Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, H^2(g,M)=0. The solution of this question gives a counterexample to Whitehead's second lemma for infinite dimensional g-module.
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2$\begingroup$ I've voted to close as "too localized", as I think this is homework; but even if it isn't, this question is certainly not on the level appropriate for this site. See FAQ for further information. $\endgroup$– Leonid PositselskiDec 11, 2011 at 9:58
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$\begingroup$ The question makes no sense to me, until the OP makes precise what is the augmentation ideal of a Lie algebra - I know of augmentation ideals for group rings, see en.wikipedia.org/wiki/Augmentation_ideal $\endgroup$– Alain ValetteDec 11, 2011 at 11:07
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1$\begingroup$ Alain- it's the ideal generated by the Lie algebra in the UEA (the kernel of the action on the trivial representation). Every Hopf algebra has an augmentation ideal, which includes the group algebra and UEA examples. $\endgroup$– Ben Webster ♦Dec 11, 2011 at 14:28
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1$\begingroup$ Sounds like homework to me. $\endgroup$– Vladimir DotsenkoDec 11, 2011 at 22:23
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2 Answers
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It seems that the idea of the question partly was to have a
counterexample to Whitehead's second lemma. However this can be easily given by considering Verma modules (which are infinite-dimensional).
F. L. Williams has computed the structure of the cohomology of a finite-dimensional complex semisimple Lie algebra with coefficients in an arbitrary Verma module, in the paper The cohomology of semisimple Lie algebras with coefficients in a Verma module, 1978. The results show that
the cohomology $H^2(\mathfrak{g},M)$ does not vanish in general for Verma modules $M$.
answered Dec 21, 2013 at 19:34
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This is false. Consider the example of $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$, and let $J$ be the augmentation ideal of $U(\mathfrak{g})$. Firstly $H^2(\mathfrak{g}, J) = \operatorname{Ext}^2_{U(\mathfrak{g})}(\mathbb{C},J) \cong \operatorname{Ext}^1_{U(\mathfrak{g})}(J,\mathbb{C})$ by Corollary 7.2 of Auslander-Gorenstein Rings by Ajitabh, Smith and Zhang. Then by dimension shifting $\operatorname{Ext}_{U(\mathfrak{g})}^1(J,\mathbb{C}) = \operatorname{Ext}_{U(\mathfrak{g})}^2(\mathbb{C},\mathbb{C})$ which is zero by the second Whitehead lemma.
The question appears as Exercise 6.3 in Hilton and Stammbach's A course in homological algebra. I guess they meant to ask about $\operatorname{Ext}^2_{U(\mathfrak{g})}(J, \mathbb{C})$ since they say to use the previous exercise which shows $\operatorname{Ext}^3_{U(\mathfrak{g})}(\mathbb{C}, \mathbb{C})\neq 0$
