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.
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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
This identity can be seen as the invariance of the associator
Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically.
Examples
If we denote by
If we study the difference between
Let
One can introduce a bilinear product
where
Then