Chain spaces via Clifford algebras
type: Item
instance of: scholarly article
title: Chain spaces via Clifford algebras
zbMATH Open document ID: 0872.51006
DOI: 10.1007/BF01305964
author: Andrea Blunck
published in: Monatshefte für Mathematik
publication date: 1997-09-23T00:00:00Z
full work available at URL: https://eudml.org/doc/178762
review text: The author continues her investigation of chain spaces. She considers the chain space \(\Sigma({\mathcal Q})\) associated with a proper non-degenerate quadric \({\mathcal Q}\) in projective space of dimension at least 3 over a commutative field \(K\). The point set of \(\Sigma({\mathcal Q})\) contains all points of \({\mathcal Q}\) except its double points and the chain set consists of all non-trivial plane sections of \({\mathcal Q}\). This generalises the familiar models of the Miquelian Möbius, Laguerre and Minkowski planes in case \(K\) being the field of reals and \({\mathcal Q}\) being an elliptic quadric, an elliptic cylinder or a ruled quadric in 3-dimensional real projective space, respectively. Another algebraic description of these three Miquelian Benz planes is one as a chain geometry belonging to the 2-dimensional real kinematic algebras, that is, the complex numbers, the dual numbers and the anomal complex numbers in the respective cases. The author extends the algebraic descriptions of Miquelian Benz planes to one for chain spaces \(\Sigma({\mathcal Q})\) in case \(K\) has at least 5 elements. The underlying vector space \(W\) can be written as the orthogonal sum of a metric vector space \(V\) with quadratic form \(Q\) and a hyperbolic plane. Let \(A\) be the Clifford algebra associated with \((V,Q)\) and let \(B=Vw\) for some vector \(w\in V\) with \(Q(w)=1\). Then \(B\) is a strong Jordan system in \(A\) and gives rise to a subspace \(\Sigma(K,A,B)\) of the chain geometry \(\Sigma(K,A)\) (the projective line over \(A\) with chains being the images of the projective line over \(K\) under the general linear group over \(A\)). In fact, up to isomorphism, \(\Sigma(K,A,B)\) is independent of the choice of \(w\) since different vectors yield isotopic Jordan systems and these in turn yield isomorphic chain spaces, see the author [Abh. Math. Semin. Univ. Hamb. 64, 33-49 (1994; Zbl 0817.51004)]. It is shown that \(\Sigma({\mathcal Q})\) and \(\Sigma(K,A,B)\)…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51B05
zbMATH DE Number: 986868
zbMATH Keywords: chain space
zbMATH Keywords: chain geometry
zbMATH Keywords: Clifford algebra
zbMATH Keywords: Jordan system
MaRDI profile type: Publication
cites work: Q5674022
cites work: Chain spaces over Jordan systems
cites work: Klingenberg chain spaces
cites work: Q3285650
cites work: On sets of subspaces closed under reguli
cites work: Q4764150
cites work: A categorical glimpse at the reconstruction of geometries
cites work: Q4004234
Recommended article: Some aspects of the embeddings of chain geometries
Recommended article: Q3373757
Recommended article: Q4306035
Recommended article: The Double Tangency Symmetries in Laguerre Plane
Recommended article: Affine circle geometry over quaternion skew fields
Recommended article: Generalized affine chain geometries
Recommended article: Remarks on the proof of the representation theorem for Miquelian Möbius planes by A. Lenard
Recommended article: Q3830119
Recommended article: Q4026391
Recommended article: Chain geometries over local alternative algebras
description: scientific article; zbMATH DE number 986868