Groups with an involutory antiautomorphism and \(K\)-loops; application to space-time-world and hyperbolic geometry. I
type: Item
instance of: scholarly article
title: Groups with an involutory antiautomorphism and \(K\)-loops; application to space-time-world and hyperbolic geometry. I
zbMATH Open document ID: 0788.20034
DOI: 10.1007/BF03322306
author: Helmut Karzel
author: Heinrich Wefelscheid
published in: Results in Mathematics
publication date: 1993-10-11T00:00:00Z
review text: If \((G,\cdot)\) is a group with involutory automorphism \(*\) then under certain conditions the set \(P = \{xx^*\mid x\in G\}\) can be provided with an operation \(+\) such that \((P,+)\) is a \(K\)-loop. \(K\)-loops are generalisations of neardomains. The affine group \(\text{Aff}(F,+)\) of a weak \(K\)-loop \((F,+)\) is defined and it is shown that \(\text{Aff}(F,+)\) can be expressed in terms of certain quasidirect products, such as \((F,+)\rtimes_ Q\text{Aut}(F,+)\). These results are then applied to the special linear group \(SL(2,L)\) where \(L\) is quadratic field extension of a commutative field. For these cases it is shown that the associated loops \((P,+)\) are \(K\)-loops and for \(L = \mathbb{R}\), \(P\) is the unit sphere in the future cone of the associated Minkowski-space-time-world.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 53Z05
zbMATH DE Number: 428634
zbMATH Keywords: kinematic spaces
zbMATH Keywords: involutory automorphisms
zbMATH Keywords: \(K\)-loops
zbMATH Keywords: generalisations of neardomains
zbMATH Keywords: affine groups
zbMATH Keywords: quasidirect products
zbMATH Keywords: special linear groups
zbMATH Keywords: Minkowski-space-time
Wikidata QID: Q59383316
reviewed by: W. Kerby
MaRDI profile type: Publication
cites work: Q4004345
cites work: On loops of odd order
cites work: Q5662517
cites work: Bemerkungen über Fastbereiche und scharf zweifach transitive Gruppen. (Remarks on near-domains and sharply 2-transitive groups.)
cites work: On \(K\)-loops of finite order
cites work: Group-like structure underlying the unit ball in real inner product spaces
cites work: Fibered indidence loops and kinematic loops
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description: scientific article; zbMATH DE number 428634