Inmathematics, theMaurer–Cartan form for aLie groupG is a distinguisheddifferential one-form onG that carries the basic infinitesimal information about the structure ofG. It was much used byÉlie Cartan as a basic ingredient of hismethod of moving frames, and bears his name together with that ofLudwig Maurer.

As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in theLie algebra associated to the Lie groupG. The Lie algebra is identified with thetangent space ofG at the identity, denotedT_eG. The Maurer–Cartan formω is thus a one-form defined globally onG, that is, a linear mapping of the tangent spaceT_gG at eachg ∈G intoT_eG. It is given as thepushforward of a vector inT_gG along the left-translation in the group:

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

A Lie group acts on itself by multiplication under the mapping

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

A question of importance to Cartan and his contemporaries was how to identify aprincipal homogeneous space ofG. That is, amanifoldP identical to the groupG, but without a fixed choice of unit element. This motivation came, in part, fromFelix Klein'sErlangen programme where one was interested in a notion ofsymmetry on a space, where the symmetries of the space weretransformations forming a Lie group. The geometries of interest werehomogeneous spacesG/H, but usually without a fixed choice of origin corresponding to thecoseteH.

A principal homogeneous space ofG is a manifoldP abstractly characterized by having afree and transitive action ofG onP. TheMaurer–Cartan form^[1] gives an appropriateinfinitesimal characterization of the principal homogeneous space. It is a one-form defined onP satisfying anintegrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define theexponential map of the Lie algebra and in this way obtain, locally, a group action onP.

Letg ≅ T_eG be the tangent space of a Lie groupG at the identity (itsLie algebra).G acts on itself by left translation

${\displaystyle L:G\times G\to G}$

such that for a giveng ∈G we have

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

and this induces a map of thetangent bundle to itself:${\displaystyle (L_g{)}_{\ast }:T_{h}G\to T_{gh}G.}$A left-invariantvector field is a sectionX ofTG such that^[2]

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

TheMaurer–Cartan formω is ag-valued one-form onG defined on vectorsv ∈ T_gG by the formula

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

IfG is embedded inGL(n) by a matrix valued mappingg =(g_ij), then one can writeω explicitly as

${\displaystyle {\omega }_g=g^{-1}dg.}$

In this sense, the Maurer–Cartan form is always the leftlogarithmic derivative of the identity map ofG.

If we regard the Lie groupG as aprincipal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the uniqueprincipal connection on the principal bundleG. Indeed, it is the uniqueg = T_eG valued1-form onG satisfying

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

whereR_h* is thepullback of forms along the right-translation in the group andAd(h) is theadjoint action on the Lie algebra.

IfX is a left-invariant vector field onG, thenω(X) is constant onG. Furthermore, ifX andY are both left-invariant, then

${\displaystyle \omega ([X,Y])=[\omega (X),\omega (Y)]}$

where the bracket on the left-hand side is theLie bracket of vector fields, and the bracket on the right-hand side is the bracket on the Lie algebrag. (This may be used as the definition of the bracket ong.) These facts may be used to establish an isomorphism of Lie algebras

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

By the definition of theexterior derivative, ifX andY are arbitrary vector fields then

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

Hereω(Y) is theg-valued function obtained by duality from pairing the one-formω with the vector fieldY, andX(ω(Y)) is theLie derivative of this function alongX. SimilarlyY(ω(X)) is the Lie derivative alongY of theg-valued functionω(X).

In particular, ifX andY are left-invariant, then

${\displaystyle X(\omega (Y))=Y(\omega (X))=0,}$

so

${\displaystyle d\omega (X,Y)+[\omega (X),\omega (Y)]=0}$

but the left-invariant fields span the tangent space at any point (the push-forward of a basis inT_eG under a diffeomorphism is still a basis), so the equation is true for any pair of vector fieldsX andY. This is known as theMaurer–Cartan equation. It is often written as

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

Here[ω, ω] denotes thebracket of Lie algebra-valued forms.

One can also view the Maurer–Cartan form as being constructed from aMaurer–Cartan frame. LetE_i be abasis of sections ofTG consisting of left-invariant vector fields, andθ^j be thedual basis of sections ofT^*G such thatθ^j(E_i) =δ_i^j, theKronecker delta. ThenE_i is a Maurer–Cartan frame, andθⁱ is aMaurer–Cartan coframe.

SinceE_i is left-invariant, applying the Maurer–Cartan form to it simply returns the value ofE_i at the identity. Thusω(E_i) =E_i(e) ∈g. Thus, the Maurer–Cartan form can be written

${\displaystyle \omega =\sum _i{E}_i(e)\otimes {\theta }^i.}$

1

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

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

The quantitiesc_ij^k are thestructure constants of the Lie algebra (relative to the basisE_i). A simple calculation, using the definition of the exterior derivatived, yields

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

so that by duality

${\displaystyle d{\theta }^i=-\frac{1}{2}\sum _{jk}{c_{jk}}^i{\theta }^j\wedge {\theta }^k.}$

2

This equation is also often called theMaurer–Cartan equation. To relate it to the previous definition, which only involved the Maurer–Cartan formω, take the exterior derivative of(1):

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

The frame components are given by

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

which establishes the equivalence of the two forms of the Maurer–Cartan equation.

Maurer–Cartan forms play an important role in Cartan'smethod of moving frames. In this context, one may view the Maurer–Cartan form as a1-form defined on the tautologicalprincipal bundle associated with ahomogeneous space. IfH is aclosed subgroup ofG, thenG/H is a smooth manifold of dimensiondimG − dimH. The quotient mapG →G/H induces the structure of anH-principal bundle overG/H. The Maurer–Cartan form on the Lie groupG yields a flatCartan connection for this principal bundle. In particular, ifH = {e}, then this Cartan connection is an ordinaryconnection form, and we have

${\displaystyle d\omega +\omega \wedge \omega =0}$

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, says :G/H →G. (If working on asubmanifold of the homogeneous space, thens need only be a local section over the submanifold.) Thepullback of the Maurer–Cartan form alongs defines a non-degenerateg-valued1-formθ =s^*ω over the base. The Maurer–Cartan equation implies that

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

Moreover, ifs_U ands_V are a pair of local sections defined, respectively, over open setsU andV, then they are related by an element ofH in each fibre of the bundle:

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

The differential ofh gives a compatibility condition relating the two sections on the overlap region:

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

whereω_H is the Maurer–Cartan form on the groupH.

A system of non-degenerateg-valued1-formsθ_U defined on open sets in a manifoldM, satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifoldM locally with the structure of the homogeneous spaceG/H. In other words, there is locally adiffeomorphism ofM into the homogeneous space, such thatθ_U 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 theDarboux derivative.

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

• Lie groups
• Equations
• Differential geometry

• Articles with short description
• Short description is different from Wikidata