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arXiv · 1004.1204

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

Nantel Bergeron·Jean-Louis Loday
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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.

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Nantel Bergeron, Jean-Louis Loday. 2010-04-07. The symmetric operation in a free pre-Lie algebra is magmatic. https://doi.org/10.1090/s0002-9939-2010-10813-4

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