Pre-Lie algebras and the rooted trees operad

TL;DR

This paper provides a combinatorial description of the operad associated with Pre-Lie algebras using rooted trees and establishes its Koszul property, advancing the understanding of their algebraic structure.

Contribution

It introduces a rooted tree-based combinatorial model for the Pre-Lie operad and proves its Koszulity, offering new insights into their algebraic and operadic properties.

Findings

Explicit combinatorial description of the Pre-Lie operad

Proof that the Pre-Lie operad is Koszul

Enhanced understanding of Pre-Lie algebra structures

Abstract

A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.