On the antipode of a quasitriangular Hopf algebra

Abstract

Let (A, R) be a quasitriangular Hopf algebra with antipode s in the category of vector spaces over a field k. In this paper we show that s2 is inner, which means s is bijective, and when A is finite-dimensional there is a grouplike element h E A such that s4(a) = h&r-” for all a E A. Suppose that A is any Hopf algebra with antipode s over the field k. Proposition 1 of Section 1 gives a sufficient condition for 3’ to be inner which we apply to quasitriangular Hopf algebras in Section 2. Its statement and proof are based on calculations found in [2, pp. 66671. Generally s2 need not be inner. Suppose that A is finite-dimensional. If s2 were inner, then all ideals of A would be invariant under s2, and hence all subcoalgebras of the dual Hopf algebra A* would be invariant under S*, where S = s* is the antipode of A*. There are examples [5] over algebraically closed lields in which the latter is not the case. Section 2 begins with a discussion of properties of quasit~angular Hopf algebras (A, R) over the held k used in this paper. Write R=CR(‘)C@R(~)EA@A and set u=CS(R’*‘)R(~). The main result of Section 2 is that u is invertible and s2(a) = uau-’ for all a E A. This result was established in [Z, Exercise 7.3.61 under the hypothesis that s is bijective. If A is any finite-dimensional Hopf algebra with antipode s over the field k, there are distinguished grouplike elements g E A and tl E A * which relate left and right integrals and which together describe s4. In Section 3 we consider s4 when (A, R) is a unite-dimensional quasit~angular Hopf algebra. Let h = IIU, where u =C s(R’*‘) R(l) is defined as above and u= s(u) -I. By relating h to g and ~1, we are able to show that h is a grouplike element and that s4(a) = hah-’ for all a E A. When A is unimodular we prove that h =g. Thus the product vu does not depend on R in the unimodular case. We assume that the reader has some familiarity with the elementary aspects of the theory of Hopf algebras. A recommended reference is [6].