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This volume introduces mathematicians and physicists to a crossing point of algebra, physics, differential geometry and complex analysis. The book follows the French tradition of Cartan, Chevalley and Crumeyrolle and summarizes Crumeyrolle's own work on exterior algebra and spinor structures. The depth and breadth of Crumeyrolle's research interests and influence in the field is investigated in a number of articles.
Of interest to physicists is the modern presentation of Crumeyrolle's approach to Weyl spinors, and to his spinoriality groups, which are formulated with spinor operators of Kustaanheimo and Hestenes. The Dirac equation and Dirac operator are studied both from the complex analytic and differential geometric points of view, in the modern sense of Ryan and Trautman.
For mathematicians and mathematical physicists whose research involves algebra, quantum mechanics and differential geometry.
Preface/Avant-propos. La démarche algébrique d'un géomètre; A. Micali. List of publications of Albert Crumeyrolle (1919—1992). Ph.D. theses written under the supervision of Albert Crumeyrolle. Historical Survey:- Some Clifford algebra history; A. Diek, R. Kantowski. Clifford Algebras:- Tensors and Clifford algebra; A. Charlier, M.-F. Charlier, A. Roux. Sur les algèbres de Clifford III; A. Micali. Finite geometry, Dirac groups and the table of real Clifford algebras; R. Shaw. Clfford algebra techniques in linear algebra; G. Sobczyk. Crumeyrolle/Chevalley, Weyl, Pure and Majorana Spinors:- Construction of spinors via Witt decomposition and primitive idempotents: a review; R. Ablamowicz. Crumeyrolle-Chevalley-Riesz spinors and covariance; G. Jones, W.E. Baylis. Twistors as geometric objects in spacetime; J. Keller. Crumeyrolle's bivectors and spinors; P. Lounesto. On the relationships between the Dirac spinors and Clifford subalgebra Cl+1,3; F. Piazzese. Spinor fields and superfields as equivalence classes of exterior algebra fields; W.A. Rodrigues Jr., Q.A.G. de Souza, J. Vaz Jr. Chevalley-Crumeyrolle spinors in McKane-Parisi-Sourlas theorem; S. Rodríguez-Romo. Spinors from a differential point of view; M. Rosenbaum, C.P. Luehr, H. Harleston. Dirac Operator, Maxwell's Equations, and Conformal Covariance:- Eigenvalues of the Dirac operator, twistors and Killing spinors on Riemannian manifolds; H. Baum, Th. Friedrich. Dirac's field operator ; H.T. Cho, A. Diek, R. Kantowski. Biquaternionic formulation of Maxwell's equations and their solutions; K. Imaeda. The massless Dirac equation, Maxwell's equations, and the application of Clifford algebras; P. Morgan. The conformal covariance of Huygens' principle-type formulae in Clifford analysis; J. Ryan. Clifford Analysis, Boundary Value Problems, Hermite Interpolants, and Padé Approximants:- Clifford-valued functions in Cl3; W.E. Baylis, B. Jancewicz. Clifford analysis and elliptic boundary value problems; K. Gürlebeck, W. Sprösig. A complete boundary collocation system; F. Kippig. On the algebraic foundations of the vector &egr;-algorithm; D.E. Roberts. Clifford Algebras and Generalizations:- Classical spinor structures on quantum spaces; M. urðevic. A unified metric; B.M. Kemmell. Quantum braided Clifford algebras; J. Ławrynowicz, L.C. Papaloucas, R. Rembielinski. Clifford algebra for Hecke braid; Z. Oziewicz. Index.