Symmetry and pseudosymmetry of v-Yetter–Drinfeld categories for Hom–Hopf algebras

Abstract

The purpose of this paper is to introduce the category of [Formula: see text]-Yetter–Drinfeld modules ([Formula: see text]) over a Hom–Hopf algebra. We first prove that every category of [Formula: see text]-Yetter–Drinfeld modules over a Hom–Hopf algebra with a bijective antipode [Formula: see text] is a braided tensor category and that every [Formula: see text]-Yetter–Drinfeld module can provide the solution of the Hom–Yang–Baxter equation. Secondly, we find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of [Formula: see text]-Yetter–Drinfeld modules by a quasitriangular Hom–Hopf algebra and study their relationship.