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

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.

arXiv commentView in color, 12 pages

References in corpus (4)

• Pre-Lie algebras and the rooted trees operad
• Lie elements in pre-Lie algebras, trees and cohomology operations
• A rigidity theorem for preLie algebras
• The non-symmetric operad pre-Lie is free

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• The Lie algebra of classical mechanics
• Fine structures inside the PreLie operad
• Clones of pigmented words and realizations of special classes of monoids