Читать книгу Algebra and Applications 2 - Группа авторов - Страница 42

Cayley (1857) discovered a link between rooted trees and vector fields on the manifold ℝⁿ, endowed with its natural flat torsion free connection, which can be described in modern terms as follows: let be the free pre-Lie algebra on the space of vector fields on ℝⁿ. A basis of is given by rooted trees with vertices decorated by some basis of χ(ℝⁿ). There is a unique pre-Lie algebra morphism , the Cayley map, such that for any vector field X.

PROPOSITION 1.15.– For any rooted tree t, with each vertex v being decorated by a vector field Xv, the vector field is given at x ∈ ℝⁿ by the following recursive procedure (Hairer et al. 2002): if the decorated tree t is obtained by grafting all of its branches tk on the root r decorated by the vector field , that is, ifit writes , then:

[1.95]

[1.96]

where stands for the k^th differential of fi.

PROOF. – From equation [1.94], for any vector field X and any other vector field :

[1.97]

In other words, X ⊳ Y is the derivative of Y along the vector field X, where Y is viewed as a C^∞ map from ℝⁿ to ℝⁿ. We prove the result by induction on the number k of branches: for k = 1, we check:

Now, we can compute, using the Leibniz rule and the induction hypothesis (we drop the point x ∈ ℝⁿ where the vector fields are evaluated):

COROLLARY 1.2 (closed formula).– For any rooted tree t with set of vertices and root r, each vertex v being decorated by a vector field , the vector field is given at x ∈ ℝn by the following formula:

[1.98]

with the shorthand notation:

[1.99]

where the product runs over the incoming vertices of v.

Now fix a vector field X on ℝⁿ and consider the map dX from undecorated rooted trees to vector field-decorated rooted trees, which decorates each vertex by X. It is obviously a pre-Lie algebra morphism, and is the unique pre-Lie algebra morphism that sends the one-vertex tree • to the vector field X.