Pre-Lie algebra
Encyclopedia
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.
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a pre-Lie algebra is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
on a vector space, that describes some properties of objects such as rooted trees
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
and vector fields on affine space.
The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber
Murray Gerstenhaber
Murray Gerstenhaber is an American mathematician, professor of mathematics at the University of Pennsylvania, best known for his contributions to theoretical physics with his discovery of Gerstenhaber algebra.- About :...
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 bilinear map , satisfying the relationThis identity can be seen as the invariance of the associator
Associator
In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure.-Ring theory:...
under the exchange of the two variables and .Every associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
is hence also a pre-Lie algebra, as the associator vanishes identically.
Examples
- Vector fields on the affine space
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 havewhich 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.