Cocycle Deformations and Brauer Group Isomorphisms

Let $H$ be a Hopf algebra over a commutative ring $k$ with unity and $\sigma :H\otimes H⟶k$ be a cocycle on $H$. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra $H^{\sigma }$ is equivalent to the Yetter-Drinfeld module category of $H$. As a result of the equivalence, the “quantum Brauer” group BQ$(k,H)$ is isomorphic to BQ$(k,H^{\sigma })$. Moreover, the group $\text{Gal}(\text{HR})$ constructed in \cite{Z} is studied under a cocycle deformation.