Symmetries in Yetter-Drinfel'd-Long categories

Symmetric categories have been of great interest in quantum algebra and mathematical physics. Cohen and Westreich in 1998 studied symmetries in the Yetter-Drinfel'd category over a Hopf algebra under some conditions. Pareigis in 2001 found the necessary and sufficient condition for $\!^{H}_{H}\mathcal{YD}$ to be symmetric. Later, Panaite et al. in 2010 proposed the definition of pseudosymmetric braided categories which can be viewed as a kind of weakened symmetric braided categories, and showed that the category $\!_{H}\mathcal{YD}^{H}$ is pseudosymmetric if and only if is commutative and cocommutative. Let $H$ be a Hopf algebra and $\mathcal{LR}(H)$ the category of Yetter-Drinfel'd-Long bimodules over $H$. We first show that the Yetter-Drinfel'd-Long category $\mathcal{LR}(H)$ is symmetric if and only if $H$ is trivial in four different methods, and that $\mathcal{LR}(H)$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We then introduce the definition of the $u$-condition in $\mathcal{LR}(H)$ and give a necessary and sufficient condition for $H_{i}$ $(i=1,2,3,4)$ to satisfy the $u$-condition. Then we study the relation between the $u$-condition and the symmetry of $\mathcal{LR}(H)$.