Spinors and calibrations

TABLE OF CONTENTS......Page 5
PREFACE......Page 9
PART I: CLASSICAL GROUPS AND NORMED ALGEBRAS......Page 13
1. CLASSICAL GROUPS I ......Page 15
The General Linear Groups ......Page 16
Groups Defined by Bilinear Forms ......Page 18
Other Miscellaneous Groups ......Page 20
Isomorphisms ......Page 25
Summary ......Page 27
2. THE EIGHT TYPES OF INNER PRODUCT SPACES ......Page 31
The Standard Models ......Page 34
Orthogonality ......Page 35
A Canonical Form (The Basis Theorem) ......Page 41
The Parts of an Inner Product ......Page 43
Group Representations and Orbits ......Page 53
Generalized Spheres ......Page 54
The Basis Theorem Revisited ......Page 58
Adjoints ......Page 59
Lie Algebras ......Page 60
4. EUCLIDEAN/LORENTZIAN VECTORS SPACES ......Page 69
The Cauchy-Schwarz Equality ......Page 70
Special Relativity ......Page 74
The Cartan-Dieudonne Theorem ......Page 79
Grassmannians and SOT(p, q), the Reduced Special Orthogonal Groups ......Page 87
Real n-Manifolds: The Group GL(n, R) ......Page 93
Oriented Real n-Manifolds: The Group GL+(n, R) ......Page 94
Complex (and Almost Complex) n-Manifolds: The Group GL(n, C) ......Page 95
Quaternionic (and Almost Quaternionic) n-Manifolds: The Group GL(n, H)  H* ......Page 97
Manifolds with Volume: The Groups SL(n, R) and SL(n, C) ......Page 98
Real Symplectic Manifolds: The Group Sp(n, R) ......Page 99
13. INNER PRODUCTS ON THE SPACES OF SPINORS......Page 0
Complex Symplectic Manifolds: The Group Sp(n, C) ......Page 100
Kahler Manifolds (of Signature p, q): The Group U(p, q) ......Page 101
Special Kahler Manifolds (of Signature p, q): The Group SU(p, q) ......Page 102
HyperKiihler Manifolds (of Signature p, q): The Group HU(p, q) ......Page 103
Quaternionic Kahler Manifolds (of Signature p, q): The Group HU(p,q) HU(1) ......Page 104
Quaternionic Skew Hermitian Manifolds: The Group SK(n, H) ......Page 105
Coincidences of Geometries in Low Dimensions ......Page 106
6. NORMED ALGEBRAS ......Page 113
The Cayley-Dickson Process ......Page 116
The Hurwitz Theorem ......Page 119
Cross Products ......Page 122
The Exceptional Lie Group G2 ......Page 125
The Fundamental Theorem ......Page 137
The Kahler Case ......Page 139
The Special Lagrangian Calibration ......Page 142
The Special Lagrangian Differential Equation ......Page 146
Examples of Special Lagrangian Submanifolds ......Page 150
Associative Geometry ......Page 156
The Angle Theorem ......Page 158
Generalized Nance Calibrations and Complex Structures ......Page 168
Representations ......Page 175
Uniqueness of Intertwining Operators ......Page 179
Automorphisms ......Page 181
Inner Products ......Page 182
PART II: SPINORS......Page 187
9. THE CLIFFORD ALGEBRAS ......Page 189
The Clifford Automorphisms ......Page 193
The Clifford Involutions ......Page 194
The Clifford Inner Product ......Page 196
The Main Symmetry ......Page 197
The Clifford Center ......Page 199
Self Duality ......Page 200
Trace ......Page 201
The Complex Clifford Algebras ......Page 202
10. THE GROUPS SPIN AND PIN ......Page 207
The Grassmannians and Reflections ......Page 209
Additional Groups for Nondefinite Signature ......Page 211
The Conformal Pin (or Clifford) Group ......Page 213
Determinants ......Page 215
11. THE CLIFFORD ALGEBRAS Cl(r, s) AS ALGEBRAS ......Page 219
The Pinor Representations ......Page 222
The Spinor Representations ......Page 225
The First Proof ......Page 227
Even Dimensions ......Page 229
Odd Dimensions ......Page 232
A Model for C1(p, p) ......Page 239
Pinor Inner Products for Cl(p, p) = EndR(P(p, p)) ......Page 242
The Complex Clifford Algebras (Continued from Chapter 9) ......Page 245
Cl(r, s)(r + s = 2p) as a Subalgebra of Clc (2p) = Cl(p, P) OR C ......Page 247
The Pinor Reality Map ......Page 249
A Second Proof of the Classification Theorem ......Page 252
Pure Spinors ......Page 253
The Spinor Inner Product ......Page 259
The Spin Representation and the (Reduced) Classical Companion Group Cp?r, s) ......Page 262
The Pinor Inner Products t and e ......Page 264
Pinor Multiplication ......Page 278
Signature ......Page 280
Cartan's Isomorphisms ......Page 283
Triality ......Page 287
Transitive Actions on Spheres ......Page 295
The Cayley Plane and the Exceptional Group F4 ......Page 301
Clifford Algebras in Low Dimensions ......Page 310
Squares of Spinors and Calibrations ......Page 320
REFERENCES ......Page 327
INDEX ......Page 329