Biperfect Hopf Algebras
Abstract
Recall that a finite group is called perfect if it does not have nontrivial 1dimensional 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 1dimensional Hmodule 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 nontrivial grouplike element, fails in the quasitriangular case. The counterexample is the quantum double D(H).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 1999
 arXiv:
 arXiv:math/9912068
 Bibcode:
 1999math.....12068E
 Keywords:

 Quantum Algebra
 EPrint:
 5 pages, latex