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This section exposes a result by Guin and Oudom (2005).

THEOREM 1.3.– Let A be any left pre-Lie algebra, and let S(A) be its symmetric algebra, that is, the free commutative algebra on A. Let A_Lie be the underlying Lie algebra of A, that is, the vector space A endowed with the Lie bracket given by [a, b] = a ⊳ b − b ⊳ a for any a, b ∈ A, and let be the enveloping algebra of A_Lie, endowed with its usual increasing filtration. Let us consider the associative algebra as a left module over itself.

There exists a left -module structure on S(A) and a canonical left -module isomorphism , such that the associated graded linear map Gr is an isomorphism of commutative graded algebras.

PROOF.– The Lie algebra morphism

extends by the Leibniz rule to a unique Lie algebra morphism L : A → Der S(A). Now we claim that the map M : A → End S(A) defined by:

[1.53]

is a Lie algebra morphism. Indeed, for any a, b ∈ A and u ∈ S(A) we have:

Hence

which proves the claim. Now M extends, by universal property of the enveloping algebra, to a unique algebra morphism . The linear map:

is clearly a morphism of left -modules. It is immediately seen by induction that for any a₁,…,an ∈ A, we have η(a₁ … an) = a₁ … an + v, where v is a sum of terms of degree < n – 1. This proves the theorem. □

REMARK 1.3.– Let us recall that the symmetrization map , uniquely determined by σ(an) = an for any a ∈ A and any integer n, is an isomorphism for the two A_Lie-module structures given by the adjoint action. This is not the case for the map η defined above. The fact that it is possible to replace the adjoint action of on itself by the simple left multiplication is a remarkable property ofpre-Lie algebras, and makes Theorem 1.3 different from the usual Lie algebra PBW theorem.

Let us finally note that, if p stands for the projection from S(A) onto A, for any a₁,…, ak ∈ A, we easily get:

[1.54]

by a simple induction on k. The linear isomorphism η transfers the product of the enveloping algebra into a noncommutative product * on defined by:

[1.55]

Suppose now that A is endowed with a complete decreasing compatible filtration as in section 1.4.2. This filtration induces a complete decreasing filtration S(A) = S(A)₀ ⊃ S(A)₁ ⊃ S(A)₂ ⊃ …, and the product * readily extends to the completion . For any a ∈ A, the application of equation [1.54] gives:

[1.56]

as an equality in the completed symmetric algebra .

According to equation [1.48], we can identify the pro-unipotent group {e^*a, a ∈ A} ⊂ and the group of formal flows of the pre-Lie algebra A by means of the projection p, namely:

[1.57]

for any a, b ∈ A.