories, including a discussion of the propagation of strings on orbifolds. String interactions in a subclass of such models naturally involve twisted vertex operators. The centrally extended loop algebra realization of Kac-Moody algebras is used to explain why the inequivalent gradations of basic representations of Kac-Moody algebras g(^r) associated with g are in one-to-one correspondence with the conjugacy classes of the automorphism group of the root system, aut Ф(_g).The structure of the automorphism groups of the simple Lie algebra root systems are examined. A method of classifying the conjugacy classes of the Weyl groups is explained and then extended to cover the whole automorphism group in cases where there are additional Dynkin diagram symmetries. All possible automorphisms, a, that have the property that det (1 – σ(^r)) ≠ 0, r = 1, ….. , n - 1 where n is the order of a, are determined. Such automorphisms lead to interesting orbifold models in which some of the calculations are simplified. A thorough exposition of the twisted vertex operator representation is given including a detailed explanation of the zero-mode Hilbert space and the construction of the required cocycle operators. The relation of the vacuum degeneracy to the number of fixed subspace singularities in the orbifold construction is discussed. Explicit examples of twisted vertex operators and their associated cocycles are given. Finally it is shown how the twisted and an alternative shifted vertex operator representation of the same gradation may be identified. This is used to determine the invariant subalgebras of the gradations along with the vacuum degeneracies and conformal weights of the representations. The results of calculations for inequivalent gradations of the simply laced exceptional algebras are given.

Involutive automorphisms of complex affine Kac-Moody algebras (in particular, their conjugacy classes within the group of all automorphisms) and their compact real forms are studied, using the matrix formulation which was developed by Cornwell. The initial study of the a(1) series of affine untwisted Kac-Moody algebras is extended to include the complex affine untwisted Kac-Moody algebras B(1), C(1) and D(1). From the information obtained, explicit bases for real forms of these Kac-Moody algebras are then constructed. A scheme for naming some real forms is suggested. Further work is included which examines the involutive automorphisms and the real forms of A2(2)and the algebra G(1)2 (which is based upon an exceptional simple Lie algebra). The work involving the algebra A2(2)is part of work towards extending the matrix formulation to twisted Kac-Moody algebras. The analysis also acts as a practical test of this method, and from it we may infer different ways of using the formulation to eventually obtain a complete picture of the conjugacy classes of the involutive automorphisms of all the affine Kac-Moody algebras.

