Braided autoequivalences and quantum commutative bi-Galois objects

Abstract

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 $_H^H{YD)$ over H and the category of comodules over some braided Hopf algebra _RH in the category $_HM$. Based on this isomorphism, we prove that every braided bi- Galois object A over the braided Hopf algebra $_RH$ defines a braided autoequivalence of the category $_H^H{YD}$ 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 $_H^H{YD}$ 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 [21] in the Hopf algebra case.