Symmetries in Yetter-Drinfel'd-Long categories

TL;DR

This paper investigates the conditions under which the category of Yetter-Drinfel'd-Long bimodules over a Hopf algebra exhibits symmetry or pseudosymmetry, introducing new concepts and exploring their implications.

Contribution

It provides necessary and sufficient conditions for symmetry and pseudosymmetry in $ ext{LR}(H)$, introduces the $u$-condition, and analyzes symmetric subcategories over triangular Hopf algebras.

Findings

$ ext{LR}(H)$ is symmetric under specific conditions.

The $u$-condition relates to the symmetry of $ ext{LR}(H)$.

Triangular Hopf algebras contain rich symmetric subcategories.

Abstract

Let $H$ be a Hopf algebra and $\mathcal{LR}(H)$ the category of Yetter-Drinfel'd-Long bimodules over $H$. We first give sufficient and necessary conditions for $\mathcal{LR}(H)$ to be symmetry and pseudosymmetry, respectively. We then introduce the definition of $u$-condition in $\mathcal{LR}(H)$ and discuss the relation between the $u$-condition and the symmetry of $\mathcal{LR}(H)$. Finally, we show that $\mathcal{LR}(H)$ over a triangular (cotriangular, resp.) Hopf algebra contains a rich symmetric subcategory.