Maurer-Cartan form
= Maurer-Cartan form
{c}
{wiki=Maurer–Cartan_form}
For a <Lie group> $G$ with <Lie algebra> $\mathfrak g$, the left Maurer-Cartan form is the $\mathfrak g$-valued <differential 1-form>
$$
\theta_g(v)=(dL_{g^{-1}})_g v.
$$
It translates a <tangent vector> at any point back to the identity, giving a <linear isomorphism> $T_gG\to\mathfrak g$. In particular, $\theta(X^L)=X$ for the <left-invariant vector field> determined by $X$. This makes $\theta$ left-invariant and supplies a global coframe after choosing a <basis> of $\mathfrak g$.
The <exterior derivative> on <left-invariant vector fields> gives $d\theta(X^L,Y^L)=-[X,Y]$, since the two derivative terms of constant coefficients vanish. Thus $d\theta+\tfrac12[\theta,\theta]=0$, the <Maurer-Cartan equation>. For a <Matrix Lie group>, $\theta=g^{-1}dg$ and this becomes $d\theta+\theta\wedge\theta=0$, where the <exterior product> includes <matrix multiplication>. The additive <real numbers> have $\theta=dx$, while the positive multiplicative <real numbers> have $\theta=da/a$. For the <real affine group>, the <Maurer-Cartan coframe of the real affine group> makes the nonzero <Lie bracket> visible as $d(db/a)=-(da/a)\wedge(db/a)$.