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-formIt translates a tangent vector at any point back to the identity, giving a linear isomorphism $T_{g}G\to \mathfrak{g}$. 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 $d\theta +\frac{1}{2}[\theta ,\theta ]=0$, 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 $d(db/a)=-(da/a)\wedge (db/a)$.

The Maurer-Cartan equation is $d\rho +\rho \wedge \rho =0$ for the left-invariant form $\rho =g^{-1}dg$.

If $\rho ={\sigma }^{\alpha }{T}_{\alpha }$ and $[T_{\alpha },T_{\beta }]=c^{\gamma }{}_{\alpha \beta }{T}_{\gamma }$, then

When $d{\sigma }^{\gamma }={\sum }_{\alpha ,\beta }{f}^{\gamma }{}_{\alpha \beta }{\sigma }^{\alpha }\wedge {\sigma }^{\beta }$ sums over ordered pairs, the Maurer-Cartan equation determines only $f^{\gamma }{}_{[\alpha \beta ]}=-c^{\gamma }{}_{\alpha \beta }/2$. Symmetric additions vanish in the exterior product. The usual coefficients are the unique antisymmetric representatives; summing instead over $\alpha <\beta$ removes the factor one half.