Factorizations of Möbius gyrogroups.
type: Item
instance of: scholarly article
title: Factorizations of Möbius gyrogroups.
zbMATH Open document ID: 1197.20056
author: N. E. Zubov
published in: Advances in Applied Clifford Algebras
publication date: 2009-08-11T00:00:00Z
full work available at URL: http://hdl.handle.net/10400.8/3819
review text: The Möbius addition on the unit ball \(\mathbf B_1\) in a real Hilbert space \(H\) is a famous example of a gyrogroup which arose from the addition of velocities in the theory of relativity. A. Ungar has introduced and studied this class of loops which have appeared in various other areas of mathematics. The author of the paper under review describes most of the other instances where these loops play a role in Geometry and Algebra. In his work he follows the definitions and notations of Ungar. Considering on \(H\) the quadratic form \(Q(v)=\langle v\mid v\rangle\) the author uses the Clifford algebra \(\text{Cl}[H,Q]\). The canonical embedding of \(H\) in \(\text{Cl}[H,Q]\) defines the so called geometric multiplication of vectors in \(H\) which is used to give a neat form of the operation of the gyrogroup \(\mathbf B_1\). For a vector \(\omega\) on the unit sphere of \(H\) put \(L_\omega=\{t\omega\mid -1\leq\omega\leq 1\}\) and \(D_\omega=\omega^\perp\cap\mathbf B_1\). Then \(L_\omega\) and \(D_\omega\) are gyrosubgroups of \(\mathbf B_1\) and \(L_\omega\) is even a subgroup of \(\mathbf B_1\). The author shows (Theorem 2) that for a given \(\omega\) every element \(c\in\mathbf B_1\) has unique decompositions \(c=d_1l_1=l_2d_2\) with \(l_1,l_2\in L_\omega\), \(d_1,d_2\in D_\omega\). In the sequel it is shown that in \(\mathbf B_1\) the decompositions \(\mathbf B_1=D_\omega L_\omega=L_\omega D_\omega\) define coset partitions \(\{bD_\omega\mid b\in L_\omega\}\), \(\{D_\omega b\mid b\in L_\omega\}\), \(\{aL_\omega\mid a\in D_\omega\}\), \(\{D_\omega a\mid a\in L_\omega\}\). In the paper the members of these partitions are computed in coordinates. Furthermore, dualities between them are described.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 83A05
Mathematics Subject Classification ID: 53A60
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 22E43
zbMATH DE Number: 5593785
zbMATH Keywords: Bruck loops
zbMATH Keywords: gyrogroups
zbMATH Keywords: Clifford algebras
reviewed by: Liudmila Sabinina
MaRDI profile type: Publication
OpenAlex ID: W2000733306
Wikidata QID: Q114232398
DOI: 10.1007/S00006-009-0154-7
description: scientific article; zbMATH DE number 5593785