Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map

from the Lie algebra ${\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}$ to the space of differential forms on M that are equivariant; i.e.,

In other words, an equivariant differential form is an invariant element of

For an equivariant differential form ${\displaystyle \alpha }$, the equivariant exterior derivative ${\displaystyle d_{\mathfrak {g}}\alpha }$ of ${\displaystyle \alpha }$ is defined by

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