Braided autoequivalences and quantum commutative bi-Galois objects

Let (H, R) be a quasitriangular weak Hopf algebra over a field k. We show that there is a braided monoidal isomorphism between the Yetter-Drinfeld module category (HYD)-Y-H over H and the category of comodules over some braided Hopf algebra H-R in the category M-H. Based on this isomorphism, we prove that every braided bi-Galois object A over the braided Hopf algebra H-R defines a braided autoequivalence of the category (HYD)-Y-H if and only if A is quantum commutative. In case H is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of (HYD)-Y-H trivializable on M-H is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in M-H form a group measuring the Brauer group of (H, R) as studied in [21] in the Hopf algebra case. (C) 2015 Elsevier B.V. All rights reserved.