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 YDHH over H and the category of comodules over some braided Hopf algebra HR in the category MH. Based on this isomorphism, we prove that every braided bi-Galois object A over the braided Hopf algebra HR defines a braided autoequivalence of the category YDHH 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 YDHH trivializable on MH is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in MH form a group measuring the Brauer group of (H,R) as studied in [21] in the Hopf algebra case.