On the Lie enveloping algebra of a pre-Lie algebra

Abstract

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 enveloping algebra of L Lie . Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. 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.