McKay matrices for finite-dimensional Hopf algebras

Finite Dimensional Pointed Hopf Algebras over S

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 ...

Hopf Algebras and Biunitary Matrices

Actually to any spin model one can associate a vertex model (this is clear from V. Jones’ initial interpretation – in terms of statistical mechanics – of these objects) and the construction of Hopf algebras from complex Hadamard matrices is a particular case of the construction of Hopf algebras from biunitary matrices. The construction of Hopf algebras from biunitary matrices is a particular ca...

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...