Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. / Made available in DSpace on 2012-10-18T11:44:29Z (GMT). No. of bitstreams: 0Bitstream added on 2014-09-25T22:54:30Z : No. of bitstreams: 1 177271.pdf: 2955474 bytes, checksum: e3ebdd318db782e99a6896cb10ffe886 (MD5) / Estudo sobre o plano quântico e sua geometria. Dá-se uma visão geral do que é uma álgebra de Hopf e introduz-se os conceitos de ação, coação e estruturas quasi-triangulares. Constrói-se também a álgebra de Hopf U_q(sl(2)) e sua ação no plano quântico com uma realização matricial e através de operadores diferenciais. Finalmente é feito um estudo do cálculo diferencial e do complexo de Wess-Zumino, para q raiz n-ésima da unidade, e também para q raiz cúbica da unidade.

Nesta dissertação realizamos um estudo sobre dois resultados clássicos da literatura referentes à dualidade para ação de álgebras de Hopf: a dualidade de Cohen-Montgomery e a dualidade de Blattner-Montgomery. No primeiro caso estendemos o resultado de Cohen-Montgomery ao contexto de ação parcial de grupos e no segundo caso estendemos o resultado de Blattner-Montgomery ao contexto de ação parcial de álgebras de Hopf. Esta dissertação foi elaborada tendo como base o artigo de C. Lomp, "Duality for partial groups actions", arXiv: 0711.0849v1[math.RA], 2007. / In this dissertation we are concerned with the following two classical results on duality: the Cohen-Montgomery duality and the Blattner-Montgomery duality; and we present their corresponding versions in the context of partial action of groups and partial actions of Hopf algebras, respectively. The subject of this dissertation is based on the C. Lomp’s paper: "Duality for partial group actions", arXiv: 0711.0849v1 [math.RA], 2007.

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Neste trabalho realizamos um estudo de algebras, co algebras e algebras de Hopf, introduzindo estas no c~oes e algumas de suas propriedades e exemplos. Al em disso, aprofundamos o estudo apresentando as algebras de Hopf quase cocomutativas e quasitriangulares, demonstrando que a ant poda dessas algebras cumpre certas condi c~oes. / In this work we present a study of algebras, coalgebras and Hopf algebras, introducing examples and some of its properties. Also, we expand this study presenting almost cocommutative and quasitriangular Hopf algebras, showing that the antipode of these algebras satis es determined conditions.

Nesta dissertação realizamos um estudo sobre dois resultados clássicos da literatura referentes à dualidade para ação de álgebras de Hopf: a dualidade de Cohen-Montgomery e a dualidade de Blattner-Montgomery. No primeiro caso estendemos o resultado de Cohen-Montgomery ao contexto de ação parcial de grupos e no segundo caso estendemos o resultado de Blattner-Montgomery ao contexto de ação parcial de álgebras de Hopf. Esta dissertação foi elaborada tendo como base o artigo de C. Lomp, "Duality for partial groups actions", arXiv: 0711.0849v1[math.RA], 2007. / In this dissertation we are concerned with the following two classical results on duality: the Cohen-Montgomery duality and the Blattner-Montgomery duality; and we present their corresponding versions in the context of partial action of groups and partial actions of Hopf algebras, respectively. The subject of this dissertation is based on the C. Lomp’s paper: "Duality for partial group actions", arXiv: 0711.0849v1 [math.RA], 2007.

For an extension of local fields, a scaffold is shown to be a powerful tool for dealing with the problem of the freeness of fractional ideals over their associated orders (Byott, Childs and Elder: \textit{Scaffolds and Generalized Integral Galois Module Structure}, Ann. Inst. Fourier, 2018). The first class of field extensions admitting scaffolds is \enquote*{near one-dimensional elementary abelian extension}, introduced by Elder (\textit{Galois Scaffolding in One-dimensional Elementary Abelian Extensions}, Proc. Amer. Math. Soc. 2009). However, the scaffolds constructed in Elder's paper arise only from the classical Hopf-Galois structure. Therefore, the study in this thesis aims to investigate scaffolds in non-classical Hopf-Galois structures. Let $L/K$ be a near one-dimensional elementary abelian extension of degree $p^2$ for a prime $p \geq 3.$ We show that, among the $p^2-1$ non-classical Hopf-Galois structures on the extension, there are only $p-1$ of them for which scaffolds may exist, and these exist only under certain restrictive arithmetic condition on the ramification break numbers for the extension. The existence of scaffolds is beneficial for determining the freeness status of fractional ideals of $\mathfrak{O}_L$ over their associated orders. In almost all other cases, there is no fractional ideal which is free over its associated order. As a result, scaffolds fail to exist.

Univ., Diss--München, 1999. / Literaturverz. S. [147] - 150.

Cette thèse se situe au carrefour de la combinatoire et de l'algèbre. Elle se consacre d'une part à traduire des problèmes algébriques en des problèmes combinatoires, et inversement, utilise le formalisme algébrique pour traiter des questions combinatoires. Après un rappel des notions classiques de combinatoire et d'algèbres de Hopfavec quelques applications, nous abordons l'étude de certaines statistiques définies sur les permutations : les pics, les vallées, les doubles montées et les doubles descentes, qui sont à la base de la bijection de Françon-Viennot, elle-même débouchant sur une étude combinatoire des polynômes orthogonaux. Nous montrons qu'à partir de ces statistiques, il est possible de construire diverses sous-algèbres ou algèbres quotients de FQSym, une algèbre dont une base est indexée par les permutations. Puis, nous étudions deux suites classiques de combinatoire par une démarche non commutative : les polynômes de Gandhi, un raffinement polynomial des nombres de Genocchi, et les nombres d'Euler, une suite recelant de nombreuses propriétés combinatoires. Nous nous attachons à montrer que l'approche non commutative permet, dans la majeure partie des cas, d'obtenir de manière directe des interprétations d'identités combinatoires. Enfin, inversement, certaines questions de nature algébrique peuvent être abordées d'un point de vue combinatoire. Ainsi, à travers l'étude des algèbres dendriformes, des algèbres tridendriformes, et des quadrialgèbres, nous prouvons des questions de liberté à propos de ces algèbres grâce à la combinatoire des arbres étiquetés / This thesis is at the crossroads between combinatorics and algebra. It studies some algebraic problems from a combinatorial point of view, and conversely, some combinatorial problems have an algebraic approach which enables us tosolve them. In the first part, some classical statistics on permutations are studied: the peaks, the valleys, the double rises, and the double descents. We show that we can build sub algebras and quotients of FQSym, an algebra which basis is indexed by permutations. Then, we study classical combinatorial sequences such as Gandhi polynomials, refinements of Genocchi numbers, and Euler numbers in a non commutative way. In particular, we see that combinatorial interpretations arise naturally from the non commutative approach. Finally, we solve some freeness problems about dendriform algebras, tridendriform algebras and quadrialgebras thanks to combinatorics of some labelled trees

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