O objetivo principal desta tese e a construção do semigrupo inverso das classes de isomorfismo das extensões abelianas parciais de um anel comutativo. / The main purpose of this thesis is the construction of the inverse semigroup of the isomorphism classes of the partial abelian extensions of a commutative ring.

by Lo Kwok Kei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 97-98). / Chapter 1 --- Introduction --- p.2 / Chapter 1.1 --- Algebra and Coalgebra --- p.2 / Chapter 1.2 --- Bialgebra and Hopf Algebra --- p.18 / Chapter 1.3 --- Integral and Semisimplicity --- p.26 / Chapter 2 --- Order of antipode --- p.44 / Chapter 2.1 --- An Isomorphism between H and H* --- p.44 / Chapter 2.2 --- Opposite-co-opposite Hopf algebras --- p.50 / Chapter 2.3 --- Antipode has finite order --- p.57 / Chapter 3 --- Larson's Characters --- p.65 / Chapter 3.1 --- Theory of Characters for comodules --- p.66 / Chapter 3.2 --- Simple subcoalgebras under the action s2 --- p.75 / Chapter 4 --- Semisimple and Cosemisimple Hopf Algebras --- p.78 / Chapter 4.1 --- Trace functions for Hopf algebras --- p.78 / Chapter 4.2 --- Cosemisimple Hopf algebras are semisimple --- p.86 / Chapter 4.3 --- Semisimple Hopf algebras are involutory --- p.90

Neste trabalho discutimos a semissimplicidade de álgebras de Hopf finito-dimensionais e construímos o Duplo de Drinfeld D(H) de uma tal álgebra H. Além disso, apresentamos um resultado mostrando a equivalência entre as categorias de representações dos módulos sobre D(H) e dos módulos de Yetter-Drinfeld sobre Hcop. Como consequência deste estudo, apresentamos um resultado que caracteriza uma álgebra de Hopf quase triangular. / In this work we discuss the semisimplicity of some finite-dimensional Hopf Algebras and we set up the Drinfel’d double D(H) of such an algebra H. In addiction, we present a result showing the equivalence between the representation category of modules over D(H) and the Yetter-Drinfeld modules over Hcop. As a consequence of this, we present a result that features a quasitriangular Hopf algebra.

In 2003, Radford introduced a new method to construct simple modules for the Drinfel’d double of a graded Hopf algebra. Until then, simple modules for such algebras were usually constructed by taking quotients of Verma modules by maximal submodules. This new method gives a more explicit construction, in the sense that the simple modules are given as subspaces of the Hopf algebra and one can easily find spanning sets for them. I use this method to study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel’d double of rank one pointed Hopf algebras of nilpotent type. The groups of group-like elements of these Hopf algebras are abelian; hence, they fall among those Hopf algebras classified by Andruskiewitsch and Schneider. I study, in particular, under what conditions a simple module can be factored as the tensor product of a one dimensional module with a module that is naturally a module for a special quotient. For restricted two-parameter quantum groups, given θ a primitive ℓth root of unity, the factorization of simple uθy,θz (sln)-modules is possible, if and only if gcd((y − z)n, ℓ) = 1. I construct simple modules using the computer algebra system Singular::Plural and present computational results and conjectures about bases and dimensions. For rank one pointed Hopf algebras, given the data D = (G, χ, a), the factorization of simple D(HD)-modules is possible if and only if |χ(a)| is odd and |χ(a)| = |a| = |χ|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or equal than |χ(a)| + 1.

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Thesis (Ph. D.)--Colorado State University, 2004. / Includes bibliographical references.

Düsseldorf, Universiẗat, Diss., 2002.

Düsseldorf, Universiẗat, Diss., 2001.

O objetivo principal desta tese e a construção do semigrupo inverso das classes de isomorfismo das extensões abelianas parciais de um anel comutativo. / The main purpose of this thesis is the construction of the inverse semigroup of the isomorphism classes of the partial abelian extensions of a commutative ring.

O objetivo deste trabalho é mostrar que o conjunto de todas G-super palavras monótonas restritas escritas com super letras duras forma uma base para a álgebra de Hopf H, onde H é gerada por um conjunto skew-primitivo semi-invariante {a1, ..., an} e um grupo abeliano G de todos elementos grouplike. / The objective of this work is to show that the set of all monotonic restricted G-super-words written with hard super-letters form a basis for the Hopf algebra H, where H is generated by a skew-primitive semi-invariant set {a1, ..., an} and an abelian group G of all group-like elements.