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 linear map
, satisfying the relation
This identity can be seen as the invariance of the associator
under the exchange of the two variables
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.
be an open neighborhood of
, parameterised by variables
defines a pre-Lie algebra structure.
to overlapping open neighborhoods of
, they each define a pre-Lie algebra structure
on vector fields defined on the overlap.
, their commutators do agree:
, the Lie bracket of
be the free vector space spanned by all rooted trees.
One can introduce a bilinear product
be two rooted trees.
is the rooted tree obtained by adding to the disjoint union of
an edge going from the vertex
to the root vertex of
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.