Neste trabalho, construímos duas generalizações de uma classe de modelos discretos bidimensionais, assim chamados \"Quantum Double Models\", definidos em variedades orientáveis, compactas e sem fronteiras. Na primeira generalização, introduzimos campos de matéria aos vértices e, na segunda, às faces. Além das propriedades básicas dos modelos, estudamos como se comporta a sua ordem topológica sob a hipótese de que os estados de base são indexados por grupos Abelianos. Na primeira generalização, surge um novo fenômeno de confinamento. Como consequência, a degenerescência do estado fundamental se torna independente do grupo fundamental sobre o qual o modelo está definido, dependendo da ação do grupo de calibre e do segundo grupo de homologia. A segunda generalização pode ser vista como o dual algébrico da primeira. Nela, as mesmas propriedades de confinamento de quasipartículas está presente, mas a degenerescência do estado fundamental continua dependendo do grupo fundamental. Além disso, degenerescências adicionais aparecem, relacionadas ao homomorfismo de coação entre os grupos de matéria e de calibre. / In this work, we constructed two generalizations of a class of discrete bidimensional models, the so called Quantum Double Models, defined in orientable, compact and boundaryless manifolds. In the first generalization we introduced matter fields to the vertices and, in the second one, to the faces. Beside the basic model properties, we studied its topological order behaviour under the hypothesis that the basic states be indexed by Abelian groups. In the first generalization, appears a new phenomenon of quasiparticle confinement. As a consequence, the ground state degeneracy becomes independent of the fundamental group of the manifold on which the model is defined, depending on the action of the gauge group and on the second group of homology. The second generalization can be seen as the algebraic dual of the first one. In it, the same quasiparticle confinement properties are present, but the ground state degeneracy stay dependent on the fundamental group. Besides, additional degeneracies appear, related to a coaction homomorphism between matter and gauge groups.

La théorie de Galois a eu un impact sur les mathématiques plus important que ce qu'elle laissait présager au départ. Son résultat le plus important est le théorème de correspondance qui s'énonce de la manière suivante :si L/K est une extension de corps finie galoisienne et si G = Gal(L/K) est son groupe de Galois, alors il existe une correspondance biunivoque entre les corps intermédiaires de L/K et les sous-groupes de G. Explicitement, si G_0 est un sous-groupe de G, alors on lui associe l'ensemble des G_0-invariants L^(G_0) qui est un corps intermédiaire de L/K. D'autre part, si L_0 est un corps intermédiaire de L/K, alors on lui associe le groupe de Galois Gal(L/L_0) qui est un sous-groupe de G.Il existe de nombreuses manières de généraliser la théorie de Galois, celle que nous avons choisie utilise les algèbres de Hopf. L'idée, introduite par Chase et Sweedler, est de remplacer l'action de groupe G par une action d'algèbre de Hopf H. De telles extensions sont appelées Hopf-galoisiennes.La première étape vers la généralisation du théorème de correspondance est due à Chase et Sweedler :si L/K est une extension Hopf-galoisienne d'algèbre de Hopf H et si H_0 est une sous-algèbre de Hopf de H, alors on peut construire l'ensemble des H_0-invariants L^(H_0) qui est un corps intermédiaire de L/K. Malheureusement, contrairement au cas des extensions galoisiennes, tous les corps intermédiaires de L/K ne s'obtiennent pas de cette manière et une caractérisation des corps de la forme L^(H_0) ne semble pas être connue.Le but de cette thèse est de généraliser le théorème de correspondance pour des extensions Hopf-galoisiennes finies séparables. Dans ce but, nous avons caractérisé de manière naturelle et intrinsèque les corps intermédiaires de L/K qui peuvent s'écrire sous la forme L^(H_0) pour une certaine sous-algèbre de Hopf H_0 de H. Ainsi, nous avons pu prouver un théorème de correspondance tout à fait analogue à celui de la théorie de Galois. Nous avons également établi, à l'instar de la théorie de Galois, une variante du théorème de correspondance pour les sous-algèbres de Hopf qui sont normales.Un apport essentiel à cette thèse est fourni par les travaux de Greither et Pareigis. Ceux-ci ont associé un groupe à une extension Hopf-galoisienne finie séparable. Nous avons prouvé qu'il était possible de traduire le théorème de correspondance en termes de ce groupe. De plus, ce groupe nous a permis de construire une structure Hopf-galoisienne alternative nous aidant à mieux comprendre le théorème de correspondance.Enfin, nous avons proposé une définition d'extensions Hopf-galoisiennes pour des extensions de corps infinies séparables et avons obtenu des résultats encourageants. Cela ouvre un nouveau champ de possibilités pour des recherches futures. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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After presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order. / text

