On the quasitriangular structures of a semisimple Hopf algebra

Abstract

In this paper we show that the theorem of [4] on the order of the group of Hopf algebra automorphisms Aut,,,, (A) of certain semisimple Hopf algebras over a field k has an interesting implication for quantum groups. If A is a finite-dimensional Hopf algebra over the field k, there may be infinitely many RE A @A such that (A, R) is a quasitriangular Hopf algebra. However, if A is semisimple and the characteristic of k is 0, or if A is semisimple, cosemisimple, and involutory and the characteristic of k is p > dim A, we show in the theorem of this paper that there are at most finitely R EA @A such that (A, R) is quasitriangular. Our proof is based on the fact that the group Aut,,(A) is finite [4, Theorem 33 for such a Hopf algebra A. We will assume that the reader is familiar with the elementary aspects of the theory of Hopf algebras. A good general reference is [S].