Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992) PDF
Preview Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992)
CliffordAlgebras and SpinorStructures Mathematics and Its Applications ManagingEditor: M.HAZEWINKEL CentreforMathematicsandComputerScience, Amsterdam,TheNetherlands Volume321 Clifford Algebras and Spinor Structures A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919-1992) editedby Rafal Ablamowicz Gannon University, Erie,Pennsylvania, U.S.A. and PerttiLounesto Helsinki University ofTechnology, Espoo, Finland Springer-Science+Business Media, B.Y: AC.I.P.Cataloguerecordforthisbookisavailablefrom theLibraryofCongress. ISBN978-90-481-4525-6 ISBN978-94-015-8422-7(eBook) DOI 10.1007/978-94-015-8422-7 Printedonacid-freepaper AllRights Reserved ©1995SpringerScience+BusinessMedia Dordrecht OriginallypublishedbyKluwer AcademicPublishersin1995. Softcoverreprint ofthehardcover1stedition 1995 Nopart ofthematerialprotectedbythiscopyrightnotice maybereproducedor utilizedinanyform orbyanymeans, electronicormechanical, includingphotocopying,recordingorbyanyinformationstorageand retrievalsystem,withoutwritten permissionfrom thecopyrightowner. TABLE OF CONTENTS/TABLE DES MATIERES Preface/Avant-Propos viii A. Micali, Albert Crumeyrolle, la demarche algebriqued'un geometre. ix List of publications of Albert Crumeyrolle (1919-1992). xv Ph.D. Theses written under the supervision of Albert Crumeyrolle. xix HISTORICAL SURVEY 1 A. Diek and R. Kantowski,Some Clifford algebra history. 3 CLIFFORD ALGEBRAS 13 A. Charlier, M.-F. Charlier and A. Roux, Tensors and Clifford algebra. 15 A. Micali, Sur les algebres de CliffordIII. 39 R. Shaw, Finite geometry, Diracgroups and the table ofreal Clifford algebras. 59 G. Sobczyk, Clifford algebra techniques in linear algebra. 101 CRUMEYROLLE/CHEVALLEY, WEYL, PURE AND MAJORANA SPINORS 111 R. Ablamowicz, Construction ofspinors via Witt decomposition and primitive idempotents: a review. 113 G. Jones and W.E. Baylis, Crumeyrolle-Chevalley-Rieszspinors and covariance. 125 J. Keller, Twistors as geometricobjects in spacetime. 133 P. Lounesto, Crumeyrolle's bivectors and spinors. 137 F. Piazzese, On the relationships between the Dirac spinors and Clifford subalgebra Cet,3' 167 W.A. Rodrigues, Jr., Q.A.G. de Souza, and J. Vaz, Jr., Spinor fields and superfields as equivalence classes ofexterior algebrafields. 177 S. Rodriguez-Romo, Chevalley-Crumeyrollespinors in McKane-Parisi- Sourlas theorem. 199 M. Rosenbaum, C.P. Luehr, and H. Harleston, Spinors from a differential geometric point ofview. 205 v DIRAC OPERATOR, MAXWELL'S EQUATIONS, AND CONFORMAL COVARIANCE 241 H. Baum and Th. Friedrich, Eigenvalues of the Dirac operator, twistors and Killing spinors on Riemannian manifolds. 243 H.T. Cho, A. Diek, and R. Kantowski, Dirac's field operator W. 257 K. Imaeda, Biquaternionic formulation of Maxwell's equations and their solutions. 265 P. Morgan, The massless Dirac equation, Maxwell's equations, and the application of Clifford algebras. 281 J. Ryan, The conformal covariance of Huygens' principle-type integral formulae in Clifford analysis. 301 CLIFFORD ANALYSIS, BOUNDARY VALUE PROBLEMS, HERMITE INTERPOLANTS, AND PADE APPROXIMANTS 311 W.E. Baylis and B. Jancewicz, Cliffor-valued functions in C£3' 313 K. Giirlebeck and W. Sprossig, Clifford analysis and elliptic boundary value problems. 325 F. Kippig, A complete boundary collocation system. 335 D.E. Roberts, On the algebraic foundations of the vector €-algorithm. 343 CLIFFORD ALGEBRAS AND GENERALIZATIONS 363 M. Durdevic, Classical spinor structures on quantum spaces. 365 B.M. Kemmell, A unified metric. 379 J. Lawrynowicz, L.C. Papaloucas, and J. Rembieliriski, Quantum braided Clifford algebras. 387 Z. Oziewicz, Clifford algebrafor Heeke braid. 