Combinatorics of pre-Lie products sharing a Lie bracket

TL;DR

This paper explores the operad governing multiple pre-Lie algebra structures sharing a common Lie bracket, providing a combinatorial description and revealing algebraic properties similar to the pre-Lie operad.

Contribution

It introduces a combinatorial framework for the operad controlling shared-bracket pre-Lie structures, extending known properties of the pre-Lie operad.

Findings

Operad admits a combinatorial description akin to Chapoton and Livernet's for pre-Lie operad

Shares many algebraic properties with the pre-Lie operad

Provides detailed analysis of operad structure for shared-bracket pre-Lie algebras

Abstract

We study in detail the operad controlling several pre-Lie algebra structures sharing the same Lie bracket. Specifically, we show that this operad admits a combinatorial description similar to that of Chapoton and Livernet for the pre-Lie operad, and that it has many of the remarkable algebraic properties of the pre-Lie operad.