Burnside’s theorem for Hopf algebras

Abstract

A classical theorem of Burnside asserts that if X is a faithful complex character for the finite group G, then every irreducible character of G is a constituent of some power Xn of X . Fifty years after this appeared, Steinberg generalized it to a result on semigroup algebras K[G] with K an arbitrary field and with G a semigroup, finite or infinite. Five years later, Rieffel showed that the theorem really concerns bialgebras and Hopf algebras. In this note, we simplify and amplify the latter work. Let K be a field and let A be a K-algebra. A map A: A -+ A A is said to be a comultiplication on A if A is a coassociative K-algebra homomorphism. For convenience, we call such a pair (A, A) a b-algebra. Admittedly, this is rather nonstandard notation. One is usually concerned with bialgebras, that is, algebras which are endowed with both a comultiplication A and a counit e: A -K. However, semigroup algebras are not bialgebras in general, and the counit rarely comes into play here. Thus it is useful to have a name for this simpler object. Now a b-algebra homomorphism 6: A -k B is an algebra homomorphism which is compatible with the corresponding comultiplications, and the kernel of such a homomorphism is called a b-ideal. It is easy to see that I is a b-ideal of A if and only if I ' A with A(I) C I A+A +A I. Of course, the b-algebra structure can be used to define the tensor product of A-modules. Specifically, if V and W are left A-modules, then A acts on V X W via a(vow) =A(a)(v?w) for all a e A, v E V, we W. Notice that if I is a b-ideal of A, then the set of all A-modules V with annA V D I is closed under tensor product. Conversely, we have Proposition 1. Let A be a b-algebra and let Y be a family of A-modules closed under tensor product. Then I= n annA V VEY