Maurer–Cartan equation lemma

A computational identity: the exterior derivative of the Maurer–Cartan form is the negative bracket.

Maurer–Cartan equation lemma

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ , and let $\theta$ be the left Maurer–Cartan form .

Lemma

For any smooth vector fields $X,Y$ on $G$ ,

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

Equivalently,

d\theta + \frac12[\theta,\theta]=0,

which is the Maurer–Cartan equation .

Proof idea (invariant-field reduction)

It suffices to verify the identity on left-invariant vector fields because both sides are left-invariant 2-forms.

If $X=X^L$ and $Y=Y^L$ are left-invariant with $X,Y\in \mathfrak g$ , then $\theta(X^L)=X$ and $\theta(Y^L)=Y$ are constant (as $\mathfrak g$ -valued functions). Using the definition of exterior derivative,

(d\theta)(X^L,Y^L) = X^L(\theta(Y^L)) - Y^L(\theta(X^L)) - \theta([X^L,Y^L]).

The first two terms vanish since $\theta(Y^L)$ and $\theta(X^L)$ are constant, and the last term becomes $-\theta([X^L,Y^L])$ . By the identification of brackets of left-invariant fields with the Lie bracket , $\theta([X^L,Y^L])=[X,Y]$ . Hence $(d\theta)(X^L,Y^L)=-[X,Y]$ , giving the desired formula.

Context

This lemma is the workhorse behind computations with invariant forms and is the differential-geometric source of the Lie bracket, complementary to the flow-based viewpoint via one-parameter subgroups as integral curves .