There is an exact sequence

$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \mathfrak{g}^*) \to H^1(\mathfrak{g}, S^2\mathfrak{g}),$$

where $\mathfrak{g}$ is a Lie algebra over a field $k$ and $H^i(\mathfrak{g}, M)$ are Lie algebra cohomology. It is interesting because of a map $d$, a Koszul homomorphism, which sends an invariant symmetric bilinear form $\left< \cdot, \cdot \right>$ to a 3-cocycle $\left<[\cdot, \cdot], \cdot\right>$ and is an isomorphism for a semi-simple Lie algebra.

The question is whether it is a part (namely, lower-degree terms) of some spectral sequence? If not, what is a natural way to obtain it? An unclear proof may be found in Neeb, Wagemann The second cohomology of current algebras of general Lie algebras, proposition 7.2.


It's part of the Pirashvili exact sequence, relating Lie algebra and Leibniz cohomologies of Lie algebras. This is discussed (in homology terms) in the end of p2 of this paper of mine on Koszul's homomorphism (arxiv link). Pirashvili's paper is freely accessible here on Numdam; it's also written in terms of homology but the cohomological statement follows.

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