Este trabalho é essencialmente sobre pontos racionais em curvas algébricas sobre corpos finitos ou, equivalentemente, lugares racionais em corpos de funções algébricas em uma variável sobre corpos finitos. O objetivo é a demonstração da existência de constantes aq e bq ∈ R> 0 tais que se g ≥ aq. N + bq, então existe uma curva sobre Fq de gênero g com N pontos racionais. / This work is essentially about rational points on algebraic curves over finite fields or, equivalently, rational places on algebraic function fields of one variable over finite fields. The aim is the proof of the existence of constants aq and bq ∈ R> 0 such that if g ≥ aq ∈ aq . N+bq then there exists a curve over Fq of genus g with N rational points.

Document intends to re-establish Hom-Lie algebra theory for a wider class of morphisms on the underlying coefficient algebra. A look is taken into deformed Witt and Virasoro algebras and a new direction is taken into further quasi-Hom-Lie VIrasoro-type extensions for different Witt algebras.

Extensões de Ore são anéis de polinômios, denotados por R[x o &], nos quais a variável x e os elementos de R não comutam necessariamente. Algebras de Hopf fracas são algebras que tamb em são coálgebras e satisfazem um conjunto de axiomas de compatibilidade entre essas estruturas. Neste trabalho investigamos extensões de Ore cujo anel base e uma algebra de Hopf fraca. Mais especi camente, dada uma algebra de Hopf fraca R, estudamos sob quais condições R[x o &] e uma algebra de Hopf fraca com uma estrutura que estende a estrutura de R. Sob certas hipóteses, obtemos condições necessárias e su cientes para que a extensão de Ore seja uma álgebra de Hopf fraca, obtendo assim um resultado que generaliza um teorema de Panov para o contexto de algebras de Hopf fracas. / Ore extensions are polynomial rings, denoted by R[x o &], in which the variable x and the elements of R do not commute necessarily. Weak Hopf algebras are algebras which are also coalgebras and satisfy a set of axioms of compatibility betweem these structures. In this work, we investigate Ore extensions whose base ring is a weak Hopf algebra. More speci cally, if R is a weak Hopf algebra then we study under what conditions R[xo &] is a weak Hopf algebra extending the structure of R. Under certain hypotheses, we obtain necessary and su cient conditions for an Ore extension to be a weak Hopf algebra, obtaining a result that generalizes a Panov's theorem to the setting of weak Hopf algebras.

Este trabalho tem como principal objetivo estudar extensões de aplicações multilineares, de polinômios homogêneos e de funções analíticas entre espaços de Banach. Desta maneira, nos baseamos em importantes trabalhos sobre o assunto. Inicialmente apresentamos o produto de Arens para álgebras de Banach, extensões de Aron-Berner e de Davie-Gamelin para aplicações multilineares e provamos que todas estas extensões coincidem. A partir destes resultados, apresentamos a extensão de polinômios homogêneos e o Teorema de Davie-Gamelin que afirma que, assim como no caso de aplicações multilineares, as extensões de polinômios preservam a norma e, como consequência deste teorema, apresentamos uma generalização do Teorema de Goldstine. Em seguida estudamos espaços de Banach regulares e simetricamente regulares, que são propriedades relacionadas com a unicidade de extensão e são definidas a partir do ideal de operadores lineares fracamente compactos K^w(E, F) . Finalmente apresentamos a extensão de uma função de H_b(E) para H_b(E\'\') e o resultado, de Ignacio Zalduendo, que caracteriza esta extensão em termos da continuidade fraca-estrela do operador diferencial de primeira ordem. / The main purpose of this work is to study extensions of multilinear mappings, homogeneous polynomials and analytic functions between Banach Spaces. In this way, we rely on important works on the subject. Firstly we present the Arens-product for Banach algebras, the Aron-Berner and Davie-Gamelin extensions for multilinear mappings and we prove that all these extensions are the same. From these results, we present an extension for homogeneous polynomials and the Davie-Gamelin theorem which asserts that, as in the case of multilinear mappings, the polynomial extension is norm-preserving and, as a consequence of this theorem, we present a generalization of the Goldstine theorem. After that we study regular and symmetrically regular Banach spaces which are properties related to the uniqueness of the extension and are defined in the setting of weakly compact linear operators K^w(E, F) . Lastly, we present the extension of a function of H_b(E) to one in H_b(E\'\') and the result, according to Ignacio Zalduendo, which characterizes this extension in terms of weak-star continuity of the first order differential operator.

