Purely Hom-Lie bialgebras

TL;DR

This paper develops the theory of Hom-Lie bialgebras, establishing their structure, dual representations, and connections to Manin triples and the Hom-Yang-Baxter equation, expanding the algebraic framework in this area.

Contribution

It introduces the concept of purely Hom-Lie bialgebras, links them with Manin triples, and constructs solutions to the classical Hom-Yang-Baxter equations.

Findings

Hom-Lie algebra structure on $(\sigma,\sigma)$-derivations

One-to-one correspondence between Manin triples and purely Hom-Lie bialgebras

Construction of solutions to the classical Hom-Yang-Baxter equations

Abstract

In this paper, first we show that there is a Hom-Lie algebra structure on the set of $(\sigma,\sigma)$-derivations of a commutative algebra. Then we construct dual representations of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-$\mathcal O$-operators.