Table Of ContentCliffordAlgebras 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
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©1995SpringerScience+BusinessMedia Dordrecht
OriginallypublishedbyKluwer AcademicPublishersin1995.
Softcoverreprint ofthehardcover1stedition 1995
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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.
