Braided autoequivalences and quantum commutative bi-Galois objects

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Let (H,R) be a quasitriangular weak Hopf algebra over a field k. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category ʰₕYD over H and the category of comodules over some braided Hopf algebra ᵣH in the category ₕM. Based on this equivalence, we prove that every braided bi-Galois object A over the braided Hopf algebra ᵣH defines a braided autoequivalence of the category ʰₕYD 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 ʰₕYD trivializable on ₕM is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in ₕM form a group measuring the Brauer group of (H,R) as studied in [20] in the Hopf algebra case.