Symmetries and the u-condition in weak monoidal Hom–Yetter–Drinfeld categories

Abstract

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