Quasitriangular and Differential Structures on Bicrossproduct Hopf Algebras

Abstract

Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra H=kM▷◀k(G) is itself a bicrossproduct kX▷◀k(Y) associated to a group YX, where Y=G×Mop. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup YθXθ associated to every order-reversing automorphism θ of X. The corresponding Hopf algebra kXθ▷◀k(Yθ) has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H in terms of orbits in a certain quotient space of X.