Maurer–Cartan form

Remarks

Example: for $G = GL_n(\R)$, left translation by $g^{-1}$ is the linear map $V \mapsto g^{-1}V$ of $M_n(\R)$. So $\omega_g(V) = g^{-1} V$, written $\omega = g^{-1}\, dg$. Then $d\omega = -g^{-1}\,dg\, g^{-1} \wedge dg = -\omega \wedge \omega$, which is the Maurer–Cartan equation for matrix groups.

Non-example: the form $\omega'_g = (dR_{g^{-1}})_g$, with $R_g(h) = hg$, also takes values in $\mathfrak{g}$, but it is not the left Maurer–Cartan form. For $GL_n(\R)$ it is $dg\, g^{-1}$, and at $g = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$, $V = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ one gets $V g^{-1} = \tfrac12 V \ne V = g^{-1}V$. It is the right Maurer–Cartan form.

The form $\omega$ is left-invariant: $L_h^*\omega = \omega$ for all $h \in G$. For each left-invariant vector field $X$, the function $\omega(X)$ is constant, equal to $X_e$. In general it satisfies $d\omega + \tfrac12[\omega \wedge \omega] = 0$.