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We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category., Comment: 44 pages

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair of characters $\rho \in \Alg(U, k)$ and $\chi \in \Alg(A, k)$ we define a left $H$-module $L(\rho, \chi)$. Under reasonable hypotheses the correspondence $(\rho, \chi) \mapsto L(\rho, \chi)$ determines a bijection between character pairs and the isomorphism classes of objects in a certain category ${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases the finite-dimensional objects of ${}_H\underline{\mathcal M}$ are the finite-dimensional irreducible left $H$-modules. In math.QA/0603269 we apply the results of this paper and show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters., Comment: amstex, 33 pages

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work of N. Andruskiewitsch and the second author. Special cases are the multiparameter deformations of the enveloping algebras of semisimple Lie algebras where the deforming parameters are not roots of unity and some of their finite-dimensional versions in the root of unity case., Comment: amstex, 35 pages

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras $(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$ and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in ${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in ${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products., Comment: amstex, 29 pages

In our previous paper math/0502157 we classified a large class of finite-dimensional pointed Hopf algebras up to isomorphism. However the following problem was left open for Hopf algebras of of type $A,D$ or $E_6$, that is whose Cartan matrix is connected and allows a non-trivial automorphism of the corresponding Dynkin diagram. In this case we described the isomorphisms between two such Hopf algebras with the same Cartan matrix only implicitly. The problem is whether it is possible to compute the isomorphisms in terms of the defining families of parameters. In the present paper we solve this problem for type $A$., Comment: 25 pages. The second version is final, to appear in Proceedings of the XVI Coloquio Latinoamericano de Algebra (Colonia, Uruguay, 2005)

We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group., Comment: 28 pages