Möbius groups over general fields using Clifford algebras associated with spheres
type: Item
instance of: scholarly article
title: Möbius groups over general fields using Clifford algebras associated with spheres
zbMATH Open document ID: 0702.51003
DOI: 10.1007/BF00673627
author: Jay P. Fillmore
author: Arthur Springer
published in: International Journal of Theoretical Physics
publication date: 1990-01-01T00:00:00Z
review text: The space H(2,\({\mathbb{C}})\) of 2-by-2 Hermitian matrices is isometric to Minskowski space \({\mathbb{R}}^{1,3}\). This is commonly used to exhibit the group SL(2,\({\mathbb{C}})\) as a twofold cover of the identity component of the Lorentz group O(1,3). That the Hermitian matrices of H(2,\({\mathbb{C}})\) also represent equations of circles in the Euclidean plane \(E^ 2\) leads to the group PSL(2,\({\mathbb{C}})\) as the Möbius group - a group of complex point transformations that send ``circles'' to ``circles'' - of \(E^ 2\). Clifford algebras naturally arise in the construction of covers of the orthogonal group of a vector space V by spin groups. The authors here extend the classical case \(V={\mathbb{C}}\)- using a new approach which is quite different from that of Vahlen as revived recently by Ahlfors (1986) and others - to the case of a finite-dimensional vector space V over an arbitrary commutative field of characteristic \(\neq 2\) as follows: the equation of a ``sphere'' is determined by a nondegenerate quadratic form of V (instead of by the Hermitian form associated with \({\mathbb{C}})\); the cover of the full orthogonal group of V is constructed by using the whole Clifford algebra of the space of equations of spheres.
Mathematics Subject Classification ID: 51B10
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 51P05
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 51F25
Mathematics Subject Classification ID: 83E30
zbMATH DE Number: 4151235
zbMATH Keywords: cover of the orthogonal group by spins
zbMATH Keywords: Möbius group of a finite- dimensional vector space over a general commutative field
zbMATH Keywords: Clifford algebra
MaRDI profile type: Publication
cites work: Q3951174
cites work: Q3684278
cites work: Q3937844
cites work: Q5675788
cites work: Conformal Transformations and Clifford Algebras
cites work: Q3810082
cites work: Q3324593
description: scientific article; zbMATH DE number 4151235