On some examples of simple quantum groups

Abstract

The aim of this note is to construct noncocommutative finite dimensional simple Hopf algebras by twisting the group algebras of finite simple groups. In [N1] Nikshych described those constructions and proved that if the twist is possible then the resulting Hopf algebra will remain simple and the Grothendieck rings of the two Hopf algebras will be the isomorphic as ordered rings with involution. We show that the twisting is always possible and describe all the cases in which his construction can be done. Recently in [N2] these methods have been used to describe all simple Hopf algebras of dimension at most 60. To fix the notation let k be an algebraically closed field of characteristic zero. We will only be interested in finite dimensional Hopf algebras over k. The easiest example is provided by the group algebra kG for G a finite group and with comultiplication ∆(g) = g⊗g, counit e(g) = 1 and antipode S(g) = g−1. We will call such an example a trivial Hopf algebra. A Hopf algebra H is simple if it has no Hopf quotients. In particular in the case of a group algebra kG the normal Hopf subalgebras are in 11 correspondence with the normal subgroups of G and in particular kG is simple if and only if G is simple. Also A is cocommutative (respectively commutative) iff A is a trivial (respectively the dual of a trivial algebra). We will prove the following :