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 $\C=_H ^H\YD$, 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 $\BQ(k; B\rtimes H)$ of the Radford biproduct Hopf algebra $B\rtimes H$. 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 $\BM(k; B\rtimes H)$. We apply our recent results on the latter group and generate a new subgroup of the quantum Brauer group of $B\rtimes H$. In particular, we get new information on the quantum Brauer groups of some known Hopf algebras.