Maurer–Cartan form

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

\omega(v) = (L_{g^{-1}})_* v,\quad v\in T_gG.

Motivation and interpretation

Construction of the Maurer–Cartan form

Intrinsic construction

Let $g ≅ T e G$ be the tangent space of a Lie group $G$ at the identity (its Lie algebra). $G$ acts on itself by left translation

L : G \times G \to G

such that for a given $g \in G$ we have

L_g : G \to G \quad \mbox{where} \quad 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 $X$ of $T G$ such that ^[2]

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

The Maurer–Cartan form $ω$ is a $g$ -valued one-form on $G$ defined on vectors $v \in T g G$ by the formula

\omega_g(v)=(L_{g^{-1}})_*v.

Extrinsic construction

If $G$ is embedded in $GL(n)$ by a matrix valued mapping $g =(g ij)$ , then one can write $ω$ 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 $G$ .

Characterization as a connection

If we regard the Lie group $G$ 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 $G$ . Indeed, it is the unique $g = T e G$ valued $1$ -form on $G$ satisfying

$\omega_e = \mathrm{id} : T_eG\rightarrow {\mathfrak g},\text{ and}$
$\forall g \in G \quad \omega_g = \mathrm{Ad}(h)(R_h^*\omega_e),\text{ where }h=g^{-1},$

where $R h *$ is the pullback of forms along the right-translation in the group and $Ad(h)$ is the adjoint action on the Lie algebra.

Properties

If $X$ is a left-invariant vector field on $G$ , then $ω (X)$ is constant on $G$ . Furthermore, if $X$ and $Y$ 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 $g$ . (This may be used as the definition of the bracket on $g$ .) These facts may be used to establish an isomorphism of Lie algebras

\mathfrak{g}=T_eG\cong \{\hbox{left-invariant vector fields on G}\}.

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

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

Here $ω (Y)$ is the $g$ -valued function obtained by duality from pairing the one-form $ω$ with the vector field $Y$ , and $X (ω (Y))$ is the Lie derivative of this function along $X$ . Similarly $Y (ω (X))$ is the Lie derivative along $Y$ of the $g$ -valued function $ω (X)$ .

In particular, if $X$ and $Y$ are left-invariant, then

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

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

but the left-hand side is simply a $2$ -form, so the equation does not rely on the fact that $X$ and $Y$ are left-invariant. The conclusion follows that the equation is true for any pair of vector fields $X$ and $Y$ . This is known as the Maurer–Cartan equation. It is often written as

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

Here $[ω, ω]$ 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 $E i$ be a basis of sections of $T G$ consisting of left-invariant vector fields, and $θ j$ be the dual basis of sections of $T * G$ such that $θ j (E i) = δ i j$ , the Kronecker delta. Then $E i$ is a Maurer–Cartan frame, and $θ i$ is a Maurer–Cartan coframe.

Since $E i$ is left-invariant, applying the Maurer–Cartan form to it simply returns the value of $E i$ at the identity. Thus $ω (E i) = E i (e) \in g$ . Thus, the Maurer–Cartan form can be written

$\omega=\sum_iE_i(e)\otimes\theta^i.$

(1)

Suppose that the Lie brackets of the vector fields $E i$ are given by

[E_i,E_j]=\sum_k{c_{ij}}^kE_k.

The quantities $c ij k$ are the structure constants of the Lie algebra (relative to the basis $E i$ ). A simple calculation, using the definition of the exterior derivative $d$ , 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

$d\theta^i=-\frac12 \sum_{jk} {c_{jk}}^i\theta^j\wedge\theta^k.$

(2)

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

d\omega = \sum_i E_i(e)\otimes d\theta^i\,=\,-\frac12 \sum_{ijk}{c_{jk}}^iE_i(e)\otimes\theta^j\wedge\theta^k.

The frame components are given by

d\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.

Maurer–Cartan form 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 $1$ -form defined on the tautological principal bundle associated to a homogeneous space. If $H$ is a closed subgroup of $G$ , then $G / H$ is a smooth manifold of dimension $dim G - dim H$ . The quotient map $G \to G / H$ induces the structure of an $H$ -principal bundle over $G / H$ . The Maurer–Cartan form on the Lie group $G$ yields a flat Cartan connection for this principal bundle. In particular, if $H = {e$ }, 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 $s : G / H \to G$ . (If working on a submanifold of the homogeneous space, then $s$ need only be a local section over the submanifold.) The pullback of the Maurer–Cartan form along $s$ defines a non-degenerate $g$ -valued $1$ -form $θ = s * ω$ over the base. The Maurer–Cartan equation implies that

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

Morever, if $s U$ and $s V$ are a pair of local sections defined, respectively, over open sets $U$ and $V$ , then they are related by an element of $H$ in each fibre of the bundle:

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

The differential of $h$ 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 $ω H$ is the Maurer–Cartan form on the group $H$ .

A system of non-degenerate $g$ -valued $1$ -forms $θ U$ defined on open sets in a manifold $M$ , satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifold $M$ locally with the structure of the homogeneous space $G / H$ . In other words, there is locally a diffeomorphism of $M$ into the homogeneous space, such that $θ U$ 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 the Darboux derivative.

Notes

↑ Introduced by Cartan (1904).
↑ Subtlety: $(L_g)_{*}X$ gives a vector in $T_{gh}G \text{ if } X\in T_h G$

References

Cartan, Élie (1904). "Sur la structure des groupes infinis de transformations" (PDF). Annales Scientifiques de l'École Normale Supérieure 21: 153–206.
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.

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