Algebra and Applications 2. Группа авторов

where B is any pre-Lie algebra, and Aj = hjB[[h]]. This group admits a more transparent presentation as follows: introduce a fictitious unit 1, such that 1 ⊳ a = a ⊳ 1 = a for any a ∈ A, and define W : A → A by:

      [1.45]

      The application W is clearly a bijection. The inverse, denoted by Ω, also appears under the name “pre-Lie Magnus expansion” in Ebrahimi-Fard and Manchon (2009b). It verifies the equation:

      [1.46]

      where the Bis are the Bernoulli numbers. The first few terms are:

      [1.47]

By transferring the BCH product by means of the map W, namely:

[1.48]

      we have W(a) # W(b) = W(C(a, b)) = eLa e Lb 1 – 1, hence W(a)#W(b) = W(a) + eLa W(b). The product # is thus given by the simple formula:

      [1.49]

      The inverse is given by a^#–1 = W(–Ω(a)) = e^–LΩ(a) 1 – 1. If (A, ⊳) and (B, ⊳) are two such pre-Lie algebras and ψ : A → B is a filtration-preserving pre-Lie algebra morphism, we should immediately check that for any a, b ∈ A we have:

      [1.50]

      In other words, the group of formal flows is a functor from the category of complete filtered pre-Lie algebras to the category of groups.

      When the pre-Lie product ⊳ is associative, all of this simplifies to:

      [1.51]

      and

      [1.52]

      1.4.3. The pre-Lie Poincaré–Birkhoff–Witt theorem

      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