On the Antipode of a Co-Frobenius (Co)Quasitriangular Hopf Algebra

For H a quasitriangular Hopf algebra, S ², the square of the antipode is the inner automorphism induced by the Drinfeld element u, and S ⁴ is the inner automorphism induced by the grouplike element g = uS(u)^?1. For H finite dimensional, results of Drinfeld and Radford express g in terms of the modular elements of H. This note supplies another proof which replaces the requirement of finite dimensionality with existence of a nonzero integral for H in H*. Similar results hold for the infinite dimensional coquasitriangular case; here we supply some interesting examples.