The algebraic theory of spinors and Clifford algebras: Collected works, volume 2

Description

This volume is the first in a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a much sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J.-P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. An insightful review of "Spinors" by J. Dieudonne is also made available to the reader in this new edition. THE CONSTRUCTION AND STUDY OF CERTAIN IMPORTANT ALGEBRAS Preface .............................................................. 3 Conventions ......................................................... 4 CHAPTER I. GRADED ALGEBRAS ................................ 5 1. Free Algebras ......................................... 5 2. Graded Algebras ...................................... 7 3. Homogeneous Linear Mappings ........................ 10 4. Associated Gradations and the Main Involution ........ 11 5. Derivations ............................................ 13 6. Existence of Derivations in Free Algebras .............. 18 CHAPTER II. TENSOR ALGEBRAS ............................... 20 1. Tensor Algebras ............................ ' ........... 20 2. Graded Structure of Tbnsor Algebras .................. 23 3. Derivations in a Tensor Algebra ....................... 27 4. Preliminaries About Tensor Product of Modules ....... 29 5.Tensor Product of Semi-Graded Algebras ............... 32 CHAPTER III. CLIFFORD ALGEBRAS ............................ 35 1. Clifi'ord Algebras ...................................... 35 2. Exterior Algebras ..................................... 37 3. Structure of the Clifford Algebra when M has a Base .. . 38 4. Canonical Anti-Automorphism ........................ 43 5. Derivations in the Exterior Algebras; Trace ............ 45 6. Orthogonal Groups and Spinors (a Review) ............ 48 CHAPTER IV. SOME APPLICATIONS OF EXTERIOR ALGEBRAS ............................................ 52 1. Pliicker Coordinates ................................... 52 2. Exponential Mapping ................................. 53 3. Determinants ......................................... 57 4. An application to Combinatorial Topology ............. 62 THE ALGEBRAIC THEORY OF SPINORS INTRODUCTION ................................................... 67 PRELIMINARIES ................................................... 69 1. Terminology ........................................... 69 2. Associative Algebras .................................. 70 3. Exterior Algebras ..................................... 70 CHAPTER I. QUADRATIC FORMS ................................ 72 1.1. Bilinear Forms ...................................... 72 1.2. Quadratic Forms .................................... 75 1.3. Special Bases ........................................ 77 1.4. The Orthogonal Group .............................. 79 1.5. Symmetries .......................................... 83 1.6. Representation of G on the Multivectors ............. 86 CHAPTER II. THE CLIFFORD ALGEBRA ......................... 101 2.1. Definition of the Clifford Algebra .................... 101 2.2. Structure of the Clifford Algebra ..................... 106 2.3. The Group of Clifford ............................... 113 2.4. Spinors (Even Dimension) ........................... 119 2.5. Spinors (Odd Dimension) ............................ 121 2.6. Imbedded Spaces .................................... 122 2.7. Extension of the Basic Field ......................... 124 2.8. The Theorem of Hurwitz ............................ 125 2.9. Quadratic Forms over the Real Numbers ............. 129 CHAPTER III. FORMS OF MAXIMAL INDEX ..................... 134 3.1. Pure Spinors ........................................ 135 3.2. A Bilinear Invariant ................................. 141 3.3. The Tensor Product of the Spin Representation with Itself ................................................ 148 - 3.4. The Tensor Product of the Spin Representation with Itself (Characteristic 9E 2) ............................ 153 3.5. Imbedded Spaces .................................... 161 3.6. The Kernels of the Half—Spin Representations ........ 165 3.7. The Case m = 6 ..................................... 166 3.8. The Case of Odd Dimension ......................... 170 CHAPTER IV. THE PRINCIPLE OF TRIALITY ................... 176 4.1. A New Characterization of Pure Spinors ............. 177 4.2. Construction of an Algebra .......................... 177 4.3. The Principle of 'Ilriality ............................. 181 4.4. Geometric Interpretation ............................ 185 4.5. The Octonions ....................................... 187 BOOK REVIEW (J. DIEUDONNE) .............................. 193 SPINORS IN 1995 (J.-P. BOURGUIGNON) ...................... 199 SUBJECT INDEX ................. . ............................... 211 SYMBOL INDEX ................................................. ' 213