TRANSMUTATION THEORY AND RANK FOR QUANTUM BRAIDED GROUPS

Let f: H-1 --> H-2 be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H-1, f, H-2) in the braided monoidal category C of H-1-modules. It consists in the same algebra as H-2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H-1 and H-2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of Z2 becomes transmuted to a super-Hopf algebra.