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 $h\in G$ .

Remarks

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

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

where, in this page's convention, $[\theta^L,\theta^L](X,Y)=[\theta^L(X),\theta^L(Y)]$ .

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.