Quasitriangular and differential structures on bicrossproduct Hopf algebras

Let X = GM be a finite group factorisation. It is shown that the quantum do uble D(H) of the associated bicrossproduct Hopf algebra H = kM --> <-- k(G) is itself a bicrossproduct kX --> <-- k(Y) associated to a group IX, where Y = G x M-op. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup (YXtheta)-X-theta associa ted to every order-reversing automorphism theta of X. The corresponding Hop f algebra kX(theta) --> <-- k(Y-theta) has the same coalgebra as H. Using r elated results, we classify the first order bicovariant differential calcul i on H in terms of orbits in a certain quotient space of X. (C) 1999 Academ ic Press.