Finite Group Factorizations and Braiding Original Research Article

We compute the quantum double, braiding, and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebraHassociated to the factorization of a finite groupXinto two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that basis-preserving self-duality structures for the bicrossproduct Hopf algebras are in one–one correspondence with factor-reversing group isomorphisms. The example image6image6is given in detail. We show further that the quantum doubleD(H) is the twisting ofD(X) by a nontrivial quantum cocycle.