Averaging pre-Lie bialgebras

TL;DR

This paper develops the theory of averaging pre-Lie bialgebras, establishing their connections with Rota-Baxter operators, Yang-Baxter equations, and Lie bialgebras, advancing the algebraic understanding of these structures.

Contribution

It introduces representations, matched pairs, and bialgebra theories for averaging pre-Lie algebras, and links them with Rota-Baxter operators and Yang-Baxter equations.

Findings

Averaging pre-Lie bialgebras are equivalent to certain matched pairs and Manin triples.

Averaging operators on quadratic Rota-Baxter pre-Lie algebras generate averaging pre-Lie bialgebras.

Every averaging pre-Lie bialgebra induces an averaging Lie bialgebra.

Abstract

In this paper, we first introduce representations of averaging pre-Lie algebras and study their matched pairs, Manin triples, and bialgebra theories. We prove that these three notions are equivalent under certain conditions. Moreover, by introducing averaging operators on quadratic Rota-Baxter pre-Lie algebras, we show that such operators give rise to averaging pre-Lie bialgebras. Then we introduce the notion of admissible classical Yang-Baxter equations in averaging pre-Lie algebras, as well as the relative Rota-Baxter operators on averaging pre-Lie algebras, and show that the relative Rota-Baxter operators on averaging pre-Lie algebras yield symmetric solutions of admissible classical Yang-Baxter equations in averaging pre-Lie algebras. Finally, we show that every averaging pre-Lie bialgebra induces an averaging Lie bialgebra.