The constructions of 3-Hom-Lie bialgebras

TL;DR

This paper introduces 3-Hom-Lie bialgebras, explores their relation to Manin triples, and constructs solutions to the 3-Lie classical Hom-Yang-Baxter equation using $\\mathcal{O}$-operators and 3-Hom-pre-Lie algebras.

Contribution

It defines 3-Hom-Lie bialgebras, establishes their equivalence to Manin triples, and links phase spaces of 3-Hom-Lie algebras to 3-Hom-pre-Lie algebras.

Findings

3-Hom-Lie bialgebras are equivalent to Manin triples.

Constructed solutions to the 3-Lie classical Hom-Yang-Baxter equation.

Characterized phase spaces of 3-Hom-Lie algebras via 3-Hom-pre-Lie algebras.

Abstract

In this paper, we first introduce the notion of a 3-Hom-Lie bialgebra and prove that it is equivalent to a Manin triple of 3-Hom-Lie algebras. Also, we study the $\mathcal{O}$-operator and construct solutions of the 3-Lie classical Hom-Yang-Baxter equation interms of $\mathcal{O}$-operators and 3-Hom-pre-Lie algebras. Finally, we show that a 3-Hom-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom-pre-Lie algebra.