Symmetries in Yetter-Drinfel'd-Long categories

Abstract

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