Yetter-Drinfeld modules for weak Hom-Hopf algebras

TL;DR

This paper introduces and analyzes Yetter-Drinfeld modules over weak Hom-Hopf algebras, establishing their categorical properties and deriving solutions to the quantum Hom-Yang-Baxter equation.

Contribution

It defines Yetter-Drinfeld modules for weak Hom-Hopf algebras and proves their category is rigid and braided, also providing new solutions to the quantum Hom-Yang-Baxter equation.

Findings

The category of Yetter-Drinfeld modules is a rigid braided monoidal category.

Derived a new solution to the quantum Hom-Yang-Baxter equation.

Showed subcategories for quasitriangular and coquasitriangular weak Hom-Hopf algebras.

Abstract

The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid category and a braided monoidal category, and obtain a new solution of quantum Hom-Yang-Baxter equation. It turns out that, If $H$ is quasitriangular (respectively, coquasitriangular)weak Hom-Hopf algebras, the category of modules (respectively, comodules) with bijective structure maps over $H$ is a braided monoidal subcategory of the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules over weak Hom-Hopf algebras.