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 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 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 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's Erlangen programme where one was interested in a notion of symmetry on a space, where the symmetries of the space were transformations forming a Lie group. The geometries of interest were homogeneous spaces , but usually without a fixed choice of origin corresponding to the coset .
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
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 on the principal bundle . Indeed, it is the unique valued -form on satisfying
where is the pullback of forms along the right-translation in the group and is the adjoint action 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
where the bracket on the left-hand side is the Lie bracket of vector fields, and the bracket on the right-hand side is the bracket on the Lie algebra . (This may be used as the definition of the bracket on .) These facts may be used to establish an isomorphism of Lie algebras
By the definition of the exterior derivative, if and are arbitrary vector fields then
Here is the -valued function obtained by duality from pairing the one-form with the vector field , and is the Lie derivative of this function along . Similarly is the Lie derivative along of the -valued function .
In particular, if and are left-invariant, then
so
but the left-invariant fields span the tangent space at any point (the push-forward of a basis in under a diffeomorphism is still a basis), so the equation is true for any pair of vector fields and . This is known as the Maurer–Cartan equation. It is often written as
Here denotes the bracket of Lie algebra-valued forms.
Maurer–Cartan frame
One can also view the Maurer–Cartan form as being constructed from a Maurer–Cartan frame. Let be a basis of sections of consisting of left-invariant vector fields, and be the dual basis of sections of such that , the Kronecker delta. Then is a Maurer–Cartan frame, and is a Maurer–Cartan coframe.
Since is left-invariant, applying the Maurer–Cartan form to it simply returns the value of at the identity. Thus . Thus, the Maurer–Cartan form can be written