On the identification of a Lie algebra given by its structure constants. I: Direct decompositions, Levi decompositions, and nilradicals
type: Item
instance of: scholarly article
title: On the identification of a Lie algebra given by its structure constants. I: Direct decompositions, Levi decompositions, and nilradicals
zbMATH Open document ID: 0668.17004
DOI: 10.1016/0024-3795(88)90210-8
author: D. W. Rand
author: Pavel Winternitz
author: H. Zassenhaus
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: In this nice article the authors give the theoretical foundation for an algorithmic identification of finite-dimensional Lie algebras starting with their commutator relations and structure constants. This algorithm is the basis for corresponding computer programs presented in the paper reviewed above and answers the following questions for a Lie algebra L: - Is L decomposable as the direct sum of Lie algebras ? - If not so, is L simple or solvable ? - If not so, how to find the radical and the Levi decomposition of L ? - If L is solvable, how to find the nilradical? The answers are given by a transformation to a canonical basis, where basis elements are grouped together corresponding to decompositions, ideals etc. The explanation at first is rather elementary and then based on representation theory, linear algebra and Lie algebra methods. For the determination of nilradicals a new algorithm working without any irrational calculations is used. Pregnant examples are given, the algorithm seems to be very useful with respect to the determination of symmetry Lie algebras of differential equations.
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 68W99
zbMATH DE Number: 4093693
zbMATH Keywords: computer algebra
zbMATH Keywords: Lie algebras
zbMATH Keywords: radical
zbMATH Keywords: Levi decomposition
zbMATH Keywords: nilradical
zbMATH Keywords: algorithm
reviewed by: Günter Czichowski
describes a project that uses: LIE0
describes a project that uses: MACSYMA
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90210-8
OpenAlex ID: W2024734434
Wikidata QID: Q115364353
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description: scientific article; zbMATH DE number 4093693