A lot of developments made during the last years show that Kac-Moody algebras play an important role in the algebraic structure of some supergravity theories. These algebras would generate infinite-dimensional symmetry groups. The possible existence of such symmetries have motivated the reformulation of these theories as non-linear sigma-models based on the Kac-Moody symmetry groups. Such models are constructed in terms of an infinite number of fields parametrizing the generators of the corresponding algebra. If these conjectured symmetries are indeed actual symmetries of certain supergravity theories, a meaningful question to elucidate will be the interpretation of this infinite tower of fields. Another substantial problem is to find the correspondence between the sigma-models, which are explicitly invariant under the conjectured symmetries, and these corresponding space-time theories. The subject of this thesis is to address these questions in certain cases. This dissertation is divided in three parts. In Part I, we first review the mathematical background on Kac-Moody algebras required to understand the results of this thesis. We then describe the investigations of the underlying symmetry structure of supergravity theories. In Part II, we focus on the bosonic sector of eleven-dimensional supergravity which would be invariant under the extended symmetry E_{11}. We study its subalgebra E_{10} and more precisely the real roots of its affine subalgebra E_9. For each positive real roots of E_9 we obtain a BPS solution of eleven-dimensional supergravity or of its exotic counterparts. All these solutions are related by U-dualities which are realized via E_9 Weyl transformations. In Part III, we study the symmetries of pure N=2 supergravity in D=4. As is known, the dimensional reduction of this model with one Killing vector is characterized by a non-linearly realized symmetry SU(2,1). We consider the BPS brane solutions of this theory preserving half of the supersymmetry and the action of SU(2,1) on them. Infinite-dimensional symmetries are also studied and we provide evidence that the theory exhibits an underlying algebraic structure described by the Lorentzian Kac-Mody group SU(2,1)^{+++}. This evidence arises from the correspondence between the bosonic space-time fields of N=2 supergravity in D=4 and a one-parameter sigma-model based on the hyperbolic group SU(2,1)^{++}. It also follows from the structure of BPS brane solutions which is neatly encoded in SU(2,1)^{+++}. As a worthy by-product of our analysis, we obtain a regular embedding of su(2,1)^{+++} in E_{11} based on brane physics./ Nombreuses sont les recherches récentes indiquant que différentes théories de gravité couplée à un certain type de champs de matière pourraient être caractérisées par des algèbres de Kac-Moody. Celles-ci généreraient des symétries infinies-dimensionnelles. L'existence possible de ces symétries a motivé la reformulation de ces théories par des actions explicitement invariantes sous les transformations du groupe de Kac-Moody. Ces actions sont construites en termes d'une infinité de champs associés à l'infinité de générateurs de l'algèbre correspondante. Si la conjecture de ces symétries est exacte, qu'en est-il de l'interprétation de l'infinité de champs? Qu'en est-il d'autre part de la correspondance entre ces actions explicitement invariantes sous les groupes de Kac-Moody et les théories d'espace-temps correspondantes? C'est autour de ces questions que gravite cette thèse. Nous nous sommes d'abord focalisés sur le secteur bosonique de la supergravité à 11 dimensions qui possèderait selon diverses études une symétrie étendue E_{11}. Nous avons étudié la sous-algèbre E_{10} et plus particulièrement les racines réelles de sa sous-algèbre affine E_9. Pour chacune de ces racines, nous avons obtenu une solution BPS de la supergravité à 11 dimensions dépendant de deux dimensions d'espace non-compactes. Cette infinité de solutions résulte de transformations de Weyl successives sur des champs dont l'interprétation physique d'espace-temps était connue. Nous avons ensuite analysé les symétries de la supergravité N=2 à 4 dimensions dont le secteur bosonique contient la gravité couplée à un champ de Maxwell. Cette théorie réduite sur un vecteur de Killing est caractérisée par la symétrie SU(2,1). Nous avons considéré les solutions de brane BPS qui préservent la moitié des supersymétries ainsi que l'action du groupe SU(2,1) sur ces solutions. Les symétries infinies-dimensionnelles ont également été étudiées. D'une part, la correspondance entre les champs d'espace-temps de la théorie N=2 et le modèle sigma basé sur le groupe hyperbolique SU(2,1)^{++} est établie. D'autre part, on montre que la structure des solutions de brane BPS est bien encodée dans SU(2,1)^{+++}. Ces considérations argumentent le fait que la supergravité N=2 possèderait une structure algébrique décrite par le groupe de Kac-Moody Lorentzien SU(2,1)^{+++}.