Dans ce travail, nous nous intéressons aux problèmes relatifs aux polylogarithmes et aux sommes harmoniques pris en les multiindices négatifs(au sens large, appelés dans la suite non-positifs) et en les indices mixtes. Notre étude donnera des résultats généraux sur ces objets en relation avec les algèbres de Hopf. Les techniques utilisées sont basées sur la combinatoire des séries formelles non commutatives, formes linéaires sur l’algèbre de Hopf de φ−Shuffle. Notre travail donnera aussi un processus global pour renormaliser les polyzetâs divergents. Enfin, nous appliquerons les structures mises en évidence aux systèmes dynamiques non linéaires avec entrées singulières. / In this memoir are studied the polylogarithms and the harmonic sums at non-positive (i.e. weakly negative) multi-indices. General results about these objects in relation with Hopf algebras are provided. The technics exploited here are based on the combinatorics of non commmutative generating series relative to the Hopf φ−Shuffle algebra. Our work will also propose a global process to renormalize divergent polyzetas. Finally, we will apply these ideas to non-linear dynamical systems with singular inputs.

Working over an arbitrary field F of characteristic not 2, we consider linear algebraic groups over F. We view these as functors, represented by finitely generated F-Hopf algebras, from the category of commutative, associative, F-algebras Alg_F, to the category of groups. Classical examples of these groups, such as the special linear group SL_n are split, however there are also linear algebraic groups arising from central simple F-algebras which are non-split. For example, associated to a non-split central simple F-algebra A of degree n is a non-split special linear group SL(A). It is well known that central simple algebras are twisted forms of matrix algebras. This means that over the separable closure of F, denoted F_sep, we have A⊗_F F_sep ∼= M_n(F_sep) and that there is a twisted Gal(F_sep/F)-action on M_n(F_sep) whose fixed points are A. We show that a similar method of twisted Galois descent can be used to obtain all non-split semisimple linear algebraic groups associated to central simple algebras as fixed points within their split counterparts. In particular, these techniques can be used to construct the spin and half-spin groups Spin(A, τ ) and HSpin(A, τ ) associated to a central simple F-algebra of degree 4n with orthogonal involution. Furthermore, we develop a theory of twisted Galois descent for Hopf algebras and show how the fixed points obtained this way are the representing Hopf algebras of our non-split groups. Returning to the view of group schemes as functors, we discuss how the group schemes we consider are sheaves on the étale site of Alg_F whose stalks are Chevalley groups over local, strictly Henselian F-algebras. This allows us to use the generators and relations presentation of Chevalley groups to explicitly describe group scheme morphisms. After showing how the Kronecker tensor product of matrices induces maps between simply connected groups, we give an explicit description of these maps in terms of Chevalley generators. This allows us to compute the kernel of these new maps composed with standard isogenies and thereby construct new tensor product maps between non-simply connected split groups. These new maps are Gal(F_sep/F)-morphisms and so we apply our techniques of twisted Galois descent to also obtain new tensor product morphisms between non-split groups schemes. Finally, we use one of our new split tensor product maps to compute the degree three cohomological invariants of HSpin_4n for all n.

Incidence coalgebras of categories as defined by Joni and Rota are studied, specifically in cases where a strict monoidal product on the underlying category turns the incidence coalgebra into a bialgebra or weak bialgebra. Examples of these incidence bialgebras in combinatorics are given, and include rooted trees and forests, skew shapes and bigraphs. The relations between incidence bialgebras of monoidal categories, incidence bialgebras of operads and posets, combinatorial Hopf algebras and the quiver Hopf algebras of Cibils and Rosso are discussed. Building on a result of Bergbauer and Kreimer, incidence bialgebras are seen as a useful setting in which to study aspects of combinatorial Dyson-Schwinger equations. The possibility of defining a grafting operator B+ and combinatorial DysonSchwinger equations for general incidence bialgebras is explored through the example of skew shapes.:1. Introduction 2. Background 3. Incidence bialgebras of monoidal categories and multicategories 4. Combinatorial Dyson-Schwinger equations

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.