BRAIDED GROUPS

We prove a highly generalized Tannaka-Krein type reconstruction theorem for a monoidal category C functored by F : C --> V to a suitably cocomplete rigid quasitensor category V. The generalized theorem associates to this a bialgebra or Hopf algebra Aut(C, F, V) in the category V. As a corollary, to every cocompleted rigid quasitensor category C is associated Aut(C) Aut(C, id, CBAR). It is braided-commutative in a certain sense and hence analogous to the ring of 'co-ordinate functions' on a group or supergroup, i.e., a 'braided group'. We derive the formulae for the transmutation of an ordinary dual quasitriangular Hopf algebra into such a braided group. More generally, we obtain a Hopf algebra B(A1, f, A2) (in a braided category) associated to an ordinary Hopf algebra map f : A1 --> A2 between ordinary dual quasitriangular Hopf algebras A1, A2.