Some problems concerning to nilpotent Lie superalgebras
type: Item
instance of: scholarly article
title: Some problems concerning to nilpotent Lie superalgebras
zbMATH Open document ID: 1069.17004
published in: Journal of Geometry and Physics
publication date: 2005-06-23T00:00:00Z
review text: Given a nilpotent Lie superalgebra \(\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus \mathfrak{g}_{\bar 1}\), the descending central sequence is defined by \(\mathcal{C}^0(\mathfrak{g})=\mathfrak{g}\) and \(\mathcal{C}^{k+1}(\mathfrak{g})=[\mathfrak{g},\mathcal{C}^k(\mathfrak{g})]\); and the least natural number \(k\) such that \(\mathcal{C}^k(\mathfrak{g})=0\) is called the nilindex of \(\mathfrak{g}\). This nilindex is at most \(\dim\mathfrak{g}-1\). The paper under review is devoted to study the function \(f(n,m)\) given by the maximal possible nilindex among the nilpotent Lie superalgebras with \(\dim\mathfrak{g}_{\bar 0}=n\) and \(\dim\mathfrak{g}_{\bar 1}=m\). It is proven that, for \(m>0\), \(f(n,m)=n+m-1\, (=\dim\mathfrak{g} -1)\) if and only if \(n=2\) and \(m\) is odd. In this case, the maximal nilindex is attained by a unique superalgebra (up to isomorphism). Sufficient conditions for \(f(n,m)=n+m-2\) are given too, and some low dimensional examples are explored.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 2182251
zbMATH Keywords: nilpotent
zbMATH Keywords: Lie
zbMATH Keywords: superalgebra
zbMATH Keywords: nilindex
zbMATH Keywords: filiform
author: J. Remedios
author: Yu. Khakimdjanov
author: R. M. Navarro
MaRDI profile type: Publication
full work available at URL: j.geomphys.2004.01.003
OpenAlex ID: W1992679721
cites work: Some remarks on solvable Lie superalgebras
cites work: The structure of Lie superalgebras with a semisimple even part
cites work: Q5597608
cites work: Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras
cites work: Lie superalgebras with semisimple even part
cites work: Perturbations des superalgèbres de Lie. (Perturbations of Lie superalgebras)
cites work: Semisimple Lie superalgebras which are not the direct sums of simple Lie superalgebras
cites work: Q5732712
cites work: Lie superalgebras
cites work: Q2994971
cites work: The theory of Lie superalgebras. An introduction
cites work: Cohomologie des algèbres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes
DOI: 10.1016/J.GEOMPHYS.2004.01.003
description: scientific article; zbMATH DE number 2182251