In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a pre-Lie algebra is an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
that describes some properties of objects such as
rooted trees and
vector fields on
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
.
The notion of pre-Lie algebra has been introduced by
Murray Gerstenhaber
Murray Gerstenhaber (born June 5, 1927) is an American mathematician and professor of mathematics at the University of Pennsylvania, best known for his contributions to theoretical physics with his discovery of Gerstenhaber algebra. He is also a ...
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 relation
This identity can be seen as the invariance of the
associator In abstract algebra, the term associator is used in different ways as a measure of the non-associativity of an algebraic structure. Associators are commonly studied as triple systems.
Ring theory
For a non-associative ring or algebra R, the associ ...
under the exchange of the two variables
and
.
Every
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
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: