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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The symmetric operation in a free pre-Lie algebra is magmatic
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by Nantel Bergeron and Jean-Louis Loday PDF
Proc. Amer. Math. Soc. 139 (2011), 1585-1597 Request permission


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:
  • Jean-Louis Loday
  • Affiliation: Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue R. Descartes, 67084 Strasbourg Cedex, France
  • Email:
  • Received by editor(s): May 25, 2010
  • Published electronically: December 15, 2010
  • Communicated by: Jim Haglund
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1585-1597
  • MSC (2010): Primary 16W30, 17A30, 18D50, 81R60
  • DOI:
  • MathSciNet review: 2763748