\(ad\)-nilpotent ideals of a Borel subalgebra
type: Item
instance of: scholarly article
title: \(ad\)-nilpotent ideals of a Borel subalgebra
zbMATH Open document ID: 0951.17003
author: Paola Cellini
author: Paolo Papi
published in: Journal of Algebra
publication date: 2000-08-13T00:00:00Z
review text: Let \({\mathfrak g}\) be a finite dimensional complex simple Lie algebra, \({\mathfrak h}\) a Cartan subalgebra of \({\mathfrak g}\), and \(\Delta^+\) a positive root system. Also let \(\widehat{\Delta}^+\) and \(\widehat{W}\) be the affine positive real root system and affine Weyl group, respectively. This paper classifies the \(ad\)-nilpotent ideals of a Borel subalgebra. Namely, given such an ideal, the authors associate to it a subset of \(\widehat{\Delta}^+\). This subset turns out to be of the form \(\{ \alpha \in \widehat{\Delta}^+ \mid w^{-1}\alpha \in -\widehat{\Delta}^+ \}\) for a unique \(w \in \widehat{W}\). Therefore the \(ad\)-nilpotent ideals are enumerated by a subset of \(\widehat{W}\). The author also observes that classifying \(ad\)-nilpotent ideals is equivalent to classifying so called increasing subsets of \(\Delta^+\). This can be accomplished via work of [\textit{J. Shi}, Q. J. Math., Oxf. II. Ser. 48, 93-105 (1997; Zbl 0889.20024)] by associating to each increasing set a suitable set of Young diagrams. This allows the author to calculate the cardinality of the set of \(ad\)-nilpotent ideals.
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 1423607
zbMATH Keywords: complex simple Lie algebra
zbMATH Keywords: root system
zbMATH Keywords: \(ad\)-nilpotent ideals
zbMATH Keywords: Borel subalgebra
zbMATH Keywords: increasing set
reviewed by: Mark R. Sepanski
MaRDI profile type: Publication
full work available at URL: jabr.1999.8099
OpenAlex ID: W4213298096
cites work: Q3724546
cites work: Q5580319
cites work: Q3693664
cites work: Q4393967
cites work: A Characterization of a Special Ordering in a Root System
cites work: Shifted Plane Partitions of Trapezoidal Shape
cites work: Q4344657
DOI: 10.1006/JABR.1999.8099
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description: scientific article; zbMATH DE number 1423607