Some applications of Frobenius algebras to Hopf algebras

TL;DR

This paper explores how Frobenius algebras can be used to unify and analyze various properties and results related to Hopf algebras through a ring theoretic approach.

Contribution

It introduces a unified Frobenius algebra framework to derive key results in Hopf algebra theory, enhancing understanding and simplifying proofs.

Findings

Proves a theorem of S. Zhu on irreducible representation degrees

Establishes the class equation for Hopf algebras

Determines the semisimplicity locus of the Grothendieck ring

Abstract

This expository article presents a unified ring theoretic approach, based on the theory of Frobenius algebras, to a variety of results on Hopf algebras. These include a theorem of S. Zhu on the degrees of irreducible representations, the so-called class equation, the determination of the semisimplicity locus of the Grothendieck ring, the spectrum of the adjoint class and a non-vanishing result for the adjoint character.