\(ad\)-nilpotent ideals of a Borel subalgebra
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.
scientific article; zbMATH DE number 1423607
scientific article