Finite-dimensional Hopf algebras of rank one in characteristic zero

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...

Representations of Finite Dimensional Hopf Algebras

Let H denote a nite dimensional Hopf algebra with antipode S over a eld |. We give a new proof of the fact, due to Oberst and Schneider OS], that H is a symmetric algebra if and only if H is unimodular and S 2 is inner. If H is involutory and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char|. In the case where |is a splitting eld for H , we give ...

Finite Dimensional Pointed Hopf Algebras over S

On Finite-dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic

Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero [Mo and references therein]. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky’s...

Hochschild Cohomology of Smash Products and Rank One Hopf Algebras

We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.