397 INDEX 413 vi Photograph: Albert Crumeyrolle, December 10, 1919- June 17, 1992. Courtesy of Mme Crumeyrolle vii PREFACE This volume is dedicated to the memory of Albert Crumeyrolle, who died on June 17, 1992. Inorganizing the volume wegave priority to: articles summarizing Crumeyrolle's own work in differential geometry, general relativity and spinors, articles which givethe readeran ideaofthedepth and breadth ofCrumeyrolle's research interests and influence in the field, articles of high scientific quality which would be ofgeneral interest. Ineach of the areas to which Crumeyrolle made significant contribution - Clifford and exterioralgebras,Weyland purespinors, spin structureson manifolds, principle of triality, conformal geometry - there has been substantial progress. Our hope is that the volume conveys the originality of Crumeyrolle's own work, the continuing vitality ofthe field he influenced, and the enduring respect for, and tribute to, him and his accomplishments in the mathematical community. It isourpleasureto thank PeterMorgan, Artibano Micali, Joseph Grifone, Marie Crumeyrolleand Kluwer AcademicPublishersfortheirhelp inpreparingthisvolume. September 1994 Rafal Ablamowicz Pertti Lounesio viii ALBERT CRUMEYROLLE, LA DEMARCHE ALGEBRIQUE D'UN GEOMETRE ARTIBANO MICALI Umiuersite Montpellier II, Departemesit des Sciences Mathematiques, Place Eugene Bataillon, 34095 Montpellier Cedex OS, France and Universite de Ouagadougou, Departement de Mathematiques et lnformatique, 03 B.P. 7021 Ouagadougou 03, Burkina Faso Abstract. Crumeyrolle's work in Mathematics has a constant line of research: in all his papers, he uses orthogonal or symplectic Clifford algebras or ideas that are closely related with Clifford algebras like spinors. In this sense, Albert Crumeyrolle is an a1gebrist. Sometimes his attention was devoted toapply Cliffordstructuresto solvegeometricalor physical problems but alwayswith good algebraic ideas. L'reuvre d'Albert Crumeyrolle (1919/1992) a 1. On ne peut jamais ecrire sur un ami qui nous a quittes jamais sans une forte a dose d'emotion. Je connaissais Albert Crumeyrolle de longue date travers ses publications car ses travaux sur les algebres de Clifford m'interessaient beaucoup. Mon premier contact avec lui eut lieu lors du premier atelier (workshop) sur les algebres de Cliffordet ses applications ala Physique Mathematique qui s'est tenu a Canterburyenseptembre1985. Onsait que Albert Crumeyrolleavait uneformation en Goometrie Differentielleet sesapplications ala PhysiqueMathematiqueobtenue dans Iesillaged'Andre Lichnerowicz. MaisdesIedebut desacarriereuniversitaire, il manifeste dans sesrecherches un net penchant pour lesmethodes algebriques. Dans Iepremier article important (d. [7]) publie apres sa these de doctorat (d. [1]),il s'attaque al'etude des variates astructure spinorielle par des methodes d'algebres de Cliffordmises au point par Claude Chevalley (1909/1984) 1 dans ses travaux sur la theorie algebrique des spineurs. Dans la suite de ses recherches sur les varietes a structure spinorielle, il introduit une importante notion algebrique, qui aura un extraordinaire developpement par la suite, celle de groupe de spinorialite (d. [9]). Bien qu'il critique Ie cadre localet purement algebriquede certaines representations matricielles de la Mecanique Quantique (d.[lO], Resume.), il utilise lui-meme des methodes algebriques dans sesrecherches tout en rappelant qu'il est, avant tout, un geometrequi regarde lemonde d'un point de vue physique (cf. [11]). 2. L'un des apports importants d'Albert Crumeyrolle, en tant qu'algebriste, a est la redaction d'un cours donne l'Universite de Toulouse sur la theorie des algebres de Clifford et I'algebrisation du concept spinoriel (cf. [12]). II convenait, 1 cf. J. Tits, La mort du mathematicien ClaudeChevalley,Le Monde 04/07/1984.
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