The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra H. We first show that the category HH𝒲ℳℋ𝒴𝒟 of Yetter–Drinfeld modules over H with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category HH𝒲ℳℋ𝒴𝒟. Finally, we prove that the representation category of triangular weak monoidal Hom–Hopf algebra is a symmetric braided monoidal subcategory of HH𝒲ℳℋ𝒴𝒟. Furthermore, a class of weak monoidal Hom–Yetter–Drinfeld modules are constructed by a quasitriangular weak monoidal Hom–Hopf algebra.

中文翻译：

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弱幺半群 Hom-Yetter-Drinfeld 范畴中的对称性和 u 条件

本文的目的是介绍和研究弱幺半 Hom-Hopf 代数上的 Yetter-Drinfeld 范畴H. 我们首先证明类别HH𝒲ℳℋ𝒴𝒟Yetter-Drinfeld 模块H具有双射对映体的是编织单曲面类别。其次，我们讨论了范畴对称性的一些性质HH𝒲ℳℋ𝒴𝒟. 最后，我们证明了三角弱幺半群 Hom-Hopf 代数的表示范畴是一个对称的编织幺半群子范畴HH𝒲ℳℋ𝒴𝒟. 此外，一类弱幺半群 Hom-Yetter-Drinfeld 模是由一个拟三角形弱幺半群 Hom-Hopf 代数构造的。