Nilpotent subalgebras of semisimple Lie algebras
type: Item
instance of: scholarly article
title: Nilpotent subalgebras of semisimple Lie algebras
zbMATH Open document ID: 1175.17006
author: Paul Levy
author: George McNinch
author: Donna M. Testerman
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2009-05-13T00:00:00Z
full work available at URL: NilCR.pdf
review text: Let \(k\) be an algebraically closed field of characteristic \(p>0.\) Let \(G\) be a semisimple linear algebraic group over \(k\) with fixed maximal torus \(T.\) Let \(\Phi=\Phi\left( G,T\right) \) be its root system, \(\Phi^{+}\) a positive system in \(\Phi,\) and \(\left\{ \alpha_{1},\dots,\alpha_{\ell}\right\} \) the corresponding basis. Suppose that is irreducible. Then there is a unique root \(\beta\) of maximal height with respect to the chosen basis. Write \(\alpha _{1}^{\vee},\dots,\alpha_{\ell}^{\vee},\beta^{\vee}\) for the corresponding coroots to \(\alpha_{1},\dots,\alpha_{\ell},\beta\) respectively, and define integers \(m_{i}^{\prime},\,1\leq i\leq\ell\) by \(\beta^{\vee}=\sum m_{i}^{\prime}\alpha_{i}^{\vee}\) We say \(p\) is \textit{torsion} for \(\Phi\) if \(m_{i}^{\prime}=p\) for some \(i\). Additionally, we say \(p\) is torsion for \(G\) if it is torsion for some irreducible component of some root system or if \(p\) divides the order of the fundamental group of \(G\). If \(p\) is a non-torsion prime for \(G\) and \(U\) is a unipotent subgroup of \(G\), then \(G\) is contained in a Borel subgroup of \(G\). For \(\mathfrak{g}\) the Lie algebra for \(G\) and \(\mathfrak{n}\) a nil subalgebra of \(\mathfrak{g},\) it follows from this result that if \(p\) it non-torsion for \(G\) then \(\mathfrak{n}\) lies in a Borel subalgebra of \(G\). Conversely, J. P. Serre asked it there exists a nil subalgebra which is not contained in a Borel subalgebra in the case \(p\) is torsion for \(G\). This note provides examples of such subalgebras, examples which are presented in two cases. As an explicit class, consider \(G=\)SL\(_{mp}\) for some positive integer \(m\). Let \(E_{ij}\) be the elementary \(mp\times mp\) matrix whose \(\left( r,s\right) \) entry is \(\delta_{ir}\delta_{js},\) and let \(X=\sum_{j=0}^{m-1}\sum _{i=1}^{p-1}E_{jp+i,jp+i+1}\) and \(Y=\sum_{j=0}^{m-1}\sum_{i=1}^{p-1} E_{jp+i+1,jp+i}.\) Then \(X^{p}=Y^{p}=0\) and \(\left[ X,Y\right] =I,\) thus the…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 5553609
zbMATH Keywords: Lie algebras
zbMATH Keywords: nilpotent subalgebras
zbMATH Keywords: Borel subalgebras
MaRDI profile type: Publication
OpenAlex ID: W2022572905
cites work: Q3999894
cites work: Q5580319
cites work: Nilpotent subspaces of maximal dimension in semi-simple Lie algebras
cites work: Q3924381
cites work: Q4670625
cites work: The maximal subgroups of positive dimension in exceptional algebraic groups
cites work: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory
cites work: Q4063618
cites work: Q5662587
cites work: Torsion in reductive groups
cites work: Normal, unipotent subgroup schemes of reductive groups
DOI: 10.1016/J.CRMA.2009.03.015
description: scientific article; zbMATH DE number 5553609