Sugawara operators and their applications to Kac-Kazhdan conjecture
type: Item
instance of: scholarly article
title: Sugawara operators and their applications to Kac-Kazhdan conjecture
zbMATH Open document ID: 0642.17010
author: Takahiro Hayashi
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: Let \(M(\lambda)\) be a Verma module for an affine Kac-Moody Lie algebra \({\mathfrak g}\) and let \(L(\lambda)\) be the unique irreducible quotient of \(M(\lambda)\). Assume that \(\lambda (c)+g=0\) where c is the canonical central element in \({\mathfrak g}\) and g is the dual Coxeter number. The author proves that the character of \(L(\lambda)\) is given by the formula \(ch(L(\lambda))=e\prod (1-e^{-\alpha})^{-1},\) where the product is over all positive real roots. The result is a more general version of the Kac-Kazhdan conjecture. To prove the result the author introduces the higher order analogy of the Sugawara operators. If \({\mathfrak g}\) is of type \(A_ n^{(1)}\), \(B_ n^{(1)}\) or \(C_ n^{(1)}\), then the author proves the existence of a `fundamental' set of Sugawara operators. Further, when \({\mathfrak g}\) is of type \(A_ n^{(1)}\) the operators are constructed explicitly as elements in a completion Û(\({\mathfrak g})\) of the universal enveloping algebra of \({\mathfrak g}\).
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4045972
zbMATH Keywords: triangular decomposition
zbMATH Keywords: character formula
zbMATH Keywords: affine Kac-Moody Lie algebra
zbMATH Keywords: Kac-Kazhdan conjecture
zbMATH Keywords: Sugawara operators
Wikidata QID: Q122871673
MaRDI profile type: Publication
full work available at URL: pjaa.63.356
OpenAlex ID: W2020430118
cites work: Sugawara operators and Kac-Kazhdan conjecture
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q4894499
cites work: Structure of representations with highest weight of infinite-dimensional Lie algebras
DOI: 10.3792/PJAA.63.356
description: scientific article; zbMATH DE number 4045972