A Frobenius-Schur theorem for Hopf algebras

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; this more general result (and its proof) may give some new insight into the classical theorem. Let G be a finite group. For h ∈ G, define θm(h) to be the number of solutions of the equation g = h, that is θm(h) = | {g ∈ G | g m = h} |. Because θm(h) is a class function it can be written as