Dedekind's eta-function and the cohomology of infinite dimensional Lie algebras.
We compute the cohomology of certain infinite dimensional Lie algebras which are subalgebras of Lie algebras introduced by Moody and Kac. We note a relation between our results and the cohomology of loop spaces of compact groups. Finally, we derive, by Euler-Poincaré, identities of Macdonald for powers of the Dedekind eta-function.
Garland
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Yale University, New Haven, Connecticut 06520.