Algebra and Applications 2. Группа авторов

, which interact as follows (Manchon and Saidi 2011):

      [1.79]

      The first pre-Lie operation ⊲ comes from the fact that is an augmented operad, whereas the second pre-Lie operation → comes from the fact that is the pre-Lie operad itself! Similarly:

      THEOREM 1.4.– The free pre-Lie algebra with d generators is the vector space of rooted trees with d colors, endowed with grafting.

1.6.2. A pedestrian approach to free pre-Lie algebra

      In this section we give a direct proof of Theorem 1.4 without using operads. It is similar to the proof of the main theorem in Chapoton and Livernet (2001) about the structure of the pre-Lie operad, except that we consider unlabeled trees. We stick to d = 1 (i.e. one generator), the proof for several generators being completely analogous. Let

be the vector space spanned by rooted trees. First, the grafting operation is pre-Lie, because for any trees s, t and u in

, the expression:

      [1.80]

      is obtained by summing up all of the possibilities of grafting s and t at some vertex of u. As such, it is obviously symmetric in s and t. Now let (A, ⊳) be any left pre-lie algebra, and choose any a ∈ A. In order to prove Theorem 1.4 for one generator, we have to show that there is a unique pre-Lie algebra morphism , such that Fa(•) = a. For the first trees, we easily obtain:

      Can we continue like this? We proceed by double induction, first, on the number of vertices, second, on the number of branches, that is, the valence of the root. Write any tree t with n vertices as t = B₊ (t₁,…, tk), where the tjs are the branches and B₊ is the operator that grafts the branches on a common extra root. By the induction hypothesis on n, the images Fa(tj) are well-defined.

Suppose first that k = 1, that is t = B₊(t₁) = t₁ → •. Then, we obviously have Fa(t) = Fa(t₁) ⊳ 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 t₁. In order to see this, choose a second branch (say t₂), and consider the expression:

      [1.82]

      which is obtained by grafting t₁ and t₂ on B₊(t₃, … , 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(T₂) + Fa(T₃), Fa(T₄) and Fa(T₅) are well-defined by the induction hypothesis on the number of branches, and are obviously symmetric in t₁ and t₂. We thus arrive at:

which is symmetric in t₁ and t₂ thanks to the left pre-Lie relation in A. The expression [1.81] is then the same if we exchange t₁ with the branch t₂ or any other branch; hence, it is invariant by any permutation of the branches t₁,…, 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 {a₁,…,ad}.

1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem

      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 ≥ 0} is graded with finite-dimensional homogeneous components. Then, the graded dual A = V⁰ is a left pre-Lie algebra, and by the Milnor-Moore theorem, the graded dual ℋ⁰ is 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_CK of 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_CK onto . 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