Left Maurer–Cartan form

The canonical g-valued 1-form θ^L = (dL_{g^{-1}})_g on a Lie group.

Left Maurer–Cartan form

Let $G$ be a Lie group with Lie algebra $\mathfrak g=T_eG$ .

Definition (Left Maurer–Cartan form).
The left Maurer–Cartan form is the $\mathfrak g$ -valued $1$ -form $\theta^L\in \Omega^1(G;\mathfrak g)$ defined by

\theta^L_g : T_gG \to \mathfrak g,\qquad \theta^L_g(v)=(dL_{g^{-1}})_g(v).

Equivalently, $\theta^L$ is characterized by:

(Normalization) $\theta^L_e=\mathrm{id}_{\mathfrak g}$ , and
(Left invariance) $(L_h)^*\theta^L=\theta^L$ for all $h\in G$ .

Maurer–Cartan equation.
$\theta^L$ satisfies the Maurer–Cartan equation

d\theta^L + \tfrac12[\theta^L,\theta^L]=0,

where the bracket combines the Lie bracket on $\mathfrak g$ with wedge product.

Context and use.
The left Maurer–Cartan form identifies tangent vectors on $G$ with elements of $\mathfrak g$ in a left-translation invariant way, and it packages all left-invariant forms as alternating tensors on $\mathfrak g$ . The right-handed analogue is the right Maurer–Cartan form .