Le travail de thèse contribue à établir des liens entre structures algébriques non-linéaires, décrites par des théories algébriques, et des structures algébriques linéaires, encodées par des algèbres sur une opérade linéaire. Pour les théories algébriques dont les modèles forment une catégorie semi-abélienne (ce qui inclut la plupart des structures intéressantes), un tel lien a été exhibé récemment par M. Hartl, au niveau des objets gradués associés à une nouvelle notion de suite centrale descendante des modèles d'une théorie donnée : il s'avère qu'ils ont une structure naturelle d'algèbre graduée sur une certaine opérade de groupes abéliens associée à la théorie. Le sujet de thèse s'inscrit dans le projet d'étendre ce lien au niveau global, c'est-à-dire d'établir des correspondances du type Mal'cev et Lazard dans le cas des groupes, à savoir entre les modèles nilpotents suffisamment radicables et les algèbres nilpotentes sur l'opérade linéaire correspondante (après tensorisation avec un sous-anneau des rationnels approprié). Ces correspondances jouent un rôle fondamental en théorie des groupes et commencent à faire leurs preuves en théorie des loops grâce au développement plus récent d'une théorie de Lie non-associative; on peut s'attendre à ce qu'il en soit de même dans un contexte plus général. Il est important de noter qu'aussi bien dans les correspondances classiques de Mal'cev et Lazard que dans leurs généralisations à des variétés multiples de loops (Moufang, Bruck, Bol etc.), le passage des algèbres (de Lie, de Mal'cev etc.) appropriées aux objets non-linéaires (groupes, voire loops) qui leur correspondent, est donné par une formule de Baker-Campbell-Hausdorff appropriée, déduite d'une étude de fonctions exponentielles et logarithmes. Dans la thèse, une nouvelle approche est développée pour construire une correspondance (en fait, une équivalence de catégories) du type Lazard entre une variété (dite aussi catégorie algébrique) 2- nilpotente 2-radicable (dans un sens approprié) C donnée et les algèbres sur une opérade symétrique unitaire linéaire et 2-nilpotente AbOp(C) dépendant de la variété, vivant dans la catégorie monoïdale des Z[1/2]-modules à gauche. L'anneau de fraction Z[1/2] apparaît car notre définition de 2-divisibilité d'objets de C se traduit par la condition de 2-divisibilité classique sur le premier terme de l'opérade. L'équivalence de type Lazard se construit grâce à la théorie des foncteurs polynomiaux (plus précisément quadratiques) et à la notion d'extension linéaire de catégories. L'idée principale est de chercher une équivalence quadratique (i.e un foncteur quadratique qui est une équivalence de catégories) entre une variété semi-abélienne 2-nilpotente 2-radicable donnée C et la catégorie des algèbres sur AbOp(C), que nous appellerons le foncteur de Lazard. La nouveauté principale de cette approche est de ne pas construire ce foncteur explicitement sur tous les objets et les morphismes, en utilisant une formule de BCH établie au préalable; mais au contraire de construire l'"ADN" du foncteur de Lazard, c'est-à-dire un ensemble de données minimales le caractérisant étudié dans ce travail de thèse, et d'en déduire une formule de type BCH dans notre contexte. Cette démarche devrait pouvoir se généraliser et ainsi fournir une approche nouvelle et intéressante même de la formule BCH classique. / The aim of this work consists of establishing the foundations and first steps of a research project which aims at a new understanding and generalization of the classical Baker-Campbell-Hausdorff formula with a conceptual approach, and its main application in group theory: refining a result of Mal'cev adapting the classical Lie correspondence to abstract groups, Lazard proved that the category of n-divisible n-step nilpotent groups is equivalent with the category of n-step nilpotent Lie algebras over the coefficient ring Z[1/2,…,1/n]. Generalizations to other algebraic structures than groups were obtained in the literature first for several varieties of loops (in particular Moufang, Bruck and Bol loops), and finally for all loops in recent work of Mostovoy, Pérez-Izquierdo and Shestakov. They invoke other types of algebras replacing Lie algebras in the respective context, namely Mal'cev algebras related with Moufang loops, Lie triple systems related with Bruck loops, Bol algebras with Bol algebras and finally Sabinin algebras with arbitrary loops. In each case, the associated type of algebras can be viewed as a linearization of the non-linear structure given by a given type of loops. This situation motivates a research program initiated by M. Hartl, namely of exhibiting suitable linearizations of all non-linear algebraic structures satisfying suitable conditions, namely all semiabelian varieties (of universal algebras, in the sense of universal algebra or of Lawvere). In fact, Hartl associated with any semi-abelian category C a multi-right exact (and hence multi-linear) functor operad on its abelian core. In the special case where C is a variety, this functor operad is even multicolimit preserving and by specialization is equivalent with an operad in abelian groups; the algebra type encoded by this operad provides a linearization of the given variety. Indeed, for each of the above-mentioned varieties of loops this algebra type coincides (over rational coefficients) with the one exhibited in the literature. These constructions and results are based on a new commutator theory in semi-abelian categories which itself relies on a calculus of functors in the framework of semi-abelian categories, both developed by Hartl in partial collaboration with B. Loiseau and T. Van der Linden. Now the project mentioned at the beginning constitutes the next major goal in this emerging general theory of linearization of algebraic structures: to generalize the Lazard equivalence and Baker- Campbell-Hausdorff formula to the context of semi-abelian varieties, and to deduce a way of explicitly computing the operad AbOp(C) from a given presentation of the variety C (more precisely, the operad obtained from AbOp(C) by tensoring its term of arity n with Z[1/2,…,1/n]). In the classical example of groups this would amount to deducing the structure of the Lie operad directly from the usual group axioms.

