A construction of Hom-Yetter-Drinfeld category

TL;DR

This paper constructs the Hom-Yetter-Drinfeld category using Radford biproduct Hom-Hopf algebras, showing it forms a pre-braided tensor category and linking it to solutions of the Hom-Yang-Baxter equation.

Contribution

It introduces the Hom-Yetter-Drinfeld category via Radford biproducts and establishes its structure as a pre-braided tensor category, extending the theory of Hom-Hopf algebras.

Findings

Hom-Yetter-Drinfeld modules solve the Hom-Yang-Baxter equation

The category $_H^H{ ext{YD}}$ is pre-braided tensor category

Characterization of Radford biproduct Hom-Hopf algebras

Abstract

In continuation of our recent work about smash product Hom-Hopf algebras in \cite{MLY}, we introduce Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ via Radford biproduct Hom-Hopf algebra, and prove that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_H^H{\mathbb{YD}}$ is a pre-braided tensor category, where $(H, \b, S)$ is a Hom-Hopf algebra. Furthermore, we obtain that $(A^{\natural}_{\diamond} H,\a\o \b)$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\a)$ is a Hopf algebra in the category $_H^H{\mathbb{YD}}$. At last, some examples and applications are given.