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 ${}_H\mathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center ${}_H^H\mathcal{Y}\mathcal{D}$ trivializable on ${}_H\mathcal{M}$. 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 ${}_{R}H$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $\mathrm{B}\mathrm{M}(k,H,R)$ 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.