Algebra and Applications 2. Группа авторов

= t and υ|_W = u. Another normalization is often employed: considering the normalized dual basis , where σ(t) = |Aut t| stands for the symmetry factor of t, we obviously have:

      [1.86]

where:

      [1.87]

where can be interpreted as the number of ways to graft the tree t on the tree u in order to get the tree v. The operation → then coincides with the grafting free pre-Lie operation introduced in section 1.6.1.2².

      The other pre-Lie operation ⊲ of section 1.6.1.2, more precisely its opposite ⊳, is associated with another right-sided Hopf algebra of forests ℋ which has been investigated in Calaque et al. (2011) and Manchon and Saidi (2011), and which can be defined by considering trees as Feynman diagrams (without loops): let be the vector space spanned by rooted trees with at least one edge. Consider the symmetric algebra , which can be seen as the k-vector space generated by rooted forests with all connected components containing at least one edge. We identify the unit of with the rooted tree •. A subforest of a tree t is either the trivial forest •, or a collection (t₁,…, tn) of pairwise disjoint subtrees of t, each of them containing at least one edge. In particular, two subtrees of a subforest cannot have any common vertex.

      Let s be a subforest of a rooted tree t. Denote by t/s the tree obtained by contracting each connected component of s onto a vertex. We turn ℋ into a bialgebra by defining a coproduct Δ : ℋ → ℋ ⊗ ℋ on each tree by :

      [1.88]

      where the sum runs over all possible subforests (including the unit • and the full subforest t). As usual we extend the coproduct Δ multiplicatively onto . In fact, coassociativity is easily verified. This makes ℋ := ⊕_{n ≥ 0} ℋ_n a connected graded bialgebra, hence a Hopf algebra, where the grading is defined in terms of the number of edges. The antipode S : ℋ → ℋ is given (recursively with respect to the number of edges) by one of the two following formulae:

      [1.89]

      [1.90]

      It turns out that ℋ_CK is left comodule-bialgebra over ℋ (Calaque et al. 2011; Manchon and Saidi 2011), in the sense that the following diagram commutes:

Here, the coaction Φ : ℋ_CK → ℋ ⊗ ℋ_CK is the algebra morphism given by Φ(1) = • ⊗ 1 and Φ(t) = Δ_ℋ(t) for any nonempty tree t. As a result, the group of characters of ℋ acts on the group of characters of ℋ_CK by automorphisms.

1.6.4. Pre-Lie algebras of vector fields

      1.6.4.1. Flat torsion-free connections

      Let M be a differentiable manifold, and let ▽ be the covariant derivation operator associated with a connection on the tangent bundle TM. The covariant derivation is a bilinear operator on vector fields (i.e. two sections of the tangent bundle): (X, Y) ↦ ▽XY, such that the following axioms are fulfilled:

      The torsion of the connection

is defined by:

      [1.91]

      and the curvature tensor is defined by:

      [1.92]

The connection is flat if the curvature R vanishes identically, and torsion-free if . The following crucial observation by Matsushima (1968, Lemma 1) is an immediate consequence of equation [1.43]:

      PROPOSITION 1.14.– For any smooth manifold M endowed with aflat torsion-free connection ▽, the space χ(M) of vector fields is a left pre-Lie algebra, with pre-Lie product given by:

      [1.93]

Note that on M = ℝⁿ, endowed with its canonical flat torsion-free connection, the pre-Lie product is given by:

[1.94]

      1.6.4.2. Relating two pre-Lie structures

      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]