by Facchini, Alberto; Herbera, Dolors

The aim of this paper is twofold. On the one hand, we show that the dual Goldie dimension codim(End(M

_R
)) of the endomorphism ring End(M

_R
) of a module M

_R
can be used as a measure of the dimension of the module M

_R
. On the other hand, we prove under suitable hypotheses the validity of the Krull–Schmidt Theorem for infinite direct sums of modules with homogeneous semilocal endomorphism rings.

DOI: 10.1023/B:ALGE.0000048320.11088.49
Print publication date: 12/1/2004
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by Müger, M.; Roberts, J. E.; Tuset, L.

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka–Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka–Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras.

DOI: 10.1023/B:ALGE.0000048337.34810.6f
Print publication date: 12/1/2004
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by Taillefer, Rachel

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.

DOI: 10.1023/B:ALGE.0000048319.07763.64
Print publication date: 12/1/2004
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by Wakui, Michihisa

We determine various additional structures on all nonsemisimple Hopf algebras of dimension 8 over an algebraically closed field k of characteristic 0, including their representation rings and quasitriangular structures. As a consequence, it is shown that for two such Hopf algebras, the tensor categories of their representations are monoidally equivalent if and only if the representation rings of them are isomorphic as rings.

DOI: 10.1023/B:ALGE.0000048318.64013.db
Print publication date: 12/1/2004
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