Self-dual modules of semisimple Hopf algebras

TL;DR

This paper proves that semisimple Hopf algebras with a nontrivial self-dual simple module must have even dimension, extending classical results and linking module properties to algebra dimension in characteristic zero.

Contribution

It generalizes Burnside's classical result by establishing a connection between self-dual modules and the even dimension of semisimple Hopf algebras in characteristic zero.

Findings

Semisimple Hopf algebra with a nontrivial self-dual simple module has even dimension

A semisimple Hopf algebra with a simple module of even dimension must have even dimension

Extension of classical Burnside's result to Hopf algebras

Abstract

We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.