The Hopf automorphism group and the quantum Brauer group in braided monoidal categories

TL;DR

This paper investigates the structure of the quantum Brauer group in braided monoidal categories, establishing an exact sequence and analyzing automorphism actions to enhance understanding of quantum symmetries in Hopf algebras.

Contribution

It constructs an exact sequence relating quantum Brauer groups and studies automorphism actions, providing new insights into their structure in braided monoidal categories.

Findings

Established an exact sequence involving quantum Brauer groups.

Proved invariance of certain subgroups under Hopf automorphisms.

Generated new subgroups of the quantum Brauer group for specific Hopf algebras.

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.