Читать книгу Algebra and Applications 2 - Группа авторов - Страница 47

We consider the flat affine n-dimensional space En although it is possible, through parallel transport, to consider any smooth manifold endowed with a flat torsion-free connection. Fix an origin in En, which will be denoted by O. For vector fields and , we set:

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where is the constant vector field obtained by freezing the coefficients of X at x = O.

PROPOSITION 1.17.– The space χ(ℝⁿ) of vector fields endowed with product ▶ O is a left NAP algebra. Moreover, for any other choice of origin O’ ∈ En, the conjugation with the translation of vector is an isomorphism from (χ(ℝⁿ), ▶O)) onto (χ(ℝⁿ), ▶_O’)).

PROOF.– Let , and be three vector fields. Then:

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is symmetric in X and Y, due to the fact that the two constant vector fields XO and YO commute. The second assertion is left as an exercise for the reader. □

With the notations of section 1.6.4, there is a unique NAP algebra morphism

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the frozen Cayley map, such that . By also considering the unique NAP algebra morphism , the maps GX,O(t) : ℝⁿ → ℝⁿ are called the frozen elementary differentials.

PROPOSITION 1.18.– For any rooted tree t, each vertex v being decorated by a vector field Xv, the vector field is given at x ∈ ℝⁿ by the following recursive procedure: if the decorated tree t is obtained by grafing all of its branches tk on the root r decorated by the vector field , that is, if it writes , then:

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with:

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where stands for the k^th differential of fi, evaluated at x.

PROOF.– We prove the result by induction on the number k of branches: for k = 1, we check:

Now we can compute using the induction hypothesis and the fact that the vector fields are constant:

COROLLARY 1.3 (closed formula).– With the notations of Corollary 1.2, 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:

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