Metric Lie algebras and quadratic extensions
type: Item
instance of: scholarly article
title: Metric Lie algebras and quadratic extensions
zbMATH Open document ID: 1112.17006
published in: Transformation Groups
publication date: 2007-01-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0312243
review text: A metric Lie algebra is a finite-dimensional real Lie algebra equipped with an invariant nondegenerate symmetric bilinear form. An example of a metric Lie algebra is the Lie algebra of the transvection group of a pseudo-Riemannian symmetric space. The paper contains a systematic study of the structure of metric Lie algebras. A metric Lie algebra \({\mathfrak g}\) is indecomposable if it does not contain a proper ideal on which the bilinear form is nondegenerate, and in this case \({\mathfrak g}\) is either simple or does not contain a simple ideal. Let \({\mathfrak l}\) be a Lie algebra and \((\rho,{\mathfrak a})\) be an orthogonal \({\mathfrak l}\)-module. A quadratic extension of \({\mathfrak l}\) by \({\mathfrak a}\) is a quadruple \(({\mathfrak g},{\mathfrak i},i,p)\), where \({\mathfrak g}\) is a metric Lie algebra, \({\mathfrak i}\) is an isotropic ideal in \({\mathfrak g}\), and \(i : {\mathfrak a} \to {\mathfrak g}/{\mathfrak i}\) and \(p : {\mathfrak g}/{\mathfrak i} \to {\mathfrak l}\) are Lie algebra homomorphisms such that \(0 \to {\mathfrak a} \to {\mathfrak g}/{\mathfrak i} \to {\mathfrak l} \to 0\) is an exact sequence, \(i(\rho(L)A) = [\tilde{L},i(A)] \in i({\mathfrak a})\) holds for all \(A \in {\mathfrak a}\) and \(\tilde{L} \in {\mathfrak g}/{\mathfrak i}\) with \(\rho(\tilde{L}) = L\), \(i({\mathfrak a}) = {\mathfrak i}^\perp/{\mathfrak i}\) and \(i : {\mathfrak a} \to {\mathfrak i}^\perp/{\mathfrak i}\) is an isometry. Such a quadratic extension does not contain a simple ideal. The authors show that any metric Lie algebra \({\mathfrak g}\) without simple ideals has the structure of a balanced quadratic extension. Here, balanced means that \({\mathfrak i}\) coincides with a canonical isotropic ideal \({\mathfrak i}({\mathfrak g})\) in \({\mathfrak g}\). For a balanced quadratic extension the representation \(\rho\) of \({\mathfrak l}\) on \({\mathfrak a}\) is semisimple.\break The balanced quadratic extensions can be identified with elements in c…
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 5118032
zbMATH Keywords: metric Lie algebras
zbMATH Keywords: quadratic extensions
zbMATH Keywords: pseudo-Riemannian symmetric spaces
author: Ines Kath
author: Martin Olbrich
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
OpenAlex ID: W2090952953
arXiv ID: math/0312243
Wikidata QID: Q115389531
DOI: 10.1007/S00031-005-1106-5
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description: scientific article