A rigidity theorem for preLie algebras

In this paper we prove a ``Leray theorem’’ for preLie algebras. We define a notion of ‘‘Hopf’’ preLie algebra: it is a preLie algebra together with a non associative permutative coproduct D and a compatibility relation between the preLie product and the coproduct D. A nonassociative permutative algebra is a vector space together with a product satisfying the relation (ab)c=(ac)b. A non associative permutative coalgebra is the dual notion. We prove that any such graded connected “Hopf” preLie algebra is a free preLie algebra. It uses the description of preLie algebras in term of rooted trees developped by Chapoton and the author. We interpret also this theorem by way of cogroups in the category of preLie algebras.