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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2013
   • (edited Aug 25th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering if the following concept has an established name in the literature: Let $\mathfrak{h} \hookrightarrow \mathfrak{g}$ be a normal incusion of Lie algebras, so that the quotient $\mathfrak{g}/\mathfrak{h}$ is again a Lie algebra. Let $$ \mu : \mathfrak{g}/\mathfrak{h} \to \mathbb{R}[n] $$ be a Lie algebra $(n+1)$-cocycle on the quotient. Then for $X$ a manifold equipped with an $(\mathfrak{h} \to \mathfrak{g})$-[[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?

   I am wondering if the following concept has an established name in the literature:

   Let $\mathfrak{h} \hookrightarrow \mathfrak{g}$ be a normal incusion of Lie algebras, so that the quotient $\mathfrak{g}/\mathfrak{h}$ is again a Lie algebra. Let

   $$\mu : \mathfrak{g}/\mathfrak{h} \to \mathbb{R}[n]$$

   be a Lie algebra $(n+1)$-cocycle on the quotient. Then for $X$ a manifold equipped with an $(\mathfrak{h} \to \mathfrak{g})$-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?