Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple

Abstract

Finite dimensional cosemisimple Hopf algebras are a natural generalization of group algebras. Much of the representation theory of finite groups can be extended to such Hopf algebras. However, there are still many unanswered basic questions for such Hopf algebras. One such question is a conjecture mentioned by Kaplansky [I]: the square of the antipode of a finite dimensional cosemisimple Hopf algebra is the identity. In this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0. In the course of proving this, we investigate two elements of the Hopf algebra which are roughly analogous to the sum of the group elements in a group algebra. We show that Kaplansky’s conjecture is equivalent to these two elements being equal. For a finite dimensional Hopf algebra A with antipode s over a field, let L(p)(q) =pq for p, q E A* denote the left module action of A* on itself. In this paper we study two elements of A defined by the trace function: the element ;i defined by p(A) = Tr(L(p) (s’)*) for all PE A*, and the element x defined by p(x) = Tr(L(p)) for all p E A*. We show that 2 is a left integral and use it to prove Theorem 3.3, the main result of this paper: a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0 is also semisimple (hence its antipode has order 1, 2, or 4). We extend the first corollary to [3, Proposition 91 in Theorems 4.3 and 4.4 by finding necessary and sufficient conditions for s to be a non-zero left integral (in which case x = A), and also study how x relates to the structure of A. In particular, when (dim A) 1 # 0 the right ideal xA may be thought of as a measure of the extent to which s* # I. The elements ji and .K are two examples of elements of A associated with an endomorphism of A. In general, for f~ End(A) we define A, E A by p(A&=Tr(L(p)cf*) for all PEA *. Thus I? = AS2 and x = A,. In Section 2 267 0021-8693/88 33.00