The algebraic theory of spinors and Clifford algebras (collected works, vol.2)(no p.206-207)

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