Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras

TL;DR

This paper explores the structure of (quasi-)twilled pre-Lie algebras, their associated $L_$-algebras, and how twisting transformations relate to solutions of Maurer-Cartan equations, with applications to (quasi-)pre-Lie bialgebras.

Contribution

It introduces a new framework linking twisting of pre-Lie algebras with Maurer-Cartan solutions and constructs quasi-pre-Lie bialgebras from symplectic Lie algebras.

Findings

Twisting transformations characterized by Maurer-Cartan solutions.

$\\mathcal{O}$-operators are solutions to Maurer-Cartan equations.

Construction of quasi-pre-Lie bialgebras from symplectic Lie algebras.

Abstract

We study (quasi-)twilled pre-Lie algebras and the associated $L_\infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_\infty$-algebras). Furthermore, we show that $\mathcal{O}$-operators and twisted $\mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_\infty$-algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.