On the quasitriangular structures of bicrossproduct Hopf algebras

In this paper we show that if B star H is a bicrossproduct Hopf algebra the n (B star H, R) is qusitriangular if and only if R has a unique decompositi on: R = Sigma Q((1))V((1))(V) over bar((1)) X (TU(1))-U-(1) X Q((2))(V-(2)- ->U-(2)) X (V) over bar((2))T((2)) such that Q, T, V and U satisfy certain compatible conditions. The result is applied to a certain bicrossproduct of H and H-op, where H is a Hopf algebra with bijective antipode.