In
mathematics, the Maurer–Cartan form for a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
is a distinguished
differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by
Élie Cartan as a basic ingredient of his
method of moving frames
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay t ...
, and bears his name together with that of
Ludwig Maurer
Ludwig Maurer (11 December 1859 – 10 January 1927) was a German mathematician and professor at University of Tübingen. He was the eldest son of Konrad Maurer (1823–1902) and Valerie Maurer, née von Faulhaber (1833–1912). His 1887 disserta ...
.
As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the
Lie algebra associated to the Lie group . The Lie algebra is identified with the
tangent space of at the identity, denoted . The Maurer–Cartan form is thus a one-form defined globally on which is a linear mapping of the tangent space at each into . It is given as the
pushforward of a vector in along the left-translation in the group:
:
Motivation and interpretation
A Lie group acts on itself by multiplication under the mapping
:
A question of importance to Cartan and his contemporaries was how to identify a
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
of . That is, a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
identical to the group , but without a fixed choice of unit element. This motivation came, in part, from
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's
Erlangen programme where one was interested in a notion of
symmetry on a space, where the symmetries of the space were
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
s forming a Lie group. The geometries of interest were
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
s , but usually without a fixed choice of origin corresponding to the
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
.
A principal homogeneous space of is a manifold abstractly characterized by having a
free and transitive action of on . The Maurer–Cartan form gives an appropriate ''infinitesimal'' characterization of the principal homogeneous space. It is a one-form defined on satisfying an
integrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the
exponential map of the Lie algebra and in this way obtain, locally, a group action on .
Construction
Intrinsic construction
Let be the tangent space of a Lie group at the identity (its
Lie algebra). acts on itself by left translation
:
such that for a given we have
:
and this induces a map of the
tangent bundle to itself:
A left-invariant
vector field is a section of such that
[Subtlety: gives a vector in ]
:
The Maurer–Cartan form is a -valued one-form on defined on vectors by the formula
:
Extrinsic construction
If is embedded in by a matrix valued mapping , then one can write explicitly as
:
In this sense, the Maurer–Cartan form is always the left
logarithmic derivative of the identity map of .
Characterization as a connection
If we regard the Lie group as a
principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique
principal connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connect ...
on the principal bundle . Indeed, it is the unique valued -form on satisfying
:#
:#
where is the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of forms along the right-translation in the group and is the
adjoint action
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
on the Lie algebra.
Properties
If is a left-invariant vector field on , then is constant on . Furthermore, if and are both left-invariant, then
: