The braided monoidal structure on the category of Hom-type Doi-Hopf modules

TL;DR

This paper constructs a braided monoidal structure on the category of Hom-type Doi-Hopf modules, unifying various Hom-Hopf algebra structures and relating the category to module categories over a smash product.

Contribution

It introduces a braided monoidal structure on Hom-type Doi-Hopf modules, unifying quasitriangular, coquasitriangular, and Hom-Yetter-Drinfeld modules, and establishes an isomorphism with a module category.

Findings

Category $_A\mathcal{M}(H)^C$ is braided monoidal.

Unification of quasitriangular and coquasitriangular Hom-Hopf algebras.

Category is isomorphic to $A\#C^*$-module category.

Abstract

Let $(H,\a_H)$ be a Hom-Hopf algebra, $(A,\a_A)$ a right $H$-comodule algebra and $(C,\a_C)$ a left $H$-module coalgebra. Then we have the category $_A\mathcal{M}(H)^C$ of Hom-type Doi-Hopf modules. The aim of this paper is to make the category $_A\mathcal{M}(H)^C$ into a braided monoidal category. Our construction unifies quasitriangular and coquasitriangular Hom-Hopf algebras and Hom-Yetter-Drinfeld modules. We study tensor identities for monoidal categories of Hom-type Doi-Hopf modules. Finally we show that the category $_A\mathcal{M}(H)^C$ is isomorphic to $A\#C^*$-module category.