On the Lie envelopping algebra of a pre-Lie algebra

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove 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, using brace algebras instead of pre-Lie algebras, we give a similar interpretation of Foissy’s Hopf algebra of planar rooted trees.