Читать книгу Algebra and Applications 2 - Группа авторов - Страница 28

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 ⊂ A_p+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:

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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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