Braided monoidal categories and Doi Hopf modules for monoidal Hom-Hopf algebras

TL;DR

This paper develops the theory of Doi Hom-Hopf modules within monoidal Hom-Hopf algebras, establishing conditions for monoidality, constructing braidings, and exploring functors between related categories, with applications to Hom-modules and Hom-Yetter-Drinfeld modules.

Contribution

It introduces Doi Hom-Hopf modules, characterizes monoidal structures, and constructs braidings in these categories, advancing the understanding of Hom-Hopf algebra representations.

Findings

Established conditions for monoidal Doi Hom-Hopf modules

Constructed braiding on the category of Doi Hom-Hopf modules

Applied theory to Hom-modules, Hom-comodules, and Hom-Yetter-Drinfeld modules

Abstract

We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Second, we give the maps between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras give rise to functors between the category of Doi Hom-Hopf modules and study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf modules. Finally, as an application of our theory, we consider the braiding on the category of Hom-modules, the category of Hom-comodules and the category of Hom-Yetter-Drinfeld modules respectively.