The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras

TL;DR

This paper introduces the classical Hom-Yang-Baxter equation as a twisted generalization of the CYBE, explores solutions induced by CYBE solutions, and develops the theory of Hom-Lie bialgebras, including duality and coboundary structures.

Contribution

It defines the classical Hom-Yang-Baxter equation and develops the theory of Hom-Lie bialgebras, extending classical concepts to the Hom-Lie setting.

Findings

Solutions of CYBE induce infinite families of solutions of CHYBE.

Hom-Lie bialgebras generalize Drinfel'd's Lie bialgebras.

Analysis of duality and coboundary structures in Hom-Lie bialgebras.

Abstract

Motivated by recent work on Hom-Lie algebras and the Hom-Yang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of Hom-Lie bialgebras, which generalize Drinfel'd's Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the sub-classes of coboundary and quasi-triangular Hom-Lie bialgebras.