A characteristic property of simple Lie algebras
type: Item
instance of: scholarly article
title: A characteristic property of simple Lie algebras
zbMATH Open document ID: 0583.17007
DOI: 10.1007/BF01078393
author: V. V. Astrakhantsev
publication date: 1985-01-01T00:00:00Z
review text: A finite-dimensional complex Lie algebra \({\mathfrak g}\) is called metrizable if on \({\mathfrak g}\) there is a nondegenerate symmetric bilinear form \(B\) invariant under the associated representation \(B([z,x],y)+B(x,[z,y])=0.\) The author studies certain properties of the decomposition of an arbitrary metrizable Lie algebra, which is not nilpotent, relative to a fixed Cartan subalgebra. The investigation follows, in the main, the standard scheme used in the theory of semisimple Lie algebras, although parts of the results and proofs of this theory are invalid or require refinements when metrizable Lie algebras are considered. The main results of this paper are : Theorem. Let \({\mathfrak g}\) be an arbitrary metrizable Lie algebra, R a system of nonzero roots relative to a fixed Cartan subalgebra, and \(R=R_ 1\cup R_ 0\) a decomposition of roots into so-called nonisotropic and isotropic parts. Then \({\mathfrak g}\) can be decomposed into the orthogonal direct sum \({\mathfrak g}={\mathfrak g}_ 1\oplus {\mathfrak g}_ 0\) of ideals \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 0\), and \(R_ 1\), \(R_ 2\) are systems of nonzero roots for \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 0\), respectively. The ideal \({\mathfrak g}_ 1\) is semisimple. Corollary 1. If all the roots of a nonsplittable metrizable Lie algebra \({\mathfrak g}\) are nonisotropic, then \({\mathfrak g}\) is simple. - Corollary 2. If a nonsplittable metrizable Lie algebra is not simple, then all its roots are isotropic. - Proofs are omitted.
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 3933282
zbMATH Keywords: decomposition
zbMATH Keywords: root system
zbMATH Keywords: isotropic roots
zbMATH Keywords: nonisotropic roots
zbMATH Keywords: metrizable Lie algebra
zbMATH Keywords: Cartan subalgebra
reviewed by: A. Fleischer
MaRDI profile type: Publication
cites work: Q5618494
cites work: Q3846750
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3933282