Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map α : g → Ω ∗ ( M ) {\displaystyle \alpha :{\mathfrak {g}}\to \Omega ^{*}(M)} from the Lie algebra g = Lie ⁡ ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} to the space of differential forms on M that are equivariant; i.e., α ( Ad ⁡ ( g ) X ) = g α ( X ) . {\displaystyle \alpha (\operatorname {Ad} (g)X)=g\alpha (X).} In other words, an equivariant differential form is an invariant element of C [ g ] ⊗ Ω ∗ ( M ) = Sym ⁡ ( g ∗ ) ⊗ Ω ∗ ( M ) .

Source: Wikipedia — Equivariant differential form (CC BY-SA 4.0)

Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map α : g → Ω ∗ ( M ) {\displaystyle \alpha :{\mathfrak {g}}\to \Omega ^{*}(M)} from the Lie algebra g = Lie ⁡ ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} to the space of differential forms on M that are equivariant; i.e., α ( Ad ⁡ ( g ) X ) = g α ( X ) . {\displaystyle \alpha (\operatorname {Ad} (g)X)=g\alpha (X).} In other words, an equivariant differential form is an invariant element of C [ g ] ⊗ Ω ∗ ( M ) = Sym ⁡ ( g ∗ ) ⊗ Ω ∗ ( M ) .

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Source: Wikipedia "Equivariant differential form" · CC BY-SA 4.0

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