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THE SYMMETRIC OPERATION IN A FREE PRE-LIE ALGEBRA IS MAGMATIC

By Nantel Bergeron, Jean-louis Loday and Communicated Jim Haglund

Abstract

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 selfcontained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.2212
Provided by: CiteSeerX
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