In
mathematics, the Maurer–Cartan form for a
Lie group is a distinguished
differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
as a basic ingredient of his
method of moving frames, and bears his name together with that of
Ludwig Maurer.
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
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
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-e ...
of . That is, a
manifold 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 grou ...
's
Erlangen programme
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
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
* Tran ...
s forming a Lie group. The geometries of interest were
homogeneous spaces , 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 In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
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
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
of the identity map of .
Characterization as a connection
If we regard the Lie group as a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique
principal connection 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: i ...
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
: