The fundamental group of a Hopf linear category

Abstract

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compare this invariant to the fundamental group of the underlying linear category, and we compute those groups for families of examples.

Computations of the fundamental group of some Hopf algebras are added. The relation with the fundamental group of the underlying associative structure is now considered. We also analyse the situation when universal covers and/or gradings exist. Dedicated to Eduardo N. Marcos for his 60th birthday. 24 pages