## 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

## 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.## References

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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
- 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: https://doi.org/10.1090/S0002-9939-2010-10813-4
- MathSciNet review: 2763748