Nesta dissertação estudamos sob que condições um módulo de comprimento finito pode ser imerso em um módulo injetivo de comprimento finito. Também apresentamos a caracterização, dada por Hirano em [8], para os anéis sobre os quais todo módulo de comprimento finito tem um fecho injetivo de comprimento finito, os chamados de ¶-V-anéis. Além disso, mostramos que as extensões normais finitas de ¶-V-anéis são também ¶-V-anéis. / In this dissertation we study under what conditions a module of finite length can be embedded in an injective module of finite length. Also, we present a charactization, given by Hirano in [8], for the rings over which all module of finite length has an injective hull of finite length, the so called ¶-V-rings. Moreover, we show that finite normalizing extensions of ¶-V-rings are also ¶-V-rings.

With the mass availability of smartphones, mobile access to e-mail is gaining importance. Over the years, the IMAP protocol has been extended with many features ranging from extensions adding new functionality to those improving efficiency over an unreliable network. This thesis evaluates the available extensions based on their suitability for use in the context of a mobile client. Three new extensions have been developed, each improving the protocol in a distinct way. The thesis also discusses how most of these extensions were implemented in Trojitá, the author's free software open source IMAP e-mail client.

Este trabalho tem como principal objetivo estudar extensões de aplicações multilineares, de polinômios homogêneos e de funções analíticas entre espaços de Banach. Desta maneira, nos baseamos em importantes trabalhos sobre o assunto. Inicialmente apresentamos o produto de Arens para álgebras de Banach, extensões de Aron-Berner e de Davie-Gamelin para aplicações multilineares e provamos que todas estas extensões coincidem. A partir destes resultados, apresentamos a extensão de polinômios homogêneos e o Teorema de Davie-Gamelin que afirma que, assim como no caso de aplicações multilineares, as extensões de polinômios preservam a norma e, como consequência deste teorema, apresentamos uma generalização do Teorema de Goldstine. Em seguida estudamos espaços de Banach regulares e simetricamente regulares, que são propriedades relacionadas com a unicidade de extensão e são definidas a partir do ideal de operadores lineares fracamente compactos K^w(E, F) . Finalmente apresentamos a extensão de uma função de H_b(E) para H_b(E\'\') e o resultado, de Ignacio Zalduendo, que caracteriza esta extensão em termos da continuidade fraca-estrela do operador diferencial de primeira ordem. / The main purpose of this work is to study extensions of multilinear mappings, homogeneous polynomials and analytic functions between Banach Spaces. In this way, we rely on important works on the subject. Firstly we present the Arens-product for Banach algebras, the Aron-Berner and Davie-Gamelin extensions for multilinear mappings and we prove that all these extensions are the same. From these results, we present an extension for homogeneous polynomials and the Davie-Gamelin theorem which asserts that, as in the case of multilinear mappings, the polynomial extension is norm-preserving and, as a consequence of this theorem, we present a generalization of the Goldstine theorem. After that we study regular and symmetrically regular Banach spaces which are properties related to the uniqueness of the extension and are defined in the setting of weakly compact linear operators K^w(E, F) . Lastly, we present the extension of a function of H_b(E) to one in H_b(E\'\') and the result, according to Ignacio Zalduendo, which characterizes this extension in terms of weak-star continuity of the first order differential operator.

There are two primary sources of motivation for the contents of this thesis. The first is an effort to generalize classical dilation theory, a brief history of which is given in Section 2.1. The second source of motivation is the study of the representation theory of tensor algebras associated to C*-correspondences; these concepts are discussed in Sections 2.2 and 2.4. Although seemingly unrelated, there is a close connection between these two motivating theories. The link between classical dilation theory and the representation theory of tensor algebras over C*-correspondences was established by Muhly and Solel in their 1998 paper Tensor Algebras over C*-Correspondences: Representations, Dilations, and C*-Envelopes. In that paper, the authors not only introduced the concept of (operator-theoretic) tensor algebras – non-selfadjoint operator algebras that generalize algebraic tensor algebras – but they also developed the representation theory of these algebras. In order to do so, they introduced and made extensive use of a generalized dilation theory for contractions on Hilbert space. In analogy with classical dilation theory, they developed notions of “isometric dilation” and “coisometric extension” for completely contractive representations of the tensor algebra. The process of forming isometric dilations proceeded smoothly, but constructing coisometric extensions proved more problematic. In contrast to the classical case, Muhly and Solel showed that there is a high degree of nonuniqueness involved when building coisometric extensions. This lack of uniqueness proved to be an impediment to developing a full generalization of the dilation and model theories of Sz.-Nagy and Foias. In this thesis, we introduce a way to manage the ambiguities that arise when forming coisometric extensions. More specifically, we show that the notion of a transfer operator from classical dynamics can be adapted to this setting, and we prove that when a transfer operator is fixed in advance, every completely contractive representation of the tensor algebra admits a unique coisometric extension that respects the transfer operator in a fashion that we describe in Chapter 5. We also prove a commutant lifting theorem in the context of coisometric extensions.

Ce document de synthèse résume les travaux de l'auteur sur les extensions et systèmes de Hopf-Galois et leurs applications en théorie des représentations des groupes quantiques, ainsi que sur les constructions d'exemples de groupes quantiques.