Quasitriangular and Differential Structures on Bicrossproduct Hopf Algebras

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.