  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
$^H_H\mathcal{YD}$ 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
$_RH$. We then apply this isomorphism to obtain a categorical interpretation of
the exact sequence of the equivariant Brauer group $\mathrm{BM}(k, H,R)$ 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.
