Maurer–Cartan equation

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

Statement

Let $\theta$ denote the left Maurer–Cartan form on $G$, i.e. the $\mathfrak g$-valued 1-form characterized by

• $\theta_e=\mathrm{id}_{\mathfrak g}$ under the identification $T_eG\cong \mathfrak g$, and
• left-invariance: $(L_g)^*\theta=\theta$ for all $g\in G$.

Then $\theta$ satisfies the Maurer–Cartan equation

Here $[\theta,\theta]$ denotes the $\mathfrak g$-valued 2-form obtained by combining wedge product with the Lie bracket : for vector fields $X,Y$ on $G$,

Equivalent form (often used in calculations)

For any vector fields $X,Y$ on $G$,

A clean proof is packaged in the Maurer–Cartan equation lemma .

Context

This equation is the differential-geometric encoding of the Lie algebra structure inside the group: it is the reason that brackets of left-invariant vector fields are controlled by the structure constants of $\mathfrak g$, and it underlies many constructions with left-invariant differential forms and bi-invariant forms .