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  is a vector space  with a linear map , satisfying the relation

This identity can be seen as the invariance of the associator  under the exchange of the two variables  and . 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  is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the  terms in the defining relation for pre-Lie algebras, above. Examples

Vector fields on an affine space 

Let  be an open neighborhood of , parameterised by variables . Given vector fields ,  we define . The difference between  and , is

which is symmetric in  and . Thus  defines a pre-Lie algebra structure. Given a manifold  and homeomorphisms  from  to overlapping open neighborhoods of , they each define a pre-Lie algebra structure  on vector fields defined on the overlap. Whilst  need not agree with , their commutators do agree: , the Lie bracket of  and . Rooted trees 

Let  be the free vector space spanned by all rooted trees. One can introduce a bilinear product  on  as follows. Let  and  be two rooted trees. where  is the rooted tree obtained by adding to the disjoint union of  and  an edge going from the vertex  of  to the root vertex of . Then  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. References
. . Lie groups
 
Non-associative algebra