On the quasitriangular structures of abelian extensions of $Z_2$

Abstract

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras k G #σ,τ kZ2 constructed through abelian extensions of Z2 by k G for an abelian group G. We prove that there are only two forms of them. Using such description together with some other techniques, we get a complete list of all universal R-matrices on Hopf algebras H 2n 2 , A 2n 2 ,t and K(8n, σ, τ ) (see Section 2 for the definition of these Hopf algebras). Then we find a simple criterion for a K(8n, σ, τ ) to be a minimal quasitriangular Hopf algebra. As a product, some minimal quisitriangular semisimple Hopf algebras are found.