Knowlpedia

Right-invariant differential form

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

Let GG be a Lie group. A differential kk-form ω∈Ωk(G)\omega\in\Omega^k(G) is right-invariant if

(Rg)∗ω=ωfor all g∈G,(R_g)^*\omega=\omega\qquad\text{for all }g\in G,

where RgR_g denotes by gg.

Right-invariant forms are completely determined by their value at the identity element e∈Ge\in G. Concretely, if ω\omega is right-invariant, then for g∈Gg\in G and v1,…,vk∈TgGv_1,\dots,v_k\in T_gG,

ωg(v1,…,vk)=ωe((dRg−1)gv1,…,(dRg−1)gvk).\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 ee gives a vector space isomorphism

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

where g=Lie(G)\mathfrak g=\mathrm{Lie}(G) (see ).

Right-invariant forms are the natural home for the , and many identities (including the ) can be expressed neatly in terms of invariant forms. Compare also with and the special case of .