On the Lie enveloping algebra of a pre-Lie algebra.

On the Lie enveloping algebra of a pre-Lie algebra. 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.