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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

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
Published electronically: December 15, 2010
MathSciNet review: 2763748
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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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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: http://dx.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.