Burnside's theorem for representations of Hopf algebras

Abstract

For a swift proof see [I]. This theorem has been generalized by Steinberg [7] to essentially the case of (possibly infinite) semigroups. Our first theorem is a generalization of Burnside’s theorem to the case of Hopf algebras, which we feel is probably the most natural setting for the theorem. Next we obtain an upper bound on the number of tensor products needed in the case of a finite dimensional Hopf algebra. Finally we show how Steinberg’s theorem, and so Burnside’s theorem, are consequences of our theorem for Hopf algebras.