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

Bergeron, Nantel; Loday, Jean-Louis

doi:10.1090/S0002-9939-2010-10813-4

The symmetric operation in a free pre-Lie algebra is magmatic
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by Nantel Bergeron and Jean-Louis Loday

Proc. Amer. Math. Soc. 139 (2011), 1585-1597

DOI: https://doi.org/10.1090/S0002-9939-2010-10813-4

Published electronically: December 15, 2010

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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. This 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 a 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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