Pre-Lie algebra

In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.
The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.
Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.
Definition.
A pre-Lie algebra formula_1 is a vector space formula_2 with a bilinear map formula_3, satisfying the relation
formula_4
This identity can be seen as the invariance of the associator formula_5 under the exchange of the two variables formula_6 and formula_7.
Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator formula_8 is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the formula_9 terms in the defining relation for pre-Lie algebras, above.
Examples.
Vector fields on an affine space.
Let formula_10 be an open neighborhood of formula_11, parameterised by variables formula_12. Given vector fields formula_13, formula_14 we define formula_15. 
The difference between formula_16 and formula_17, is
formula_18
which is symmetric in formula_19 and formula_20. Thus formula_21 defines a pre-Lie algebra structure.
Given a manifold formula_22 and homeomorphisms formula_23 from formula_24 to overlapping open neighborhoods of formula_22, they each define a pre-Lie algebra structure formula_26 on vector fields defined on the overlap. Whilst formula_21 need not agree with formula_28, their commutators do agree: formula_29, the Lie bracket of formula_19 and formula_31. 
Rooted trees.
Let formula_32 be the free vector space spanned by all rooted trees.
One can introduce a bilinear product formula_33 on formula_32 as follows. Let formula_35 and formula_36 be two rooted trees.
formula_37
where formula_38 is the rooted tree obtained by adding to the disjoint union of formula_35 and formula_36 an edge going from the vertex formula_41 of formula_35 to the root vertex of formula_36.
Then formula_44 is a free pre-Lie algebra on one generator. More generally, the free pre-Lie algebra on any set of generators is constructed the same way from trees with each vertex labelled by one of the generators.