Braided autoequivalences and the equivariant Brauer group of a quasitriangular Hopf algebra

Let (H,R) be a finite dimensional quasitriangular Hopf algebra over a field k, and MH the representation category of H. In this paper, we study the braided autoequivalences of the Drinfeld center YDHH trivializable on MH. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra HR. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group BM(k,H,R) in [18]. To this aim, we have to develop the braided bi-Galois theory established by Schauenburg in [14,15], which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category.