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:

1. $\backslash omega\_e\; =\; (\backslash hbox\{the\; identity\})\; :\; T\_eG\backslash rightarrow\; T\_eG$, and

2. $\backslash omega\_g\; =\; (R\_h)^*\backslash omega\; =\; Ad(h^\{-1\})\backslash omega$, where (R_h)^* is the pullback of forms along the right-translation in the group, 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.

Construction of the Maurer-Cartan form

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Intrinsic construction

Let $\backslash mathfrak\{g\}=T\_eG$ be the tangent space of a Lie group $G$ at the identity (its Lie algebra). $G$ acts on itself by left translation

$L\; :\; G\; \backslash times\; G\; \backslash to\; G$

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

$L\_g\; :\; G\; \backslash to\; G\; \backslash quad\; \backslash mbox\{where\}\; \backslash quad\; L\_g(h)\; =\; gh$,

and this induces a map of the tangent bundle to itself

$(L\_g)\_*:T\_hG\backslash to\; T\_\{gh\}G$.

A left-invariant vector field is a section $X$ of $TG$ such that

$(L\_g)\_\{*\}X\; =\; X\; \backslash quad\; \backslash forall\; \backslash quad\; g\; \backslash in\; G$.

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

Extrinsic construction

If $G$ is embedded in $GL(n)$ by a matrix valued mapping $g=(g\_\{i,j\})$, then one can write $\backslash omega$ explicitly as

$\backslash omega\; =\; g^\{-1\}\; dg$.

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

Properties

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If $X$ is a left-invariant vector field on $G$, then $\backslash omega(X)$ is constant on $G$. Furthermore, if $X$ and $Y$ are both left-invariant, then

$\backslash omega(*)=*$

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 $\backslash mathfrak\{g\}$. (This may be used as the definition of the bracket on $\backslash mathfrak\{g\}$.) These facts may be used to establish an isomorphism of Lie algebras

$\backslash mathfrak\{g\}=T\_eG\backslash cong\; \backslash \{\backslash hbox\{left-invariant\; vector\; fields\; on\; G\}\backslash \}$.

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

$d\backslash omega(X,Y)=X(\backslash omega(Y))-Y(\backslash omega(X))-\backslash omega(*)$.

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

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

so

$d\backslash omega(X,Y)+*=0$

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.

Maurer-Cartan form on a homogeneous space

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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 yields the structure of an H-principal bundle over G/H. The Maurer-Cartan form on the 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

$d\backslash omega+\backslash omega\backslash wedge\backslash omega=0$

which is the condition for the vanishing of the curvature.

Maurer-Cartan frame

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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 θj(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

$\backslash omega=\backslash sum\_iE\_i(e)\backslash otimes\backslash theta^i$ (1).

Suppose that the Lie brackets of the vector fields E_i are given by

$*=\backslash sum\_kc\_\{ij\}^kE\_k$.

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

$d\backslash theta^i(E\_j,E\_k)\; =\; -\backslash theta^i(*)\; =\; -\backslash sum\_r\; c\_\{jk\}^r\backslash theta^i(E\_r)$,

so that by duality

$d\backslash theta^i=-\backslash sum\_\{jk\}\; c\_\{jk\}^i\backslash theta^j\backslash wedge\backslash theta^k$ (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):

$d\backslash omega\; =\; \backslash sum\_i\; E\_i(e)\backslash otimes\; d\backslash theta^i\backslash ,=\backslash ,-\backslash sum\_\{ijk\}c\_\{jk\}^iE\_i(e)\backslash otimes\backslash theta^j\backslash wedge\backslash theta^k.$

The frame components are given by

$d\backslash omega(E\_j,E\_k)\; =\; -\backslash sum\_i\; c\_\{jk\}^iE\_i(e)\; =\; -*=-*,$

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

Lie groups | Equations

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