Frobenius Divisibility for Hopf Algebras

TL;DR

This paper provides a unified algebraic framework using Casimir elements to explain divisibility properties of irreducible representation degrees in semisimple Hopf algebras, extending Frobenius's classical results.

Contribution

It introduces a ring theoretic approach based on Casimir elements to unify and generalize known divisibility results for Hopf algebra representations.

Findings

Unified proof of divisibility results

Extension of Frobenius's theorem to Hopf algebras

Application of Casimir elements in representation theory

Abstract

We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in characteristic 0. All these results are motivated by a classical theorem of Frobenius on the degrees of irreducible complex representations of finite groups.