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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 η-function.
Keywords: Macdonald's identities, Euler-Poincaré principle, affine root system, loop space, automorphic forms
