Algebra and Applications 2. Группа авторов

15): the price to pay is that one has to replace the Hopf algebra ℋ by a non-connected bialgebra with a suitable coproduct, such that ℋ is obtained as the quotient , where is the ideal generated by • – 1. The substitution product * then coincides with the one considered in Chartier et al. (2010) via natural identifications.

1.7. Other related algebraic structures

      1.7.1. NAP algebras

      NAP algebras (NAP for Non-Associative Permutative) appear under this name in Livernet (2006), and under the name “left- (right-)commutative algebras” in Dzhumadil’daev and Löfwall (2002). They can be seen in some sense as a “simplified version” of pre-Lie algebras. Saidi showed that the pre-Lie operad is a deformation of the NAP operad in a precise sense, involving the notion of current-preserving operad (Saidi 2014).

      1.7.1.1. Definition and general properties

      A left NAP algebra over a field k is a k-vector space A with a bilinear binary composition ▶ that satisfies the left NAP identity:

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      for any a, b, c ∈ A. Analogously, a right NAP algebra is a k-vector space A with a binary composition ◀ satisfying the right NAP identity:

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      As any right NAP algebra is also a left NAP algebra with product a ▶ b := b ◀ a, we can stick to left NAP algebras, which is what we will do unless specifically indicated.

      1.7.1.2. Free NAP algebras

      The left Butcher product s ∘ t of two rooted trees s and t is defined by grafting s on the root of t. For example:

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      The following theorem is due to Dzhumadil’daev and Löfwall (2002) (see Livernet (2006) for an operadic approach):

      THEOREM 1.5.– The free NAP algebra with d generators is the vector space spanned by rooted trees with d colors, endowed with the left Butcher product.

      PROOF.– We give the proof for one generator, the case of d generators being entirely similar. The left NAP property for the Butcher product is obvious. Let (A, ▶) be any left NAP algebra, and let a ∈ A. We have to prove that there exists a unique left NAP algebra morphism Ga from to (A, ▶), such that Ga(•) = a. As in the pre-Lie case, we proceed by double induction, first on the number n of vertices, and second on the number k of branches. In the case k = 1, the tree t writes B₊(t₁) = t₁ ∘ ∙; hence, Ga(t) = Ga(s) ▶ a is the only possible choice. Now a tree with k branches writes:

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      The only possible choice is then:

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      and the result is clearly symmetric in t₁ and t₂ due to the left NAP identity in A. Using the induction hypothesis, the result is also invariant under permutation of the branches 2,3,…,k. Hence, it is invariant under the permutation of all branches, which proves the theorem. □

      Despite the similarity with the pre-Lie situation described in section 1.6.2, the NAP framework is much simpler due to the set-theoretic nature of the Butcher product: for any trees s and t, the Butcher product s ∘ t is a tree, whereas the grafting s → t is a sum of trees. We obtain for the first trees:

      1.7.1.3. NAP algebras of vector fields

      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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