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

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Let (H, R) be a finite dimensional quasitriangular Hopf algebra over a field k, and ₕM the representation category of H. In this paper, we study the braided autoequivalences of the Drinfeld center ʰₕYD trivializable on ₕM. 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 ᵣH. 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 initiated 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.