Drinfel'd double for monoidal Hom-Hopf algebras

TL;DR

This paper constructs a bicrossproduct for finite-dimensional monoidal Hom-Hopf algebras, generalizing Majid's work, and introduces a Drinfel'd double with a quasitriangular structure satisfying quantum Hom-Yang-Baxter equations.

Contribution

It extends the theory of bicrossproducts and Drinfel'd doubles to the Hom-Hopf algebra setting, providing new algebraic structures and quasitriangular properties.

Findings

Constructed bicrossproduct for monoidal Hom-Hopf algebras

Developed Drinfel'd double with quasitriangular structure

Satisfies quantum Hom-Yang-Baxter equations

Abstract

In this paper we mainly construct bicrossproduct for finite-dimensional monoidal Hom-Hopf algebra $(H,\alpha)$, generalizing the Majid's bicrossproduct. Naturally the Hom-type bicrossproduct leads to Drinfel'd double $(H^{op}\bowtie H^{\ast}, \alpha \otimes (\alpha^{-1})^\ast)$ with a quasitriangular structure $R$ satisfying the quantum Hom-Yang-Baxter equations.