# Another approach to Hom-Lie bialgebras via Manin triples

## TL;DR

This paper introduces a new framework for Hom-Lie bialgebras using Manin triples, enabling the study of coboundary structures and solutions to the Hom-Yang-Baxter equation without skew-symmetry constraints.

## Contribution

It develops a novel approach to Hom-Lie bialgebras via Manin triples and explores their coboundary cases and solutions to the Hom-Yang-Baxter equation.

## Key findings

- Defined Hom-Lie bialgebras with Manin triples and invariance conditions.
- Constructed coboundary Hom-Lie bialgebras without skew-symmetry.
- Derived solutions to the classical Hom-Yang-Baxter equation from $\\mathcal{O}$-operators and Hom-left-symmetric algebras.

## Abstract

In this paper, we study Hom-Lie bialgebras by a new notion of the dual representation of a representation of a Hom-Lie algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce the notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear form satisfying a new invariance condition. With this notion, coboundary Hom-Lie bialgebras can be studied without a skew-symmetric condition of $r\in\mathfrak{g}\otimes \mathfrak{g}$, naturally leading to the classical Hom-Yang-Baxter equation whose solutions are used to construct coboundary Hom-Lie bialgebras. In particular, they are used to obtain a canonical Hom-Lie bialgebra structure on the double space of a Hom-Lie bialgebra. We also derive solutions of the classical Hom-Yang-Baxter equation from $\mathcal{O}$-operators and Hom-left-symmetric algebras.