Complexes of Dirac operators in Clifford algebras
type: Item
instance of: scholarly article
title: Complexes of Dirac operators in Clifford algebras
zbMATH Open document ID: 1078.30045
published in: Mathematische Zeitschrift
publication date: 2002-06-27T00:00:00Z
review text: In this paper the following is studied: The Cauchy-Fueter operator in the quaternionic case was obtained by the Moisil-Theodorescu product (the case of the Dirac operator in the Clifford algebra \({\mathcal C}_3\). Thus it is motivated the choice of Dirac operators in the Clifford algebra \({\mathcal C}_m\) only in the case when \(m>4\) or \(m=4\). Another case following directly from the Cauchy-Fueter case is the case of \(n\)-Dirac operators systems in \({\mathcal C}_3\). Any procedure reduces the number of equations (without changing the nature of non-homogeneous equations). Seven methods of action of Dirac operator on a vector variable function are exposed. These methods use the facts that: the Dirac operator is a vector; Weyl equation; the case of \(m\) is even; the spinor formalism for \(m=2n\); the 4-dimensional and the 6-dimensional cases of Dirac equation; the Witt basis using the Clifford algebra conjugate. The Fischer decomposition is discussed for general Clifford polynomials of several vector variables. Rough-Fischer decomposition lemmas and theorems were formulated for all Clifford polynomials. In the case when the dimension \(m\) of the space exceeds \(2l-2\), \(l\) being the number of vector variables in the Cauchy-Dirac complex, the syzygies remain the same for all higher dimensions and they are ``purely radial'', i.e. they are polynomials belonging to the algebra generated by the vector variables alone. In section 4, the case of 3 and 4 operators acting in the Clifford algebra \({\mathcal C}_m\) \((m>4\) or \(m=4\)) is discussed, proving the authors' own results. The made running CoCoA version 3.7 on a Digital Alpha Server is being used. The resolution for the number of syzygies mimicks the quaternionic case. The case of \({\mathcal C}_5\) is being treated in two different ways. The syzygies come from the radial algebra. The authors gave several propositions refering to the compatibility conditions for each step of the proposed programme. In the ca…
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 1761097
zbMATH Keywords: Moisil-Theodorescu product
zbMATH Keywords: Dirac operators
zbMATH Keywords: Clifford algebras
zbMATH Keywords: quaternion
zbMATH Keywords: complex of Cauchy-Fueter operators
zbMATH Keywords: syzygy operator
zbMATH Keywords: syzygy polynomial
author: Irene Sabadini
author: D. C. Struppa
author: Frank Sommen
author: Peter van Lancker
reviewed by: Lidia Elena Kozma
describes a project that uses: CoCoA
MaRDI profile type: Publication
full work available at URL: s002090100297
OpenAlex ID: W2030476573
DOI: 10.1007/S002090100297
description: scientific article; zbMATH DE number 1761097