Ce travail de thèse se concentre sur l'approche non-perturbative canonique à la formulation d'une théorie quantique de la gravitation dans le cadre de la Gravitation quantique à boucles (LQG), répondant à deux problèmes majeurs. Dans la première partie, nous étudions la possible quantification, dans le cadre de la LQG, de la gravité en trois dimensions avec constante cosmologique et nous essayons de prendre contact avec autres approches de quantification déjà existantes dans la littérature. Dans la deuxième partie, nous nous concentrons sur une application très importante de la LQG: la définition et le comptage des états microscopiques d'un ensemble en mécanique statistique qui fournit une description de l'entropie des trous noirs. Notre analyse s'appuie fortement sur et s'étend à un traitement manifestement SU(2) invariant les travaux fondateurs de Ashtekar et al. / This thesis work concentrates on the non-perturbative canonical approach to the formulation of a quantum theory of gravity in the framework of Loop Quantum Gravity (LQG), addressing two major problems. In the first part, we investigate the possible quantization, in the context of LQG, of three dimensional gravity in the case of non-vanishing cosmological constant and try to make contact with alternative quantization approaches already existing in the literature. In the second part, we concentrate on a very important application of LQG: the definition and the counting of microstates of a statistical mechanical ensemble which provides a description and accounts for the black hole entropy. Our analysis strongly relies on and extends to a manifestly SU(2) invariant treatment the seminal work of Ashtekar et al.

The N= 4 Super Yang-Mills theory in four dimensions admits deformations and the exactly marginal deformations of its SU(3) R-symmetry sub-sector are known as Leigh-Strassler. Leigh-Strassler deformations break the N= 4 supersymmetry down to N= 1 while preserving conformal symmetry. With exactly marginal deformations only the F-terms are deformed thus Leigh-Strassler deformations only affect the superpotential in the Lagrangian. In this thesis we study the symmetry of the marginally deformed N= 4 SYM and demonstrate that its algebraic structure can be understood in terms of quasi-Hopf algebras. Quasi-Hopf algebras have a notion of twisting due to Drinfeld which makes them a natural mathematical language with which to treat deformations. Furthermore the deformation of the N= 4 SYM superpotential is automated by the definition of a suitable star product. / Thesis (PhD)--University of Pretoria, 2020. / NiTheP / Physics / PhD / Unrestricted

Recently Wulfman found great merit in Barut's idea on atomic super-multiplets, and he introduced the concept of the generalized Hamiltonian that is the Hamiltonian of all atoms. 24 Investigating the Schrodinger equation with this generalized Hamiltonian, it should be possible to relate the properties of different atoms and find the structure of the periodic chart from fundamental principles of dynamics and group theory. One can can use the same kinds of methods for relating the properties of different states of a single hydrogen atom with the aid of the degeneracy group S0 (4) and dynamical group SO (4, 2). These groups represent the symmetries of the time-independent and time-dependent Schrodinger equations with ordinary Hamiltonian.(25,26) The idea then is to apply these methods to the system defined by a generalized Hamiltonian. In chapter II of this thesis, we will consider the classification of chemical elements, in the light of the concept of the generalized Hamiltonian. We will make a group theoretical classification based on the characteristics of the outermost electrons in the central-field model of atomic ground states. We conclude that the classification group may be SO (p,q) with p+q≧, p ≧4. In chapter III of this thesis, we will review Wulfman's work briefly and consider an application of his idea to the ground state of helium making use of the group SO (4,1) xSO (4,1). We arrive at the conclusion that we can obtain physically significant configuration mixing using SO (4,1) xSO (4,1) or SO (4,2) xSO (4,2) in a manner analogous to the way in which SO (4) xSO (4) is used to determine configuration mixing in doubly excited states of helium-like systems.

In this thesis, I study the group theoretic structure of the Schrodinger equations of simple systems by making use of a new systematic method. Group theoretic analysis of Schrodinger equations have been made previously by numerous physicists. The groups found may be classified as: a) geometrical groups; b) dynamical degeneracy groups; c) dynamical groups The geometrical group arises simply from the spatial symmetry of the system. Although the geometrical groups are very useful, they are not very interesting from the physical viewpoint. On the other hand, the study of the dynamical degeneracy groups and the dynamical group is very attractive because it reflects the dynamic of the system. Extensive studies have previously been made by other authors on systems which exhibit nontrivial degeneracy (accidental degeneracy). It turns out that all the states which belong to the same energy level provide a basis for a unitary irreducible representation of some compact group, and the group itself is generated by a set of constants of the motion. These groups are called “dynamical degeneracy groups”. Detailed discussion on degeneracy groups will be found in the paper by McIntosh alluded to above.

