Читать книгу Algebra and Applications 2 - Группа авторов - Страница 27

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:

[1.42]

The left pre-Lie identity is rewritten as:

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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 and extend accordingly. As any right pre-Lie algebra (A, ⊲) is also a left pre-Lie algebra with product a ⊳ b := b ⊲ a, we can stick to left pre-Lie algebras, which we will do unless specifically indicated.