Symmetric, invariant, non-degenerate bilinear form on a Lie algebra
type: Item
instance of: scholarly article
title: Symmetric, invariant, non-degenerate bilinear form on a Lie algebra
zbMATH Open document ID: 0608.17007
DOI: 10.1016/0021-8693(87)90209-2
author: B. George
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Bearing in mind the importance of the Killing form, this paper is concerned with the class of regular quadratic Lie algebras \((g,\varphi)\) where \(g\) is a Lie algebra endowed with a symmetric, invariant, nondegenerate bilinear form \(\varphi\). It turns out that such Lie algebras, which are indecomposable and have nontrivial centre, are of the form \((g_ d,\varphi_ d)\) (where \(d\) is an appropriate derivation of \(g\)), and \((g_ d,\varphi_ d)\) is called the extension of \((g,\varphi)\) by \(d\). This enables classification problems to be tackled, and in particular the case when \(d\) is nilpotent. Some interesting open questions are stated, and the paper concludes with the list of all regular quadratic nilpotent Lie algebras of dimension \(\leq 7\).
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 3983438
zbMATH Keywords: regular quadratic Lie algebras
zbMATH Keywords: bilinear form
zbMATH Keywords: derivation
zbMATH Keywords: nilpotent Lie algebras
MaRDI profile type: Publication
Wikidata QID: Q115366584
full work available at URL: 0021-8693(87)90209-2
OpenAlex ID: W2024261551
cites work: Semisimple subalgebras of semisimple Lie algebras
cites work: Q3693664
cites work: KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank
cites work: Kac-Moody Lie algebras and the universal element for the category of nilpotent Lie algebras
cites work: A characteristic property of simple Lie algebras
cites work: Algèbres de Lie et produit scalaire invariant
description: scientific article; zbMATH DE number 3983438