The symmetric operation in a free pre-Lie algebra is magmatic

TL;DR

This paper investigates the symmetrization of the pre-Lie product, revealing it only satisfies commutativity and establishing an injective map from free commutative-magmatic to free pre-Lie algebras, contrasting with associative cases.

Contribution

It proves that symmetrization of a pre-Lie product only enforces commutativity, providing new insights into the algebraic structure and injectivity of related algebraic maps.

Findings

Symmetrization of pre-Lie product satisfies only commutativity.

The induced map from free commutative-magmatic to free pre-Lie algebra is injective.

Contrasts with associative case where symmetrization leads to Jordan algebras.

Abstract

A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it does not satisfy any other universal relation than commutativity. It means that the map from the free commutative-magmatic algebra to the free pre-Lie algebra induced by the symmetrization of the pre-Lie product is injective. This result is in contrast with the associative case, where the symmetrization gives rise to the notion of Jordan algebra. We first give a self-contained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras.