Symmetries and the <i>u</i>-condition in Hom-Yetter-Drinfeld categories

Abstract

Let (<i>H</i>, <i>S</i>, α) be a monoidal Hom-Hopf algebra and [Formula: see text] the Hom-Yetter-Drinfeld category over (<i>H</i>, 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 <i>u</i>-condition in [Formula: see text] and show that the Hom-Yetter-Drinfeld module (<i>H</i>, adjoint, Δ, α) (resp., (<i>H</i>, <i>m</i>, coadjoint, α)) satisfies the <i>u</i>-condition if and only if <i>S</i>² = <i>id</i>. Finally, we prove that [Formula: see text] over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.