The construction of braided $T$-category via Yetter-Drinfeld-Long bimodules

TL;DR

This paper constructs a braided $T$-category from Yetter-Drinfeld-Long bimodules over Hopf algebras, generalizing existing categories and establishing isomorphisms under certain conditions.

Contribution

It introduces a new category generalizing Yetter-Drinfeld-Long bimodules and constructs a braided $T$-category encompassing these as components.

Findings

Constructed a braided $T$-category $\\mathcal{LR}(H_1,H_2)$.

Proved isomorphism with the usual category under quadruple in involution.

Generalized the structure to non-finite dimensional Hopf algebras.

Abstract

Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite dimensional and $\alpha,\beta \in Aut_{Hopf}(H_1), \gamma,\delta \in Aut_{Hopf}(H_2)$. In this paper, we introduce a category ${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$, generalizing Yetter-Drinfeld-Long bimodules and construct a braided $T$-category $\mathcal{LR}(H_1,H_2)$ containing all the categories $_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$ as components. We also prove that if $(\alpha, \beta, \gamma, \delta)$ admits a quadruple in involution, then ${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$ is isomorphic to the usual category ${}_{H_1}\mathcal{LR}_{H_2}$ of Yetter-Drinfeld-Long bimodules.