  Recall that a finite group is called perfect if it does not have non-trivial
1-dimensional representations (over the field of complex numbers C). By
analogy, let us say that a finite dimensional Hopf algebra H over C is perfect
if any 1-dimensional H-module is trivial. Let us say that H is biperfect if
both H and H^* are perfect. Note that, H is biperfect if and only if its
quantum double D(H) is biperfect. It is not easy to construct a biperfect Hopf
algebra of dimension >1. The goal of this note is to describe the simplest such
example we know. The biperfect Hopf algebra H we construct is based on the
Mathiew group of degree 24, and it is semisimple. Therefore, it yields a
negative answer to Question 7.5 from a previous paper of the first two authors
(math.QA/9905168). Namely, it shows that Corollary 7.4 from this paper stating
that a triangular semisimple Hopf algebra over C has a non-trivial group-like
element, fails in the quasitriangular case. The counterexample is the quantum
double D(H).
