Lie algebras, 2-groups and cotriangular spaces
type: Item
instance of: scholarly article
title: Lie algebras, 2-groups and cotriangular spaces
zbMATH Open document ID: 1266.51006
author: Arjeh M. Cohen
author: Hans Cuypers
author: J. I. Hall
author: Erik Postma
author: Andries E. Brouwer
publication date: 2012-06-13T00:00:00Z
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 51A99
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 6045764
zbMATH Keywords: Kaplansky algebra
zbMATH Keywords: Lie orientation
zbMATH Keywords: binary orhogonal geometry
zbMATH Keywords: octonions
MaRDI profile type: Publication
full work available at URL: advgeom.2011.047
OpenAlex ID: W2079629526
cites work: On delta spaces satisfying Pasch's axiom
cites work: Finite geometries which contain dual affine planes
cites work: Projective planes, graphs, and simple algebras
cites work: Simple Lie algebras and graphs
Wikidata QID: Q115237052
DOI: 10.1515/ADVGEOM.2011.047
Recommended article: Lie algebras and cotriangular spaces
Recommended article: Cohomology of Lie 2-groups
Recommended article: Q4936152
Recommended article: Q3798823
Recommended article: Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34
Recommended article: Cohomology of Lie Algebras and Algebraic Groups
Recommended article: The space of Lie algebra 2-cocycles of an affine transformation group
Recommended article: Q4002018
Recommended article: Q5200212
Recommended article: Q4835654
published in: Advances in Geometry
review text: Let \((P, L)\) be a partial linear space with the point set \(P\) and the set \(L\) of lines carrying \(3\) points. If one provides each line \( l = \{x, y, z\}\) with a cyclic orientation \(\sigma ( l)\) putting \(\sigma ( l) = \{x, y, z\}\) or \(\sigma ( l) = \{y, x, z\}\), then one can construct for a field \(\mathbb F\) of characteristic different from \(2\) an algebra with bilinear multiplication taking the points of \(P\) as basis of an \(\mathbb F\)-vector space defining the multiplication on this basis by \([x, y] = 0\) if \(x = y\) or \(x, y\) noncollinear and either \([x, y] = z\) or \([x, y] = -z\) depending on whether \( l =\{x, y, z\}\in L\) and \(\sigma ( l) = \{x, y, z\}\) or \( l =\{x, y, z\}\in L\) and \(\sigma ( l) = \{y, x, z\}\). The authors call these algebras Kaplansky algebras and classify the partial linear spaces \((P, L)\), the corresponding Kaplanky algebras of which are Lie algebras. The authors determine all such Lie algebras if they are simple.
reviewed by: Karl Strambach
description: scientific article; zbMATH DE number 6045764