Читать книгу Algebra and Applications 2 - Группа авторов - Страница 38

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.