Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992) (Mathematics and Its Applications (321))

This volume is dedicated to the memory of Albert Crumeyrolle, who died on June 17, 1992. In organizing the volume we gave priority to: articles summarizing Crumeyrolle's own work in differential geometry, general relativity and spinors, articles which give the reader an idea of the depth and breadth of Crumeyrolle's research interests and influence in the field, articles of high scientific quality which would be of general interest. In each of the areas to which Crumeyrolle made significant contribution - Clifford and exterior algebras, Weyl and pure spinors, spin structures on 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 of the field he influenced, and the enduring respect for, and tribute to, him and his accomplishments in the mathematical community. It isour pleasure to thank Peter Morgan, Artibano Micali, Joseph Grifone, Marie Crumeyrolle and Kluwer Academic Publishers for their help in preparingthis volume. Front Matter....Pages i-xx Front Matter....Pages 1-1 Some Clifford Algebra History....Pages 3-12 Front Matter....Pages 13-13 Tensors and Clifford Algebra....Pages 15-38 Sur Les Algèbres de Clifford III....Pages 39-57 Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras....Pages 59-99 Clifford Algebra Techniques in Linear Algebra....Pages 101-109 Front Matter....Pages 111-111 Construction of Spinors via Witt Decomposition and Primitive Idempotents: A Review....Pages 113-123 Crumeyrolle-Chevalley-Riesz Spinors and Covariance....Pages 125-132 Twistors as Geometric Objects in Spacetime....Pages 133-135 Crumeyrolle’s Bivectors and Spinors....Pages 137-166 Spinor Fields and Superfields as Equivalence Classes of Exterior Algebra Fields....Pages 167-176 Chevalley-Crumeyrolle Spinors in McKane-Parisi-Sourlas Theorem....Pages 177-198 Spinors from a Differential Geometric Point of View....Pages 199-204 Front Matter....Pages 205-240 Eigenvalues of the Dirac Operator, Twistors and Killing Spinors on Riemannian Manifolds....Pages 241-241 Dirac’s Field Operator Ψ....Pages 243-256 Biquaternionic Formulation of Maxwell’s Equations and their Solutions....Pages 257-263 The Massless Dirac Equation, Maxwell’s Equation, and the Application of Clifford Algebras....Pages 265-280 The Conformal Covariance of Huygens’ Principle-Type Integral Formulae in Clifford Analysis....Pages 281-300 Front Matter....Pages 301-310 Cliffor-Valued Functions in Cl 3 ....Pages 311-311 Clifford-Analysis and Elliptic Boundary Value Problems....Pages 313-324 Front Matter....Pages 325-334 A Complete Boundary Collocation System....Pages 311-311 On the Algebraic Foundations of the Vector є -Algorithm....Pages 335-342 Front Matter....Pages 343-361 Classical Spinor Structures on Quantum Spaces....Pages 363-363 A Unified Metric....Pages 365-377 Quantum Braided Clifford Algebras....Pages 379-385 Clifford Algebra for Hecke Braid....Pages 387-395 Back Matter....Pages 397-411 ....Pages 413-425

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.