Lois pr\'e-Lie en interaction

TL;DR

This paper explores the algebraic structure of pre-Lie algebras related to rooted forests, establishing a derivation relation between two different pre-Lie products defined in this context.

Contribution

It proves a derivation relation linking the pre-Lie product from a Hopf algebra of rooted forests to the grafting pre-Lie product.

Findings

Established a derivation relation between two pre-Lie structures.

Connected the primitive elements' pre-Lie product with grafting operations.

Enhanced understanding of algebraic structures in combinatorial Hopf algebras.

Abstract

D. Calaque, K. Ebrahimi-Fard and D. Manchon have recently defined a Hopf algebra by introducing a new coproduct on a commutative algebra of rooted forests. The space of primitive elements of the graded dual is endowed with a left pre-Lie product defined in terms of insertion of a tree inside another. In this work we prove a ``derivation'' relation between this pre-Lie structure and the left pre-Lie product defined by grafting.