Maurer-Cartan form

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In mathematics, the Maurer-Cartan form for a Lie group G is a distinguished differential one-form on G that carries within itself the basic infinitesimal information about the structure of G. It was much used by Elie Cartan, as a basic ingredient of his method of moving frames.

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. Recall that the Lie algebra is identified with the tangent space of G at the identity, denoted T_eG. The Maurer-Cartan form is thus a one-form defined globally on G which is a linear mapping of the tangent space T_gG at each g ∈ G into T_eG. It can be characterized as the unique Lie-algebra valued 1-form ω such that:

$\omega_e = (\hbox{the identity}) : T_eG\rightarrow T_eG$ , and
$ω g = (R h) * ω = A d (h - 1)ω$ , where (R_h)^* is the pullback of forms along the right-translation in the group, $h : = g - 1$ and Ad(h^-1) is the adjoint action on the Lie algebra.

Informally, this characterization bears some resemblence to the logarithmic derivative of the identity mapping of G.

1 Construction of the Maurer-Cartan form
- 1.1 Intrinsic construction
- 1.2 Extrinsic construction
2 Properties
3 Maurer-Cartan form on a homogeneous space
4 Maurer-Cartan frame

[edit] Construction of the Maurer-Cartan form

[edit] Intrinsic construction

Let $\mathfrak{g}=T_eG$ 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

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

The Maurer-Cartan form $ω$ is a $\mathfrak{g}$ -valued one-form on $G$ defined on vectors $v\in T_g G$ by the formula $\omega(v)=(L_{g^{-1}})_*v$ .

[edit] Extrinsic construction

If $G$ is embedded in $G L (n)$ by a matrix valued mapping $g = (g i, j)$ , then one can write $ω$ explicitly as

ω = g - 1 d g

In this sense, the Maurer-Cartan form is always the left logarithmic derivative of the identity map of $G$ .

[edit] 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

ω([X, Y]) = [ω(X),ω(Y)]

where the bracket on the LHS is the Lie bracket of vector fields, and the bracket on the RHS is the bracket on the Lie algebra $\mathfrak{g}$ . (This may be used as the definition of the bracket on $\mathfrak{g}$ .) 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 differential, if $X$ and $Y$ are arbitrary vector fields then

d ω(X, Y) = X (ω(Y)) - Y (ω(X)) - ω([X, Y])

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

X (ω(Y)) = Y (ω(X)) = 0

d ω(X, Y) + [ω(X),ω(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.

[edit] Maurer-Cartan form on a homogeneous space

Another point of view uses the 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 → 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.

[edit] 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 TG 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 θⁱ 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) ∈ 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

$[E_i,E_j]=\sum_kc_{ij}^kE_k$ .

The quantities c_ij^k are constant, and called 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

$d\theta^i(E_j,E_k) = -\theta^i([E_j,E_k]) = -\sum_r c_{jk}^r\theta^i(E_r)$ ,

so that by duality

$d\theta^i=-\sum_{jk} c_{jk}^i\theta^j\wedge\theta^k$ (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):

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