Finite-dimensional pointed or copointed Hopf algebras over affine racks

Finite Dimensional Pointed Hopf Algebras over S

Representations of Finite Dimensional Pointed Hopf Algebras over S 3

The classification of finite-dimensional pointed Hopf algebras with group S3 was finished in [AHS]: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finit...

Cohomology of Finite Dimensional Pointed Hopf Algebras

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig’s small quantum groups, whose cohomology was first computed explicitly by Gin...

Pointed and Copointed Hopf Algebras as Cocycle Deformations

We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensio...

On the Classification of Finite-dimensional Pointed Hopf Algebras

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements G(A) is abelian such that all prime divisors of the order of G(A) are > 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiom...