The Symmetric Operation in a Free Pre-lie Algebra Is Magmatic

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. Introduction A pre-Lie algebra is a vector space equipped with a binary operation x ∗ y whose associator is right-symmetric. Its name comes from the fact that the antisymmetrization [x, y] := x ∗ y − y ∗ x of this binary operation is a Lie bracket. In this paper we investigate the symmetrization x#y := x ∗ y + y ∗ x of the pre-Lie product. We show that, in contrast to the bracket, it does not satisfy any universal relation (but commutativity of course). In other words the map of operads ComMag → preLie induced by the symmetrization is injective. Here ComMag stands for the operad encoding the algebras equipped with a commutative binary operation (sometimes called commutative nonassociative algebras in the literature). We give two different proofs of this result. The first one is self-contained and relies on the combinatorics of trees. The second one uses the dendriform algebras. More precisley we prove that the map of operads Mag → Dend is injective by using the theory of generalized bialgebras for duplicial algebras. Vladimir Dotsenko informed us that he found an alternative proof by using Groebner basis [3]. Received by the editors May 25, 2010. 2010 Mathematics Subject Classification. Primary 16W30, 17A30, 18D50, 81R60.