Covariant Bimodules Over Monoidal Hom-Hopf Algebras

TL;DR

This paper develops the theory of covariant Hom-bimodules over monoidal Hom-Hopf algebras, establishing their categorical structures and equivalences, extending fundamental theorems in Hom-Hopf module theory.

Contribution

It introduces covariant Hom-bimodules, studies their structure, and proves a braided monoidal equivalence with Hom-Yetter-Drinfel'd modules, extending the fundamental theorem of Hom-Hopf modules.

Findings

Category of bicovariant Hom-bimodules is braided monoidal

Category of Hom-Yetter-Drinfel'd modules is braided tensor category

Established monoidal equivalence between these categories

Abstract

Covariant Hom-bimodules are introduced and the structure theory of them in the Hom-setting is studied in a detailed way. The category of bicovariant Hom-bimodules is proved to be a (pre)braided monoidal category and its structure theory is also provided in coordinate form. The notion of Hom-Yetter-Drinfel'd modules is presented and it is shown that the category of Hom-Yetter-Drinfel'd modules is a (pre)braided tensor category as well. As one of the main results, a (pre)braided monoidal equivalence between these tensor categories is verified, which extends the fundamental theorem of Hom-Hopf modules.