Finite-dimensional semisimple Hopf algebras of a special type and matched pairs of groups
This talk is based on work of my students Julia Ludes and Matthias Klupsch and myself, and is motivated by a question of Huppert. We consider semisimple complex Hopf algebras H such that, as an algebra, H is a direct sum of k copies of the complex numbers and exactly one copy of a d-by-d matrix ring whith d > 1. The group algebras of this type have long been classified by Gary Seitz. As a post-classification result one observes that the set of parameters k and d that occur in this case is rather restricted. A matched pair af groups is a pair of finite groups acting on each other in a compatible way. Masuoka constructed a semisimple Hopf algebra from a matched pair of groups. I will discuss the conditions under which Masuoka's construction yields a Hopf algebra as above, and I propose a classification of the corresponding matched pairs. If one of the actions is trivial, this classification has already been achieved by Julia Ludes.