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 $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times M^{op}$. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup $Y^\theta X^\theta$ associated to every order-reversing automorphism $\theta$ of X. The corresponding Hopf algebra $kX^\theta\cobicross k(Y^\theta)$ 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.