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I am wondering if the following concept has an established name in the literature:
Let 𝔥↪𝔤 be a normal incusion of Lie algebras, so that the quotient 𝔤/𝔥 is again a Lie algebra. Let
μ:𝔤/𝔥→ℝ[n]be a Lie algebra (n+1)-cocycle on the quotient. Then for X a manifold equipped with an (𝔥→𝔤)-Cartan connection, we can pull back this cocycle to a differential (n+1)-form on X.
When X is the corresponding Lie group equipped with the Maurer-Cartan form connection, then this form is the left-invariant extension of the cocycle and is closed.
For general X and/or general Cartan connection on X, that pulled back cocycle form need not be closed. Imposing the condition that this form be closed is an integrability condition such as that for G2-structure and similar.
Does this general concept have an established name?
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