The fundamental group of a Hopf linear category

TL;DR

This paper introduces a new notion of the fundamental group for Hopf algebras, utilizing gradings and Galois coverings within Hopf linear categories, and compares it to existing invariants.

Contribution

It defines the fundamental group of a Hopf algebra via Hopf linear categories and explores its properties and computations for various examples.

Findings

The fundamental group of a Hopf algebra can be explicitly computed for certain families.

Hopf linear categories provide a framework for understanding coverings and gradings of Hopf algebras.

The new invariant relates to the fundamental group of the underlying 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.