Maurer–Cartan form

Template:Short description In mathematics, the Maurer–Cartan form for a Lie group Template:Math is a distinguished differential one-form on Template:Math that carries the basic infinitesimal information about the structure of Template:Math. 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 Template:Math. The Lie algebra is identified with the tangent space of Template:Math at the identity, denoted Template:Math. The Maurer–Cartan form Template:Math is thus a one-form defined globally on Template:Math, that is, a linear mapping of the tangent space Template:Math at each Template:Math into Template:Math. It is given as the pushforward of a vector in Template:Math along the left-translation in the group:

$\omega (v)=(L_{g^{-1}}{)}_{*}v,v\in T_{g}G.$

Motivation and interpretation

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

$G\times G\ni (g,h)\mapsto gh\in G.$

A question of importance to Cartan and his contemporaries was how to identify a principal homogeneous space of Template:Math. That is, a manifold Template:Math identical to the group Template:Math, 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 Template:Math, but usually without a fixed choice of origin corresponding to the coset Template:Math.

A principal homogeneous space of Template:Math is a manifold Template:Math abstractly characterized by having a free and transitive action of Template:Math on Template:Math. The Maurer–Cartan form gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on Template:Math 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 Template:Math.

Construction

Intrinsic construction

Let Template:Math be the tangent space of a Lie group Template:Math at the identity (its Lie algebra). Template:Math acts on itself by left translation

$L:G\times G\to G$

such that for a given Template:Math we have

$L_g:G\to G\text{where}{L}_g(h)=gh,$

and this induces a map of the tangent bundle to itself: $(L_g{)}_{*}:T_{h}G\to T_{gh}G.$ A left-invariant vector field is a section Template:Math of Template:Math such that

$(L_g{)}_{*}X=X\mathrm{\forall }g\in G.$

The Maurer–Cartan form Template:Math is a Template:Math-valued one-form on Template:Math defined on vectors Template:Math by the formula

${\omega }_g(v)=(L_{g^{-1}}{)}_{*}v.$

Extrinsic construction

If Template:Math is embedded in Template:Math by a matrix valued mapping Template:Math, then one can write Template:Math 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 Template:Math.

Characterization as a connection

If we regard the Lie group Template:Math 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 Template:Math. Indeed, it is the unique Template:Math valued Template:Math-form on Template:Math satisfying

1. ${\omega }_e=\mathrm{i}\mathrm{d}:T_{e}G\to 𝔤,\text{ and}$
2. $\mathrm{\forall }g\in G{\omega }_g=\mathrm{A}\mathrm{d}(h)(R_h^{*}{\omega }_e),\text{ where }h=g^{-1},$

where Template:Math is the pullback of forms along the right-translation in the group and Template:Math is the adjoint action on the Lie algebra.

Properties

If Template:Math is a left-invariant vector field on Template:Math, then Template:Math is constant on Template:Math. Furthermore, if Template:Math and Template:Math 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 Template:Math. (This may be used as the definition of the bracket on Template:Math.) These facts may be used to establish an isomorphism of Lie algebras

$𝔤=T_{e}G\cong \{\text{left-invariant vector fields on G}\}.$

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

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

Here Template:Math is the Template:Math-valued function obtained by duality from pairing the one-form Template:Math with the vector field Template:Math, and Template:Math is the Lie derivative of this function along Template:Math. Similarly Template:Math is the Lie derivative along Template:Math of the Template:Math-valued function Template:Math.

In particular, if Template:Math and Template:Math 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 Template:Math under a diffeomorphism is still a basis), so the equation is true for any pair of vector fields Template:Math and Template:Math. This is known as the Maurer–Cartan equation. It is often written as

$d\omega +\frac{1}{2}[\omega ,\omega ]=0.$

Here Template:Math 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 Template:Math be a basis of sections of Template:Math consisting of left-invariant vector fields, and Template:Math be the dual basis of sections of Template:Math such that Template:Math, the Kronecker delta. Then Template:Math is a Maurer–Cartan frame, and Template:Math is a Maurer–Cartan coframe.

Since Template:Math is left-invariant, applying the Maurer–Cartan form to it simply returns the value of Template:Math at the identity. Thus Template:Math. Thus, the Maurer–Cartan form can be written Template:NumBlk

Suppose that the Lie brackets of the vector fields Template:Math are given by

$[E_i,E_j]=\sum _k{c_{ij}}^k{E}_k.$

The quantities Template:Math are the structure constants of the Lie algebra (relative to the basis Template:Math). A simple calculation, using the definition of the exterior derivative Template:Math, yields

$d{\theta }^i(E_j,E_k)=-{\theta }^i([E_j,E_k])=-\sum _r{c_{jk}}^r{\theta }^i(E_r)=-{c_{jk}}^i=-\frac{1}{2}({c_{jk}}^i-{c_{kj}}^i),$

so that by duality Template:NumBlk This equation is also often called the Maurer–Cartan equation. To relate it to the previous definition, which only involved the Maurer–Cartan form Template:Math, take the exterior derivative of Template:EquationNote:

$d\omega =\sum _i{E}_i(e)\otimes d{\theta }^i=-\frac{1}{2}\sum _{ijk}{c_{jk}}^i{E}_i(e)\otimes {\theta }^j\wedge {\theta }^k.$

The frame components are given by

$d\omega (E_j,E_k)=-\sum _i{c_{jk}}^i{E}_i(e)=-[E_j(e),E_k(e)]=-[\omega (E_j),\omega (E_k)],$

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 Template:Nowrap defined on the tautological principal bundle associated with a homogeneous space. If Template:Math is a closed subgroup of Template:Math, then Template:Math is a smooth manifold of dimension Template:Math. The quotient map Template:Math induces the structure of an Template:Math-principal bundle over Template:Math. The Maurer–Cartan form on the Lie group Template:Math yields a flat Cartan connection for this principal bundle. In particular, if Template:Math}, 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 Template:Math. (If working on a submanifold of the homogeneous space, then Template:Math need only be a local section over the submanifold.) The pullback of the Maurer–Cartan form along Template:Math defines a non-degenerate Template:Math-valued Template:Math-form Template:Math over the base. The Maurer–Cartan equation implies that

$d\theta +\frac{1}{2}[\theta ,\theta ]=0.$

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

$h_{UV}(x)=s_V\circ s_U^{-1}(x),x\in U\cap V.$

The differential of Template:Math gives a compatibility condition relating the two sections on the overlap region:

${\theta }_V=\mathrm{Ad}(h_{UV}^{-1}){\theta }_U+(h_{UV}{)}^{*}{\omega }_H$

where Template:Math is the Maurer–Cartan form on the group Template:Math.

A system of non-degenerate Template:Math-valued Template:Math-forms Template:Math defined on open sets in a manifold Template:Math, satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifold Template:Math locally with the structure of the homogeneous space Template:Math. In other words, there is locally a diffeomorphism of Template:Math into the homogeneous space, such that Template:Math 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

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