Braided monoidal categories
I. Heckenberger, Hans-JÃ ⁄ rgen Schneider
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 = H H YD, 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.