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 Élie 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. The Lie algebra is identified with the tangent space of G at the identity, denoted TeG. The Maurer-Cartan form ω is thus a one-form defined globally on G which is a linear mapping of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in G along the left-translation in the group:

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

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[edit] Motivation and interpretation

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 G. That is, a manifold P identical to the group G, 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 motions were transformations forming a Lie group. The geometries of interest were homogeneous spaces G/H, but usually without a fixed choice of origin corresponding the coset eH.

A principal homogeneous space of G is a manifold P abstractly characterized by having a free and transitive action of G on P. The Maurer-Cartan form[1] gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on P 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 P.

[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 TG 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 GL(n) by a matrix valued mapping g = (gi,j), then one can write ω explicitly as

ω = g − 1dg.

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

[edit] Characterization as a connection

The Maurer-Cartan form can also be characterized abstractly as a kind of principal connection. It is the unique g = TeG valued 1-form on G satisfying

  1. \omega_e = (\hbox{the identity}) : T_eG\rightarrow {\mathfrak g}, and
  2. ωg = (Rh) * ω = Ad(h − 1)ω, where h = g − 1, and (Rh) * is the pullback of forms along the right-translation in the group Ad(h-1) is the adjoint action on the Lie algebra.

[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

dω(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,

so

dω(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. It is often written as

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

[edit] Maurer-Cartan frame

One can also view the Maurer-Cartan form as being constructed from a Maurer-Cartan frame. Let Ei be a basis of sections of TG consisting of left-invariant vector fields, and θj be the dual basis of sections of T*G such that θj(Ei) = δij, the Kronecker delta. Then Ei is a Maurer-Cartan frame, and θi is a Maurer-Cartan coframe.

Since Ei is left-invariant, applying the Maurer-Cartan form to it simply returns the value of Ei at the identity. Thus ω(Ei) = Ei(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 Ei are given by

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

The quantities cijk are constant, and called the structure constants of the Lie algebra (relative to the basis Ei). 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.

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.

[edit] 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 → 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 → 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 sU and sV 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 = Ad(h^{-1}_{UV})\theta_U + (h_{UV})^* \omega_H

where ωH is the Maurer-Cartan form on the group H.

A system of non-degenerate g-valued 1-forms θU 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 θU 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.

[edit] References

  1. ^ Introduced by Cartan (1904).
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