title: Symmetries and theu-condition in Hom-Yetter-Drinfeld categories.
abstract: 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 theu-condition in [Formula: see text] and show that the Hom-Yetter-Drinfeld module (H, adjoint, Δ, α) (resp., (H,m, coadjoint, α)) satisfies theu-condition if and only ifS2=id. Finally, we prove that [Formula: see text] over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.
authors.affiliation: School of Mathematics and Finance, Chuzhou University , Chuzhou 239000, China.
authors.forename: Shengxiang
authors.initials: S
authors.lastname: Wang
authors.affiliation: School of Mathematics and Statistics, Guizhou University of Finance and Economics , Guizhou 550025, China.
authors.forename: Shuangjian
authors.initials: S
authors.lastname: Guo