Classification of finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms
type: Item
instance of: scholarly article
title: Classification of finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms
zbMATH Open document ID: 1112.17013
author: Caihui Lu
published in: Journal of Algebra
publication date: 2007-05-03T00:00:00Z
review text: In the paper under review one investigates the classification of finite-dimensional complex solvable Lie algebras that allow for a nondegenerate symmetric invariant bilinear form \((\cdot\mid\cdot)\colon{\mathfrak g}\times{\mathfrak g}\to{\mathbb C}\). One proceeds by taking into account root space decompositions corresponding to a maximal abelian subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) with the property that the linear mapping \(\text{ad}_{\mathfrak g}x\colon{\mathfrak g}\to{\mathfrak g}\) is a semisimple derivation for all \(x\in{\mathfrak h}\). Specifically, one looks at minimal generating subsets of \({\mathfrak g}\) consisting of root vectors with respect to the aforementioned root space decomposition and one singles out three possible situations according to certain orthogonality relations in terms of the bilinear form \((\cdot\mid\cdot)\) on such a generating set. In each of these three situations, one provides information on the structure of the solvable Lie algebra \({\mathfrak g}\) under consideration.
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 5148859
zbMATH Keywords: solvable Lie algebra
zbMATH Keywords: invariant bilinear form
zbMATH Keywords: maximal positive root vector
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.12.028
OpenAlex ID: W2077485722
Wikidata QID: Q115351472
cites work: Q5732712
cites work: Q3693664
cites work: Q4008275
cites work: Q5295634
cites work: KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank
cites work: Q3154337
cites work: Q4796101
DOI: 10.1016/J.JALGEBRA.2006.12.028
description: scientific article; zbMATH DE number 5148859