Sur l'algèbre enveloppante d'une algèbre pré-Lie

Abstract

We construct an associative product on the symmetric module S ( L ) of any pre-Lie algebra L. It turns S ( L ) into a Hopf algebra which is isomorphic to the envelopping algebra of L Lie . Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees. To cite this article: J.-M. Oudom, D. Guin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).