Lie algebra homology and the Macdonald-Kac formulas
type: Item
instance of: scholarly article
title: Lie algebra homology and the Macdonald-Kac formulas
zbMATH Open document ID: 0358.17015
DOI: 10.1007/BF01418970
author: H. Garland
author: J. Lepowsky
published in: Inventiones Mathematicae
publication date: 1976-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/142387
review text: I. G. Macdonald's noted identities which are a highly compact generalization of famous ``theta-function Identity of Jacobi'' were originally obtained by means of ``affine root systems'' [Invent. Math. 15, 91--143 (1972; Zbl 0244.17005)]. These formulas have subsequently been interpreted by V. G. Kac and R. V. Moody as precise analogues of Weyl's denominator formula for the ``Euclidean Lie algebras''. Kac also sketches a new proof of the much wider class of identities -- the analogues of both Weyl's character and denominator formulas for a family of Lie algebras considerably more general than Euclidean Lie algebras which are introduced also by Kac and Moody by symmetrizable Cartan matrices. The main purpose of the present paper is to generalize B. Kostant's fundamental result on the homology (or cohomology) of nilradicals of parabolic subalgebras in certain modules from finite dimensional complex semisimple Lie algebras to the Kac-Moody Lie algebras. The authors obtain the results including the Macdonald identities, as immediate consequences of the Euler-Poincaré principle just as B. Kostant derived Weyl's character and denominator formulas from his homology theorem [\textit{B.~Kostant}, Ann. Math. (2) 74, 329--387 (1961; Zbl 0134.03501)]. Let \(\mathfrak g=\coprod\limits_{(n_1,\dots,n_\ell)\in\mathbb Z^\ell}\mathfrak g(n_1,\dots,n_\ell)\) be the Lie algebra defined by an \(\ell\times\ell\)-Cartan matrix \(A=(A_{ij})\). Let \(D_i\) \((1\leq i\leq\ell)\) be the \(i\)-th degree derivation of \(\mathfrak g\) that is the derivation which acts on \(\mathfrak g(n_1,\dots,n_\ell)\) as scalar multiplication by \(n_i\). Then \(D_1,\dots,D_\ell\) span an \(\ell\)-dimensional space \(\mathfrak v_0\) of commuting derivations of \(\mathfrak g\). Let \(\mathfrak v\) be a subspace of \(\mathfrak v_0\). Since \(\mathfrak v\) may be regarded as an abelian Lie algebra acting on the \(\mathfrak v\)-module \(\mathfrak g\) by derivations, we may form the semi-direct product Lie algebra \…
reviewed by: Eiichi Abe
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 3558014
zbMATH Keywords: Kac-Moody Lie algebras
zbMATH Keywords: Euler-Poincaré principle
zbMATH Keywords: Macdonald identities
Wikidata QID: Q56474332
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3558014