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

Abstract

Let (H, R)be a finite dimensional quasitriangular Hopf al-gebra over a field k, and HMthe representation category ofH. In this paper, we study the braided autoequivalences of the Drinfeld center HHYDtrivializable on HM. We establish a group isomorphism between the group of those autoequiv-alences 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 inter-pretation 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.