and deformations of algebras, under the name “pre-Lie algebras”, which is now the standard terminology. The term “chronological algebras” has also been used sometimes, for example, in the fundamental work of Agrachev and Gamkrelidze (1981). The notion itself can, however, be traced back to the work of Cayley (1857) which, in modern language, describes the pre-Lie algebra morphism Fa from the pre-Lie algebra of rooted trees into the pre-Lie algebra of vector fields on ℝn, sending the one-vertex tree to a given vector field a. For a survey emphasizing geometric aspects, see Burde (2006).
1.4.1. Definition and general properties
A left pre-Lie algebra over a field k is a k-vector space A with a bilinear binary composition ⊳ that satisfies the left pre-Lie identity:
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for a, b, c ∈ A. Analogously, a right pre-Lie algebra is a k-vector space A with a binary composition ⊲ that satisfies the right pre-Lie identity:
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The left pre-Lie identity is rewritten as:
where La: A → A is defined by Lab = a ⊳ b, and the bracket on the left-hand side is defined by [a, b] := a ⊳ b – b ⊳ a. As an easy consequence, this bracket satisfies the Jacobi identity: If A is unital (i.e. there exists 1 ∈ A, such that 1 ⊳ a = a ⊳ 1 = 1 for any a ∈ A), it is immediate thanks to the fact that L : A → End A is injective. If not, we can add a unit by considering
1.4.2. The group of formal flows
The following is taken from the paper by Agrachev and Gamkrelidze (1981). Suppose that A is a left pre-Lie algebra endowed with a compatible decreasing filtration, namely, A = A1 ⊃ A2 ⊂ A3 ⊃ …, such that the intersection of the Aj’s reduces to {0}, and such that Ap ⊳ Aq ⊂ Ap+q. Suppose, moreover, that A is complete with respect to this filtration. The Baker-Campbell-Hausdorff formula:
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then endows A with a structure of a pro-unipotent group. An example of this situation is given by A = hB[[h]], 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:
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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:
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where the Bis are the Bernoulli numbers. The first few terms are:
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By transferring the BCH product by means of the map W, namely:
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:
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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:
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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:
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and
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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 ALie 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 ALie, 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 :
