Algebra and Applications 2. Группа авторов

These two diagrams commute if and only if e verifies the unit axiom and the partial compositions verify the two associativity axioms and the equivariance axiom (Loday and Vallette 2012).

1.6. Pre-Lie algebras (continued)

      1.6.1. Pre-Lie algebras and augmented operads

      1.6.1.1. General construction

      We adopt the notations of section 1.5. The sum of the partial compositions yields a right pre-Lie algebra structure on the free -algebra with one generator, more precisely on , namely:

      [1.77]

      Following Chapoton (2002), we can consider the pro-unipotent group associated with the completion of the pre-Lie algebra for the filtration induced by the grading. More precisely, Chapoton’s group is given by the elements , such that g₁ ≠ 0, whereas is the subgroup of formed by elements g, such that g₁ = e.

Any element gives rise to an n-ary operation , and for any , we have¹ (Manchon and Saidi 2011):

[1.78]

1.6.1.2. The pre-Lie operad

      Pre-Lie algebras are algebras over the pre-Lie operad, which has been described in detail by Chapoton and Livernet (2001) as follows: is the vector space of labeled rooted trees, and the partial composition s ∘_i t is given by summing all of the possible ways of inserting the tree t inside the tree s at the vertex labeled by i. To be precise, the sum runs over the possible ways of branching on t the edges of s, which arrive on the vertex i.

The free left pre-Lie algebra with one generator is then given by the space of rooted trees, as quotienting with the symmetric group actions amounts to neglect the labels. The pre-Lie operation (s,t) ↦ (s → t) is given by the sum of the graftings of s on t at all vertices of t. As a result of [1.78], we have two pre-Lie operations on , 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

      [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: