From linear algebra via affine algebra to projective algebra
type: Item
instance of: scholarly article
title: From linear algebra via affine algebra to projective algebra
zbMATH Open document ID: 1063.17020
author: Wolfgang Bertram
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: In his fundamental paper [``Generalized projective geometries: General theory and equivalence with Jordan structures'', Adv. Geom. 2, No. 4, 329--369 (2002; Zbl 1035.17043)], the author introduced \textit{generalized projective spaces} as the global geometric objects belonging to Jordan pairs over arbitrary base rings. One basic step on the way to the axiomatic description of these objects is contained in the paper under review: Every affine space over a field of characteristic \(\not=2\) can be described in terms of the ``product maps'' \((x,y) \mapsto (1-r)x+ry\). The main part of this paper is devoted to important classical examples of generalized projective geometries: projective geometries, Grassmannian geometries, Lagrangian geometries and conformal geometries (of arbitrary dimensions and over arbitrary communtative base rings, provided that \(2\) is invertible) are discussed here. Moreover, the author considers polarities of these geometries and gives links to the theory of symmetric spaces.
Mathematics Subject Classification ID: 17C37
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51N25
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 2081750
zbMATH Keywords: affine geometry
zbMATH Keywords: projective geometry
zbMATH Keywords: polar geometry
zbMATH Keywords: Grassmannian
zbMATH Keywords: Lagrangian
zbMATH Keywords: symmetric space
zbMATH Keywords: generalized projective geometry
zbMATH Keywords: polarity
reviewed by: Harald Löwe
MaRDI profile type: Publication
full work available at URL: j.laa.2003.06.021
OpenAlex ID: W2082897362
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DOI: 10.1016/J.LAA.2003.06.021
description: scientific article; zbMATH DE number 2081750