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In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.

Does anyone know how this proof goes?

For reference, 9.11 is the statement that given a symmetrizable generalized Cartan matrix $A$, with associated lie algebra $\tilde{\mathfrak{g}}(A)$ with only the Weyl relations, the maximal ideal not intersecting $\mathfrak{h}$ is generated by the Serre relations. Thank you in advance.

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    $\begingroup$ You might check out Lectures 26 and 27 in these notes: stacky.net/files/written/LieGroups/LieGroups.pdf These are lectures of Borcherds, who bases the construction of $E_8$ on constructions of Kac-Moody algebras from vertex algebras. So this might be a good warm-up. $\endgroup$
    – Ian Agol
    Commented Apr 19, 2014 at 4:58

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The proof is written in

  • Mathieu, Olivier. Formules de caractères pour les algèbres de Kac-Moody générales. (French) [Character formulas for general Kac-Moody algebras] Astérisque No. 159-160 (1988), 267 pp. MR0980506 (90d:17024)

It uses results of Garland-Lepowsky and Garland-Kac. See Lemma 118. (I have to thank Ivan Angiono for the reference.)

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  • $\begingroup$ This is on page 222 in Mathieu's monograph, in XVI $\S4$, in case the numbering system isn't immediately clear. $\endgroup$
    – Jim Humphreys
    Commented Oct 13, 2014 at 23:02

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