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Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint representation have non-linear versions:

  • The differential of the conjugation map $GL(\mathfrak{g}) \to C^2(\mathfrak{g}, \mathfrak{g}), \phi \mapsto \phi \cdot [\,, \,]$ at the identity is the CE-differential $d: C^1 \to C^2$.
  • The differential of the Jacobi map $C^2(\mathfrak{g}, \mathfrak{g}) \to C^3(\mathfrak{g}, \mathfrak{g})$ (which assigns to $c \in C^2$ the expression that appears in the Jacobi identity $c(x, c(y,z)) + ...$) at $[\,, \,] \in C^2$ is the CE-differential $d: C^2 \to C^3$.

Question: Are there smooth maps whose differential yield the CE-differentials in higher degree (and which form a non-linear complex in some sense)?

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  • $\begingroup$ Maybe I'm missing something, but I think they do compose to zero: $J(\phi \cdot [\,,]) = \phi \cdot J([\,,]) = 0$. $\endgroup$
    – Tobias Diez
    Commented Jul 26, 2020 at 7:17
  • $\begingroup$ The action on $C^2$ is by conjugation: $(\phi \cdot c) (x, y) = \phi \, c(\phi^{-1} x, \phi^{-1}y)$. $\endgroup$
    – Tobias Diez
    Commented Jul 26, 2020 at 12:57

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