Cyclic homology and the Macdonald conjectures
type: Item
instance of: scholarly article
title: Cyclic homology and the Macdonald conjectures
zbMATH Open document ID: 0604.17007
DOI: 10.1007/BF01391498
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143390
review text: Let \(\mathfrak g\) be a complex semisimple Lie algebra of rank \(n\) with exponents \(m_ 1,\ldots,m_ n\) and for a positive integer \(k\) denote by \(A_+(k)\) the truncated polynomial ring \(\mathbb C[t]/(t^{k+1})\). In Section 1 the following Conjecture is formulated. The Lie algebra cohomology of \(\mathfrak g\otimes A_+(k)\) is an exterior algebra with \(n(k+1)\) generators. For each \(s\), there are \(k+1\) generators of degree \(2m_ s+1\). The weight of these generators -- what is inherited from the grading of \(\mathfrak g\otimes A_+(k)\) -- is 0, \((k+1)m_ s+t\) with \(t=1,2,\ldots,k\). It is shown that this conjecture implies the Macdonald root system conjectures. Section 2 is a summary on cyclic and dihedral homology of a graded \(\mathbb C\)-algebra. In Section 3 the cyclic and dihedral homologies of \(A_+(k)\) are computed. In Section 4 the case of classical Lie algebras is studied with \(\mathfrak g_ n=\mathfrak{sl}_ n(\mathbb C)\), \(\mathfrak{o}_ n(\mathbb C)\) and \(\mathfrak{sp}_{2n}(\mathbb C)\). With the help of computations from Section 3 it is shown that the stated Conjecture holds in the limit on \(n\) as \(n\) approaches infinity. Finally, the relevance of this limiting case to the case of finite \(n\) is discussed.
reviewed by: Alice Fialowski
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B22
zbMATH DE Number: 3975166
zbMATH Keywords: cyclic homology
zbMATH Keywords: complex semisimple Lie algebra
zbMATH Keywords: Lie algebra cohomology
zbMATH Keywords: exterior algebra
zbMATH Keywords: Macdonald root system conjectures
zbMATH Keywords: dihedral homology
zbMATH Keywords: classical Lie algebras
author: Philip J. Hanlon
MaRDI profile type: Publication
Wikidata QID: Q123121515
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OpenAlex ID: W2111879802
description: scientific article; zbMATH DE number 3975166