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

Authors:
Nantel Bergeron and Jean-Louis Loday

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

MSC (2010):
Primary 16W30, 17A30, 18D50, 81R60

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

Published electronically:
December 15, 2010

MathSciNet review:
2763748

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Nantel Bergeron**

Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Email:
bergeron@yorku.ca

**Jean-Louis Loday**

Affiliation:
Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue R. Descartes, 67084 Strasbourg Cedex, France

Email:
loday@math.unistra.fr

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

Keywords:
Pre-Lie algebra,
Jordan algebra,
magmatic algebra,
dendriform algebra,
duplicial algebra,
operad,
planar tree

Received by editor(s):
May 25, 2010

Published electronically:
December 15, 2010

Communicated by:
Jim Haglund

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.