Four-angle Hopf modules for Hom-Hopf algebras

TL;DR

This paper introduces four-angle Hopf modules for Hom-Hopf algebras, establishing their monoidal structure, and shows their categorical equivalence with Yetter-Drinfel'd modules, including braiding structures.

Contribution

It defines four-angle Hopf modules for Hom-Hopf algebras and proves their monoidal category is equivalent to Yetter-Drinfel'd modules, with new braiding structures.

Findings

The category of four-angle Hopf modules is monoidal with Hom-tensor and Hom-cotensor products.

Yetter-Drinfel'd modules form a braided monoidal category with a new structure.

Equivalence between four-angle Hopf modules and Yetter-Drinfel'd modules is established, including braiding.

Abstract

In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,\beta)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor product $\otimes_{H}$ or a Hom-cotensor product $\Box_{H}$ as a monoidal product. We study the category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules with bijective structure map can be organized as a braided monoidal category, in which we use a new monoidal structure. Finally, We prove an equivalence between the monoidal category $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ or $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$ of four-angle Hopf modules, and the monoidal category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules, and furthermore, we give a braiding structure of the monoidal categorys $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ (and $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$).