# Pre-Lie pairs and triviality of the Lie bracket on the twisted hairy graph complexes

## TL;DR

This paper introduces pre-Lie pairs to analyze algebraic structures in hairy graph complexes, demonstrating that twisting with specific Maurer-Cartan elements trivializes their Lie algebra structure, aiding in understanding the rational homotopy types of mapping spaces.

## Contribution

It develops the theory of pre-Lie pairs and applies it to show that certain Lie algebra structures in hairy graph complexes become trivial after twisting, advancing the understanding of $E_n$ operad mapping spaces.

## Key findings

- Twisting with specific Maurer-Cartan elements trivializes the Lie algebra structure.
- Pre-Lie pairs provide a framework to analyze homotopy Lie algebras.
- Application to rational homotopy groups of $E_n$ operad mapping spaces.

## Abstract

We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the $E_n$ operads. We show that twisting with certain Maurer-Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of the mapping spaces of $E_n$ operads.