A Frobenius-Schur theorem for Hopf algebras

TL;DR

This paper generalizes the Frobenius-Schur theorem from finite groups to semisimple Hopf algebras and related algebraic structures, extending classical results to broader algebraic contexts.

Contribution

It provides a new Frobenius-Schur theorem for semisimple Hopf algebras over algebraically closed fields of characteristic zero and extends to cosemisimple cases in positive characteristic.

Findings

Generalized Frobenius-Schur theorem for semisimple Hopf algebras

Applicable in characteristic p > 2 for cosemisimple Hopf algebras

Extended results to finite-dimensional semisimple algebras with involution

Abstract

In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic $p > 2$ if the Hopf algebra is also cosemisimple. In fact we show a more general version for any finite-dimensional semisimple algebra with an involution.