Anti-isomorphism between Brauer groups BQ(S, H) AND BQ(S^op, H∗)

Abstract

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel’d H-module algebras and the Brauer group $\mathit{BQ}(R,H^{*})$ of Hopf Yetter-Drinfel’d $H^{*}$-module algebras, where $H^{*}$ is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel’d (S, H)-module algebras and $\mathit{BQ}(S^{op},H^{*})$, the Brauer group of dyslectic Hopf Yetter-Drinfel’d $(S^{op},H^{*})$-module algebras, where S is an H-commutative Hopf Yetter-Drinfel’d H-module algebra and S^op is the opposite algebra of S.

Communicated by Alberto Elduque

KEYWORDS:

• Braided monoidal categories
• Brauer groups
• dyslectic dimodules
• dyslectic Yetter-Drinfel’d modules
• Hopf algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16W30
• Secondary: 16K50
• 16T05 18D10