REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: We work over an algebraically closed field k of characteristic zero. The purpose of this work is understand the actions of finite quantum groups, i.e. finite dimensional Hopf algebras H, on commutative domains. Two important subclasses of finite dimensional Hopf algebras are those that are semisimple (i.e., semisimple as an algebra) and those that are pointed (i.e., all simple H-comodules are 1-dimensional).

The classification of semisimple Hopf actions on commutative domains is completely understood. Namely, Pavel Etingof and I show that if a semisimple Hopf algebra acts on a commutative domain, then this action must factor through an action of finite group algebra. See arXiv:1301.4161.

On the other hand, the non-semisimple case is much more complicated. Our work continues in this case when H is finite dimensional, non-semisimple, pointed, and of finite Cartan type. The work in this direction begins by reducing to the case where H acts inner faithfully on a field (so that the action does not factor through a smaller quotient Hopf algebra). Such a Hopf algebra is referred to as Galois-theoretical. I will present examples and classification results for  these Hopf algebras; examples are discussed in arXiv:1403.4673 and the classification results is a work in preparation.