Die affinen Projektivitätengruppen der lokalkompakten zusammenhängenden Translationsebenen. (The affine groups of projectivities of locally compact connected translation planes)
type: Item
instance of: scholarly article
title: Die affinen Projektivitätengruppen der lokalkompakten zusammenhängenden Translationsebenen. (The affine groups of projectivities of locally compact connected translation planes)
zbMATH Open document ID: 0612.51003
DOI: 10.1007/BF01192005
author: Theo Grundhöfer
author: Karl Strambach
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathcal U}\) be a locally compact connected translation plane. If \({\mathcal U}\) is a Moufang plane (in which case it is coordinatized by the reals, the complexes, the quaternions, or the Cayley numbers), then the group \(\Pi^{aff}({\mathcal U})\) of affine projectivities has been determined. [See, for example, the first author, Geom. Dedicata 13, 435- 438 (1983; Zbl 0523.51006), or the second author, Math. Z. 155, 11-21 (1977; Zbl 0354.50004).] In this article the authors consider the case where \({\mathcal U}\) is not a Moufang plane and prove that \(\Pi^{aff}({\mathcal U})\), as a topological transformation group, is isomorphic to the connected component \(AGL(n,K)^ 1\) of the affine group of \(K^ n\), where K is the kernel of \({\mathcal U}\) and 2n is the dimension of \({\mathcal U}\) over K. In the process of proving the result given above, the authors characterize the quaternions among the four dimensional locally compact quasifields containing \({\mathbb{C}}\) in their kernel. The authors also give a necessary and sufficient condition involving \(\Pi^{aff}({\mathcal U})\) for a locally compact connected affine plane to be a translation plane. This sharpens a result of the second author.
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 3990224
zbMATH Keywords: group of affine projectivities
zbMATH Keywords: locally compact connected translation plane
reviewed by: Michael J. Kallaher
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192005
OpenAlex ID: W2913787760
description: scientific article; zbMATH DE number 3990224