$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of $U(\mathfrak{g})\rightarrow k$. The solution to $H^2(\mathfrak{g},J(\mathfrak{g}))\neq0$ shows that a similar statement of Whitehead's second lemma doesn't hold for infinite dimensional $\mathfrak{g}-$module.(Whitehead's second lemma: With the assumptions above, $H^2(\mathfrak{g},M)=0$ if $M$ is a finite dimensional $\mathfrak{g}-$module).
How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?
5
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lie-algebras
homology-cohomology
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0A dimension shifting argument shows that $\operatorname{Ext}^2_{U(\mathfrak{g})}(J, k) = \operatorname{Ext}^3_{U(\mathfrak{g})}(k,k)$ which is nonzero because of the Killing form. But $H^2(\mathfrak{g}, J)=\operatorname{Ext}^2_{U(\mathfrak{g})}(k,J)$ can't be dealt with this way. – 2013-12-16
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0repost @ MO: http://mathoverflow.net/q/83169/ – 2013-12-19