An introduction to commutative quaternions
type: Item
instance of: scholarly article
title: An introduction to commutative quaternions
zbMATH Open document ID: 1207.30073
author: Francesco Catoni
author: Roberto Cannata
author: Paolo Zampetti
published in: Advances in Applied Clifford Algebras
publication date: 2007-01-24T00:00:00Z
review text: Since the important articles by \textit{A. Scheffers} [Ber. Verh. Königl. Sächs. Gesellsch. Wiss. Leipzig, Math.-Phys. Kl. 45, 828--848 (1893); 46, 120--134 (1894; JFM 25.0667.03)] many authors have dealt with differentiable functions in commutative algebras leading to generalized Cauchy-Riemann equations. More success and more applications seem to be reached via non-commutative algebras, e.g. Clifford algebras. The authors of this paper deal with algebras they call commutative quaternions (a name enclosing a contradiction), which is an algebra with base elements \(1, i, j, k\) and the multiplication rules \(i^2=k^2=\alpha, j^2=1, ij=ji=k, ik=ki=\alpha j\) with \(\alpha = -1\) (elliptic case), \(\alpha=0\) (parabolic case), \(\alpha = 1\) (hyperbolic case). The generalized Cauchy-Riemann equations and some elementary functions are given. The mappings given by such functions are claimed to be conformal, so it would be very interesting to see which functions are in this class of differentiable functions (because of the theorem of Liouville).
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30A05
zbMATH DE Number: 5117978
zbMATH Keywords: functions in algebras
zbMATH Keywords: differentiation
zbMATH Keywords: Cauchy-Riemann equations
reviewed by: Klaus Habetha
MaRDI profile type: Publication
full work available at URL: s00006-006-0002-y
OpenAlex ID: W2021160337
DOI: 10.1007/S00006-006-0002-Y
description: scientific article; zbMATH DE number 5117978