The hopf automorphism group and the quantum brauer group in braided monoidal categories

Abstract

With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra H over a field k we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal category. Let B be a Hopf algebra in C = HHYD, the category of Yetter-Drinfel'd modules over H. We consider the quantum Brauer group BQ(C; B) of B in C, which is isomorphic to the usual quantum Brauer group BQ(k; B ⋊ H) of the Radford biproduct Hopf algebra B ⋊ H. We show that under certain symmetricity condition on the braiding in C there is an inner action of the Hopf automorphism group of B on the former. We prove that the subgroup BM(C; B)-the Brauer group of module algebras over B in C-is invariant under this action for a family of Radford biproduct Hopf algebras. The analogous invariance we study for BM(k; B ⋊ H). We apply our recent results on the latter group and generate a new subgroup of the quantum Brauer group of B ⋊ H. In particular, we get new information on the quantum Brauer groups of some known Hopf algebras.