In this work we set up a general framework for exact computations of the associativity, commutativity and duality morphisms in a quite general class of tortile categories. The source of the categories we study is the work of Gelfand and Kazhdan, Examples of tensor categories, Invent.Mlath. 109 (l992), 595-617. They proved that, associated to the quantized enveloping algebra of any simple Lie group at a primitive prime root of unity, there is a semisimple monoidal braided category with finite number of simple objects. The prime p needs to be greater than the Coxeter number of the corresponding Lie algebra. We show that each of the Gelfand-Kazhdan categories has at least two subcategories which are tortile, and offer algorithms for computing the associativity, commutativity and duality morphisms in any of those categories. A careful choice of the bases of the simple objects and of the product of two such objects rnake the exact computations possible. The algorithms have been implemented in Mathemetica and tested for the categories A₂,p=5, A₃,p=7, A₄.p=7, C₂,p=7, and G₂,p=11. This work was supported by the Center for Mathematical Computations through NSF grant DMS-9207973. / Ph. D.

Dans cette thèse, on étudie les propriétés combinatoires, algébriques et analytiques de certains groupes quantiques compacts libres. on prouve au chapitre 2 que les duaux des groupes quantiques de réflexions complexes possèdent, dans la plus part des cas, la propriété d'approximation de Haagerup. au chapitre 3, on décrit les règles de fusion du produit en, couronne libre d'un groupe discret par le groupe quantique des permutations. Pour cela on détermine les espaces d'entrelaceurs entre certaines coreprésentation "basiques" de ces produits en couronnes libres en termes de partitions non croisées décorées par les éléments du groupe. On peut alors identifier les coreprésentations irréductibles et décrire les règles de fusion. On propose ensuite plusieurs applications de ce résultat. On démontre premièrement que les C*-algèbres réduites de ces produits en couronnes libres sont sans la plupart des cas simples et à trace unique. Puis on prouve que les algèbres se von Neumann associées sont des facteurs de type II et que ces facteurs sont pleins. On étend finalement le résultat du chapitre 2, aux produits en couronnes libres des groupes finis par le groupe quantique de permutations. / In this thesis, we study the combinatorial and operator algebraic properties of certain free compact quantum groups. We prove in chapter 2 that the duals of the quantum reflexion groups have, in most cases, the Haagerup property. In chapter 3, we describe the fusion rules of the free wreath product of a discrete group by the quantum permutation group. To do this, we describe the interrwinner spaces berween certain “basic” corepresentations of these free wreath products in terms of non-crossing partitions decorated by the elements of the group . This provides a whole new class of compact quantum groups whose fusions rules are explicitly computed. We give several applications of this result.We prove that, in most cases, the reduced C*-algebras associates with these free wreath products are simple with unique trace. We also prove that the associated II 1 factors are full. To conclude, we extend the result of chapter 2 to the free wreath products of finite groups by the quantum permutation group.

Dans cette thèse on définit et étudie le produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes, en généralisant la notion de produit en couronne libre par le groupe quantique symétrique introduit par Bichon.Notre recherche est divisée en deux parties. Dans la première, on définit le produit en couronne libre d'un groupe discret par un groupe quantique d'automorphismes. Ensuite, on montre comment décrire les entrelaceurs de ce nouveau objet à l'aide de partitions non-croisées et décorées; à partir de cela et grâce à un résultat de Lemeux, on déduise les représentations irréductibles et les règles de fusion. Ensuite, on prouve des propriétés des algèbres d'opérateurs associées à ce groupe quantique compact, comme la simplicité de la C*-algèbre réduite et la propriété d'Haagerup de l'algèbre de von Neumann.La deuxième partie est une généralisation de la première. D'abord, on définit la notion de produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes. Après, on généralise la description des espaces des entrelaceurs donnée dans le cas discret et, en adaptant un résultat d'équivalence monoïdale de Lemeux et Tarrago, on trouve les représentations irréductibles et les règles de fusion. Ensuite, on montre des propriétés de stabilité de l'opération de produit en couronne libre. En particulier, on prouve sous quelles conditions deux produits en couronne libres sont monoïdalment équivalents ou ont le semi-anneau de fusion isomorphe. Enfin, on démontre certaines propriétés algébriques et analytiques du groupe quantique duale et des algèbres d'opérateurs associées à un produit en couronne. Comme dernier résultat, on prouve que le produit en couronne de deux groupes quantiques d'automorphismes est isomorphe à un quotient d'un particulier groupe quantique d'automorphismes. / In this thesis, we define and study the free wreath product of a compact quantum group by a quantum automorphism group and, in this way, we generalize the previous notion of free wreath product by the quantum symmetric group introduced by Bichon.Our investigation is divided into two part. In the first, we define the free wreath product of a discrete group by a quantum automorphism group. We show how to describe its intertwiners by making use of decorated noncrossing partitions and from this, thanks to a result of Lemeux, we deduce the irreducible representations and the fusion rules. Then, we prove some properties of the operator algebras associated to this compact quantum group, such as the simplicity of the reduced C*-algebra and the Haagerup property of the von Neumann algebra.The second part is a generalization of the first one. We start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group. We generalize the description of the spaces of the intertwiners obtained in the discrete case and, by adapting a monoidal equivalence result of Lemeux and Tarrago, we find the irreducible representations and the fusion rules. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.

