The homology of graded infinite-dimensional Lie algebras in connection with Macdonald identities

D. Fucks' monograph devoted to the cohomology of infinite-dimensional Lie algebras contains an error in calculating the homology of a graded affine Kac-Moody algebra of type A_n⁽¹⁾, so that the proof of the corresponding Macdonald identity, which is based on that calculation, is incorrect. In the present paper, a revised proof is suggested.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1996, pp. 5–17.The author is grateful to A. M. Vershik, who suggested the topic of this paper, for his constant attention and useful comments. The author also thanks M. A. Vsemirnov for his information on non-coincidence of the coefficients in two sides of the equality in [4, §3.2.3] for n=7, which he observed while proving Macdonald identities in a new (analytical) way, and on the sign of a summand on the right (see his paper in this issue).