Let (H, S, α) be a monoidal Hom-Hopf algebra and [Formula: see text] the Hom-Yetter-Drinfeld category over (H, α). Then in this paper, we first find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Second, we study the u-condition in [Formula: see text] and show that the Hom-Yetter-Drinfeld module (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) satisfies the u-condition if and only if S2 = id. Finally, we prove that [Formula: see text] over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.

中文翻译：

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Hom-Yetter-Drinfeld 范畴中的对称性和 theu 条件

设 (H, S, α) 是一个幺半群 Hom-Hopf 代数，[公式：见正文] 是 (H, α) 上的 Hom-Yetter-Drinfeld 范畴。那么在本文中，我们首先找到了[公式：见正文]分别是对称和伪对称的充要条件。其次，我们研究 [公式：见正文] 中的 u 条件并表明 Hom-Yetter-Drinfeld 模块 (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) 满足u 条件当且仅当 S2 = id。最后，我们证明 [公式：见正文] 在三角形（或共三角）Hom-Hopf 代数上包含丰富的对称子类别。