Maurer–Cartan form explained

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:

\omega(v) = (L _{g ^{-1}} ) _{*} v, v\in T _{gG.}

Motivation and interpretation

See also: Lie group action. A Lie group acts on itself by multiplication under the mapping

G x G \ni (g,h) \mapsto gh \in G. 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^[1] 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

L : G x G \to G such that for a given we haveL _{g} : G \to G where L _{g(h)} = gh,

and this induces a map of the tangent bundle to itself:

(L _{g)} _{*:T} _{hG\to} T _{gh} G. A left-invariant vector field is a section of such that ^[2] (L _{g)} _{*} X = X \forall g \in G.

The Maurer - Cartan form is a -valued one-form on defined on vectors by the formula

\omega _{g(v)=(L} ^{g ^{-1}} ) _{*v.}

Extrinsic construction

If is embedded in by a matrix valued mapping, then one can write explicitly as

\omega _{g} = g ^{-1} dg.

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

\omega _{e} = id : T _{eG →} { ak g}, and \forall g \in G \omega _{g} = Ad (h)(R ^{-1} _{e), where h = g} ,

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

\omega([X,Y]) = [\omega(X),\omega(Y)]

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

ak{g}=T _{eG\cong} \{\hbox{left - invariant vector fields on G}\}.

By the definition of the exterior derivative, if and are arbitrary vector fields then

d\omega(X,Y) = X(\omega(Y)) - Y(\omega(X)) - \omega([X,Y]).

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

X(\omega(Y)) = Y(\omega(X)) =0 ,

so

d\omega(X,Y) + [\omega(X),\omega(Y)] =0

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

d\omega + (1)/(2) [\omega,\omega] =0 .

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

Suppose that the Lie brackets of the vector fields are given by

[E _{i,E} _{j]=\sum} _{k{c} _{ij} }^kE_k.The quantities are the structure constants of the Lie algebra (relative to the basis). A simple calculation, using the definition of the exterior derivative, yieldsd\theta ^{i(E} _{j,E} _{k)} = - \theta ^{i([E} _{j,E} _{k])} = - \sum _{r} {c _{jk} }^r\theta^i(E_r) = -^i = -\frac(^i - ^i),so that by dualityThis equation is also often called the Maurer - Cartan equation. To relate it to the previous definition, which only involved the Maurer - Cartan form, take the exterior derivative of :d\omega = \sum _{i} E _{i(e) ⊗} d\theta ^{i =- (12)/(\sum)} _{ijk} {c _{jk} }^iE_i(e)\otimes\theta^j\wedge\theta^k.The frame components are given byd\omega(E _{j,E} _{k)} = - \sum _{i} {c _{jk} }^iE_i(e) = -[E_j(e),E_k(e)]=-[\omega(E_j),\omega(E_k)],which establishes the equivalence of the two forms of the Maurer - Cartan equation.

On a homogeneous space

Maurer - Cartan forms play an important role in Cartan's method of moving frames. In this context, one may view the Maurer - Cartan form as a defined on the tautological principal bundle associated with a homogeneous space. If is a closed subgroup of, then is a smooth manifold of dimension . The quotient map induces the structure of an -principal bundle over . The Maurer - Cartan form on the Lie group yields a flat Cartan connection for this principal bundle. In particular, if, then this Cartan connection is an ordinary connection form, and we have

d\omega + \omega\wedge\omega =0 which is the condition for the vanishing of the curvature.

In the method of moving frames, one sometimes considers a local section of the tautological bundle, say . (If working on a submanifold of the homogeneous space, then need only be a local section over the submanifold.) The pullback of the Maurer - Cartan form along defines a non-degenerate -valued -form over the base. The Maurer - Cartan equation implies that

d\theta + (1)/(2) [\theta,\theta] =0 .

Moreover, if and are a pair of local sections defined, respectively, over open sets and, then they are related by an element of in each fibre of the bundle:

h _{UV} (x) = s _{V\circ} s ^{-1} _{U} (x), x \in U \cap V. The differential of gives a compatibility condition relating the two sections on the overlap region:\theta _{V} = \operatorname{Ad}(h ^{-1} _{UV} )\theta _{U} + (h _{UV} ) ^{*} \omega _{H} where is the Maurer - Cartan form on the group .

A system of non-degenerate -valued -forms defined on open sets in a manifold, satisfying the Maurer - Cartan structural equations and the compatibility conditions endows the manifold locally with the structure of the homogeneous space . In other words, there is locally a diffeomorphism of into the homogeneous space, such that is the pullback of the Maurer - Cartan form along some section of the tautological bundle. This is a consequence of the existence of primitives of the Darboux derivative.

References

• Élie. Cartan. Sur la structure des groupes infinis de transformations. Annales Scientifiques de l'École Normale Supérieure. 1904. 21. 153–206. 10.24033/asens.538. free.
• Book: R. W. Sharpe. Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. 1996. Springer-Verlag, Berlin. 0-387-94732-9.
• Book: Shlomo Sternberg. Shlomo Sternberg. Lectures on differential geometry. registration. 1964. Prentice-Hall. 64-7993. Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra..

Notes and References

1. Introduced by Cartan (1904).
2. Subtlety: (L _{g)} _{*} X gives a vector in T _{gh} G if X\in T _{h} G