The construction of solvable Lie algebras from equidimensional nilpotent algebras
type: Item
instance of: scholarly article
title: The construction of solvable Lie algebras from equidimensional nilpotent algebras
zbMATH Open document ID: 0714.17010
DOI: 10.1016/0024-3795(90)90243-6
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The authors associate with each finite dimensional Lie algebra L with nilpotent derived algebra a unique nilpotent Lie algebra \(L^*\), called the nilpotent frame of L. The aim is to reduce the classification of solvable Lie algebras to the same problem for nilpotent Lie algebras. They begin by studying the decomposition of a matrix into its semisimple and nilpotent parts: \(a=s(a)+n(a)\). One obtains a linear epimorphism from a Cartan subalgebra of L onto a diagonal subalgebra A of the derivation algebra of L. This mapping is used to construct a monomorphism from L into the semi-direct sum of L and A, the image of which is the nilpotent frame, \(L^*\). The properties of \(L^*\) are investigated in this paper. In particular, over fields of characteristic 0, the nilpotent frame of L is unique up to an isomorphism induced by an inner automorphism of L. The reverse problem of starting with a nilpotent \(L^*\) and constructing a solvable Lie algebra with nilpotent frame \(L^*\) is also shown to have a positive solution.
reviewed by: Ernest L. Stitzinger
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 4175203
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: solvable Lie algebras
zbMATH Keywords: derivation algebra
zbMATH Keywords: nilpotent frame
Wikidata QID: Q115364342
author: Jiří Patera
author: H. Zassenhaus
MaRDI profile type: Publication
full work available at URL: 0024-3795(90)90243-6
OpenAlex ID: W1976307642
cites work: On Lie gradings. I
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cites work: Introduction to Lie Algebras and Representation Theory
cites work: On the Contraction of Groups and Their Representations
cites work: Solvable Lie algebras of dimension \(\leq 4\) over perfect fields
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description: scientific article; zbMATH DE number 4175203