Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras

TL;DR

This paper introduces the Hom-Yang-Baxter Equation (HYBE), a twisted version inspired by Hom-Lie algebras, and constructs solutions that extend to braid group representations, linking Hom-algebra structures with braid theory.

Contribution

It defines the HYBE, constructs three classes of solutions from Hom-Lie and quasi-triangular bialgebras, and connects these solutions to braid group representations.

Findings

Constructed solutions from Hom-Lie algebras and quasi-triangular bialgebras.

Extended solutions to braid group representations under invertibility.

Established a link between Hom-algebra structures and braid relations.

Abstract

We study a twisted version of the Yang-Baxter Equation, called the Hom-Yang-Baxter Equation (HYBE), which is motivated by Hom-Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom-Lie algebras and the others from Drinfeld's (dual) quasi-triangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group.