Upon dimensional reduction, certain supergravity theories exhibit symmetries otherwise undetected, called hidden symmetries. Not only do these symmetries teach us about the structure of the corresponding theories but moreover they provide methods to construct black hole solutions. In this thesis, we study the hidden symmetries of supergravity theories of particular interest and how these help constructing black hole solutions in dimensions D>4. We focus on three representative cases that are the symmetries appearing upon dimensional reduction to three, two and one dimensions. They are respectively described by finite, affine and hyperbolic algebras. In the first two cases, we develop and apply solution generating techniques. The first part of this thesis introduces the background concepts. We start with an introduction to black holes and other black objects in dimensions D>4. We present their subtleties, the known solutions and the conjectured ones. We insist on stationary axisymmetric solutions of vacuum and to the corresponding solution generating technique. The next chapter gives an introduction to Kac-Moody algebras. These indeed play a central role in this thesis as the symmetries appearing in three, two and one dimensions are described by three types of Kac-Moody algebras called respectively finite, affine and hyperbolic. In the second part, we first review the notion of dimensional reductions and how the hidden symmetries can be uncovered. The rest of the thesis contains three applications of these hidden symmetries. The first two concern five-dimensional minimal supergravity. Upon dimensional reduction to three dimensions, this theory exhibits a symmetry under the exceptional finite Kac-Moody algebra g2. This 14-dimensional algebra is the smallest exceptional finite Kac-Moody algebra. We use this duality to generate solutions while focussing mainly on black strings. After reduction to two dimensions, the symmetry becomes infinite-dimensional and is described by the affine extension of g2. Moreover, the two-dimensional theory is integrable, which allows us to develop another type of solution generating technique, hitherto applied only to vacuum gravity. In this work we generalize it to a case with matter fields. Finally, the notion of dimensional reduction to one dimension provides the necessary intuition for the conjecture of an algebraic formulation of M-theory, candidate to the unification of all interactions, based on the hyperbolic Kac-Moody algebra e10. In the last chapter of this thesis, we study an aspect of this correspondence, namely the e10 symmetry of massive type IIA supergravity in ten dimensions. / On sait depuis longtemps que par un processus appelé réduction dimensionnelle, on peut faire apparaître dans certaines théories de gravitation des symétries autrement indétectées. On les appelle des symétries cachées. La mise en évidence de ces symétries non seulement nous informe sur la structure de ces théories, mais de plus elle permet d'élaborer des méthodes de construction de solutions de trous noirs. Dans cette thèse, nous étudions les symétries cachées de certaines théories de supergravité en dimensions supérieures à quatre. Nous nous concentrons sur trois cas représentatifs que sont les symétries apparaissant après réduction à trois, deux et une dimensions. Dans les cas des symétries apparaissant à trois et à deux dimensions nous développons et appliquons des méthodes de construction de solutions. La première partie introduit les concepts préliminaires. Nous commençons par une introduction aux trous noirs et autres objets noirs en dimensions supérieures à quatre. Nous en présentons les subtilités, les solutions connues à ce jour et celles qui ne sont encore que conjecturées. Nous insistons particulièrement sur les solutions stationnaires à symétrie axiale dans le vide et à la méthode de construction de solutions correspondante. Le chapitre suivant présente une introduction aux algèbres de Kac-Moody. Celles-ci jouent en effet un rôle central dans cette thèse puisque les symétries apparaissant à trois, deux et une dimensions sont décrites par trois types d'algèbres de Kac-Moody appelées respectivement finies, affines et hyperboliques. Dans la deuxième partie, nous rentrons dans le vif du sujet, en commençant par rappeler le principe des réductions dimensionnelles et la mise en évidence des différents types de symétries cachées. Les trois derniers chapitres contiennent ensuite trois applications de ces symétries cachées. Dans deux d'entre eux, nous nous concentrons sur la théorie de supergravité minimale à cinq dimensions. Après réduction à trois dimensions, cette théorie présente un symétrie cachée sous le groupe G2 qui, avec quatorze dimensions, est le plus petit des groupes de Lie exceptionnels. Nous utilisons cette dualité pour engendrer des solutions, en nous focalisant essentiellement sur les solutions de cordes noires. A deux dimensions, la symétrie est décrite par l'extension affine de G2. De plus, la théorie est alors complètement intégrable. Cela conduit à un autre type de méthode de construction de solutions, jusqu'alors uniquement appliquée à des théories dans le vide. Dans ce travail, nous la généralisons donc à un cas avec champs de matière. Enfin, la notion de réduction à une dimension fournit l'intuition d'une conjecture selon laquelle la théorie M, candidate à l'unification de toutes les interactions, pourrait être reformulée en une théorie basée sur l'algèbre de Kac-Moody hyperbolique e10. Dans le dernier chapitre de cette thèse, nous étudions un aspect de cette correspondance, à savoir, la symétrie sous e10 de la supergravité massive de type IIA à dix dimensions.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 86-88).

Made available in DSpace on 2016-05-17T16:50:54Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-04-16. Added 1 bitstream(s) on 2016-05-17T16:54:21Z : No. of bitstreams: 1 000855807.pdf: 420265 bytes, checksum: 0d551ee445aae9709b18c8ce3eed7d19 (MD5) / A estrutura das álgebras de Kac-Moody e suas representações constituem o ingrediente básico para a construção de hierarquias integráveis e de suas respectivas soluções solitônicas (obtidas através do método de dressing). Diversos modelos contidos nas hierarquias mKdVeAKNS são discutidos em detalhe e uma nova classe de equações integráveis, correspondente a graus negativos pares da hierarquia mKdV, é proposta. Diferentes soluções e operadores de recursão são construídos para ambas as hierarquias / The structure of Kac-Moody algebras and its representations constitute a basic ingredient for the construction of integrable hierarchies and its soliton solutions (obtained from the dressing method). Several models within the mKdV and KNS hierarchies are discussed in detail and some new integrable equations, corresponding to negative even grades of the mKdV hierarchy, are proposed. Different solutions and recursion operators are constructed for both hierarchies

