On algebraic structures related to Beltrami-Klein model of hyperbolic geometry
type: Item
instance of: scholarly article
title: On algebraic structures related to Beltrami-Klein model of hyperbolic geometry
zbMATH Open document ID: 1194.51005
author: Sayed-Ghahreman Taherian
published in: Results in Mathematics
publication date: 2010-06-23T00:00:00Z
review text: In [\textit{H. Karzel}, Discrete Math. 208--209, 387--409 (1999; Zbl 0941.51002)], the algebraization of a point reflection structure is exhibited; it yields the loop of derivation at the reference point \(o\). In the present paper Karzel's procedure is executed especially for the Beltrami--Klein model of the hyperbolic plane; this is done in the more general frame of an algebra \((\mathbb K,\mathbb R)\) where \(\mathbb R\) denotes the reals and \(\mathbb K\) stands beside the complex numbers \(\mathbb C\) also for the quaternions \(\mathbb H\) or the octonions \(\mathbb O\). For \({\mathbb D}:=\{z\in{\mathbb K} \mid 1>|z|\}\) the author determines the ``relativity addition \(\oplus\)'' constructively and by formula. The author shows that \(({\mathbb D},\oplus)\) is a K-loop which is neither commutative nor associative but satisfies the Bol-identity. The paper is accompanied by six helpful figures.
reviewed by: Rolf Riesinger
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 51N25
zbMATH DE Number: 5725185
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: K-loop
zbMATH Keywords: gyrogroup
zbMATH Keywords: Beltrami--Klein model
zbMATH Keywords: special relativity
Wikidata QID: Q59383324
MaRDI profile type: Publication
full work available at URL: s00025-010-0021-9
OpenAlex ID: W1993893755
cites work: Q5462693
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DOI: 10.1007/S00025-010-0021-9
description: scientific article; zbMATH DE number 5725185