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Symmetries in Yetter-Drinfel'd-Long categories
1  School of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China

https://doi.org/10.3390/Symmetry2021-10726 (registering DOI)
Abstract:

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 $\mathcal{LR}(H)$.

Keywords: Symmetric category, Yetter-Drinfel'd-Long category, The $u$-condition, Pseudosymmetry