The doubles of a braided Hopf algebra

Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) = A⊗ A⊗C of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad dA on the center Z(B) of B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. As an endofunctor of Z(B), dA it is given by X 7→ X ⊗A⊗ ∨ A. We prove that the Hopf monad dA may not be representable by a Hopf algebra. If B has a coend C, then D(A) is the cross product of the Hopf monad dA by C. Equivalently, dA is the cross quotient of D(A) by C.