Left-invariant differential form

A differential form on a Lie group fixed by all left translations.

Left-invariant differential form

Let $G$ be a Lie group , and let $L_g:G\to G$ denote left translation by $g$ .

Definition (Left-invariant form).
A differential $k$ -form $\omega\in \Omega^k(G)$ is left-invariant if

L_g^*\omega=\omega \quad\text{for all } g\in G.

Identification with alternating forms on the Lie algebra.
Evaluation at the identity defines an isomorphism

\Omega^k(G)^{G\text{-left}} \;\cong\; \Lambda^k(\mathfrak g^*),

where $\mathfrak g=T_eG$ is the Lie algebra : a left-invariant form is uniquely determined by its value on $T_eG$ , and any alternating form on $\mathfrak g$ extends uniquely to a left-invariant form by left translation. This extension can be written succinctly using the left Maurer–Cartan form .

Context.
Left-invariant forms reduce many global computations on $G$ to multilinear algebra on $\mathfrak g$ and interact naturally with the Maurer–Cartan equation . Analogous notions include right-invariant and bi-invariant forms.