Algebra and Applications 2

Suppose first that k = 1, that is t = B +( t 1) = t 1→ •. Then, we obviously have Fa ( t ) = Fa ( t 1) ⊳ a . Suppose now that Fa ( s ) is unambiguously defined for any tree s with n vertices and k ′ branches with k ′ ≤ k – 1. The equation:

shows that, if Fa ( t ) exists, it is uniquely defined by:

[1.81]

What remains to be shown is that this expression does not depend on the choice of the distinguished branch t 1. In order to see this, choose a second branch (say t 2), and consider the expression:

[1.82]

which is obtained by grafting t 1and t 2on B +( t 3, … , tk ). This expression is the sum of five terms:

1 1) T1, obtained by grafting t1 and t2 on the root. It is nothing but the tree t itself.

2 2) T2, obtained by grafting t1 on the root and t2 elsewhere.

3 3) T3, obtained by grafting t2 on the root and t1 elsewhere.

4 4) T4, obtained by grafting t1 on some branch and t2 on some other branch.

5 5) T5, obtained by grafting t1 and t2 on the same branch.

The terms Fa ( T 2) + Fa ( T 3), Fa ( T 4) and Fa ( T 5) are well-defined by the induction hypothesis on the number of branches, and are obviously symmetric in t 1and t 2. We thus arrive at:

which is symmetric in t 1and t 2thanks to the left pre-Lie relation in A . The expression [1.81]is then the same if we exchange t 1with the branch t 2or any other branch; hence, it is invariant by any permutation of the branches t 1,…, t n. This proves Theorem 1.4 for one generator. The general case is proven similarly, except we have to replace a ∈ A by a collection { a 1,…, ad }.

Loday and Ronco (2010) have found a deep link between pre-Lie algebras and commutative Hopf algebras of a certain type: let ℋ be a commutative Hopf algebra. Following this, we say that ℋ is right-sided if it is free as a commutative algebra, that is, ℋ = S ( V ) for some k -vector space V , and if the reduced coproduct verifies:

[1.83]

Suppose that V = ⊕ n ≥ 0is graded with finite-dimensional homogeneous components. Then, the graded dual A = V 0is a left pre-Lie algebra, and by the Milnor-Moore theorem, the graded dual ℋ 0is isomorphic to the enveloping algebra as graded Hopf algebra. Conversely, for any graded pre-Lie algebra A , the graded dual is free commutative right-sided (Loday and Ronco 2010, Theorem 5.3).

The Hopf algebra ℋ CKof rooted forests enters into this framework, and, as it was first explicited in Chapoton (2001), the associated pre-Lie algebra is the free pre-Lie algebra of rooted trees with grafting: to see this, denote by ( δs ) the dual basis in the graded dual of the forest basis of ℋ CK. The correspondence δ : s ↦ δs extends linearly to a unique vector space isomorphism from ℋ CKonto . For any tree t , the corresponding δt is an infinitesimal character of ℋ CK, that is, it is a primitive element of ℋ°. We denote by * the (convolution) product of ℋ°. We have:

[1.84]

Here, t ↷ u is obtained by grafting t on u , namely:

[1.85]

where N ′( t , u , v ) is the number of partitions V ( t ) = V ∐ W, W < V , such that υ | V= 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 1.6.1.2 2 .

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 ( t 1,…, 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.