Pre-Lie algebra

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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.


A pre-Lie algebra is a vector space with a bilinear 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.


If we denote by the vector field , and if we define as , we can see that the operator is exactly the application of the field to field.

If we study the difference between and , we have which is symmetric on y and z.

  • Rooted trees

Let be the 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.