A rigidity theorem for pre-Lie algebras

In this paper we prove a “Leray theorem” for pre-Lie algebras. We define a notion of “Hopf” pre-Lie algebra: it is a pre-Lie algebra together with a non-associative permutative coproduct Δ and a compatibility relation between the pre-Lie product and the coproduct Δ. A non-associative 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 connected “Hopf” pre-Lie algebra is a free pre-Lie algebra. It uses the description of pre-Lie algebras in terms of rooted trees developed by Chapoton and the author. We also interpret this theorem by way of cogroups in the category of pre-Lie algebras.