Right-invariant differential form

A differential form on a Lie group fixed by all right translations, determined by its value at the identity.

Right-invariant differential form

Let $G$ be a Lie group. A differential $k$ -form $\omega\in\Omega^k(G)$ is right-invariant if

(R_g)^*\omega=\omega\qquad\text{for all }g\in G,

where $R_g$ denotes right translation by $g$ .

Right-invariant forms are completely determined by their value at the identity element $e\in G$ . Concretely, if $\omega$ is right-invariant, then for $g\in G$ and $v_1,\dots,v_k\in T_gG$ ,

\omega_g(v_1,\dots,v_k)=\omega_e\big((dR_{g^{-1}})_g v_1,\dots,(dR_{g^{-1}})_g v_k\big).

Thus evaluation at $e$ gives a vector space isomorphism

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

where $\mathfrak g=\mathrm{Lie}(G)$ (see Lie algebra of a Lie group ).

Right-invariant forms are the natural home for the right Maurer–Cartan form , and many identities (including the Maurer–Cartan equation ) can be expressed neatly in terms of invariant forms. Compare also with left-invariant forms and the special case of bi-invariant forms .