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

This thesis contains four related papers which study different aspects of double constructions for braided Hopf algebras. The main result is a categorical action of a braided version of the Drinfeld center on a Heisenberg analogue, called the Hopf center. Moreover, an application of this action to the representation theory of rational Cherednik algebras is considered. Chapter 1 : In this chapter, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result of Lu (1994) that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras. Chapter 2 : In this chapter, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2. Chapter 3 : The universal enveloping algebra <i>U</i>(tr_n) of a Lie algebra associated to the classical Yang-Baxter equation was introduced in 2006 by Bartholdi-Enriquez-Etingof-Rains where it was shown to be Koszul. This algebra appears as the A_<i>n</i>-1 case in a general class of braided Hopf algebras in work of Bazlov-Berenstein (2009) for any complex reection group. In this chapter, we show that the algebras corresponding to the series <i>B_n</i> and <i>D_n</i>, which are again universal enveloping algebras, are Koszul. This is done by constructing a PBW-basis for the quadratic dual. We further show how results of Bazlov-Berenstein can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra representations of different rank and type for the classical series of Coxeter groups. Chapter 4 : Quantum groups can be understood as braided Drinfeld doubles over the group algebra of a lattice. The main objects of this chapter are certain braided Drinfeld doubles over the Drinfeld double of an irreducible complex reflection group. We argue that these algebras are analogues of the Drinfeld-Jimbo quantum enveloping algebras in a setting relevant for rational Cherednik algebra. This analogy manifests itself in terms of categorical actions, related to the general Drinfeld-Heisenberg double picture developed in Chapter 2, using embeddings of Bazlov and Berenstein (2009). In particular, this work provides a class of quasitriangular Hopf algebras associated to any complex reflection group which are in some cases finite-dimensional.

L’objet de cette thèse est l’étude de l’existence d’une quantification pour les sous-algèbres de Lie coisotropes des bigèbres de Lie. Une sous-algèbre de Lie coisotrope d’une bigèbre de Lie est une sous-algèbre de Lie qui est aussi un coidéal. Le problème de quantifications d’une sous-algèbre de Lie coisotrope fut posé par V. Drinfeld, lors de son étude de la quantification des espaces de Poisson homogènes G/C. Ces deux problèmes sont liés par le principe de dualité établi par N. Ciccoli et F. Gavarini. Dans cette thèse, nous cherchons à résoudre ce problème de quantification dans différents cadres. Premièrement, nous montrons qu’une quantification existe dans le cadre des bigèbres de Lie simple. Nous trouvons une quantification aux sous-algèbres de Lie coisotropes construites par M. Zambon. Puis nous établissons un lien entre ces quantifications et une classification des sous- algèbres coidéales à droite établie par I. Heckenberger et S. Kolb. Deuxièmement, nous trouvons une obstruction à la quantification universelle en utilisant une quantification d’ordre trois construite par V. Drinfeld. Nous montrons que cette obstruction disparait dans les exemples étudiés précédemment. Finalement, nous généralisons un résultat établi par P. Etingof et D. Kazhdan sur la quantification d’espaces de Poisson homogènes, liés aux sous-algèbres Lagrangiennes du double de Drinfeld. / The aim of this thesis is the study of quantization of coisotropic Lie subalgebras of Lie bialgebras.A coisotropic Lie subalgebra of a Lie bialgebra is a Lie subalgebra which is also a Lie coideal. The problem of quantization of coisotropic Lie subalgebra was set forth by V. Drinfeld, in his study of quantization of Poisson homogeneous spaces G/C. These problems are closely related to the duality principle established by N. Ciccoli and F. Gavarini.In this thesis, we search for an answer to this quantization problem in different settings. Firstly, we show that a quantization exists for simple Lie bialgebras by constructing a quantization of examples provided by M. Zambon. We then establish a link between the quantization which we constructed and a classification of subalgebras right coideals established by I. Heckenberger and S. Kolb. Secondly, we find an obstruction to the quantization in the universal setting by using a third-order quantization constructed by V. Drinfeld. We show that this obstruction vanishes in the examples studied earlier. Finally, we generalize a result of P. Etingof and D. Kazhdan on the quantization of poisson homogeneous spaces, linked to Lagrangian Lie subalgebras of Drinfeld's double.