Orientador: José Francisco Gomes / Banca: Paulo Teotônio Sobrinho / Banca: Clisthenis Ponce Constantinidis / Resumo: A estrutura das álgebras de Kac-Moody e suas representações constituem o ingrediente básico para a construção de hierarquias integráveis e de suas respectivas soluções solitônicas (obtidas através do método de dressing). Diversos modelos contidos nas hierarquias mKdVeAKNS são discutidos em detalhe e uma nova classe de equações integráveis, correspondente a graus negativos pares da hierarquia mKdV, é proposta. Diferentes soluções e operadores de recursão são construídos para ambas as hierarquias / Abstract: The structure of Kac-Moody algebras and its representations constitute a basic ingredient for the construction of integrable hierarchies and its soliton solutions (obtained from the dressing method). Several models within the mKdV and KNS hierarchies are discussed in detail and some new integrable equations, corresponding to negative even grades of the mKdV hierarchy, are proposed. Different solutions and recursion operators are constructed for both hierarchies / Mestre

Orientador: Marcio Antonio de Faria Rosa / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica "Gleb Wataghin" / Made available in DSpace on 2018-07-21T14:47:17Z (GMT). No. of bitstreams: 1 Shimabukuro_AlexItiro_M.pdf: 4437200 bytes, checksum: 1542cdddcbae2d5fa2e25f10ffa05fb5 (MD5) Previous issue date: 1995 / Resumo: Não informado / Abstract: Not informed. / Mestrado / Física / Mestre em Física

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T13:08:41Z (GMT). No. of bitstreams: 1 Pereira_FernandadeAndrade_M.pdf: 1042187 bytes, checksum: adcbcf9ff1fb8219267fb3097af14c9d (MD5) Previous issue date: 2010 / Resumo: O conceito de afinização minimal, introduzido por V. Chari e A. 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 de muitos estudos por causa das suas aplicações em física-matemática. Um problema de interesse particular envolvendo afinizações minimais é o de descrever seus caracteres. Neste trabalho apresentamos algumas fórmulas para os caracteres de afinizações minimais quando a álgebra de Lie simples envolvida é do tipo E6. A principal técnica utilizada foi proposta por V. Chari e A. Moura ao se considerar o limite clássico das afinizações minimais. As fórmulas são obtidas através de um estudo sistemático de certos módulos graduados dados por geradores e relações para a correspodente álgebra de correntes. O ponto principal é demonstrar que estes módulos são isomorfos aos limites clássicos das afinizações minimais quando vistos como módulos para a álgebra de correntes / Abstract: The concept of minimal affinization, introduced by V. Chari and A. Pressley, arose from the impossibility of extending, in general, a representation of the quantum group associated to a simple Lie algebra to the quantum group associated to its loop algebra, which is always possible on the classical context. A special class of minimal affinizations is that of Kirillov-Reshetikhin modules, which are minimal affinizations of the irreducible modules having multiples of the fundamental weights as highest weights. These modules are objects of intensive studies because of their applications in mathematical physics. One problem of particular interest involving minimal affinizations is that of describing their characters. In this work we present some formulas for the characters of minimal affinizations when the simple Lie algebra involved is of type E6. The main strategy used here was proposed by V. Chari and A. Moura by considering the classical limit of minimal affinizations. The formulas are obtained through a systematic study of certain graded modules for the corresponding current algebra given by generators and relations. The main point is to prove that these modules are isomorphic to the classical limits of the minimal affinizations when the latter are regarded as modules for the current algebra / Mestrado / Algebra / Mestre em Matemática

Orientadores: Adriano Adrega de Moura, Vyjayanthi Chari / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T03:20:21Z (GMT). No. of bitstreams: 1 Bianchi_AngeloCalil_D.pdf: 2798335 bytes, checksum: bc4f008ee1c7b89870e5bf16c8ab0a67 (MD5) Previous issue date: 2012 / Resumo: Este trabalho é dedicado ao estudo de alguns assuntos da teoria de representações de certas álgebras que podem ser vistas como generalizações do conceito de álgebras de Kac-Moody am. De modo geral, o trabalho é dividido em duas partes: na primeira delas, abordamos questões sobre as representações de dimensão finita das hiperálgebras de laços torcidas e, na outra, abordamos certas propriedades homológicas da categoria de representações de uma álgebra de Lie multi-graduada, as quais são extremamente úteis para obter uma generalização do conceito de módulos de Kirillov-Reshetikhin / Abstract: This work is dedicated to the study of some aspects of the representation theory of certain algebras which can be regarded as generalizations of the concept of affine Kac- Moody algebras. The work is divided into two parts: the first is concerned with the finite-dimensional representations of twisted hyper loop algebras and the other focuses on certain homological properties of the category of representations of a multigraded Lie algebra which are useful to study a generalization of the concept of Kirillov-Reshetikhin modules / Doutorado / Matematica / Doutor em Matemática