Octonions, Jordan algebras, and exceptional groups

Front Cover\r......Page 1
Series title\r......Page 3
Title......Page 5
ISBN 3-540-66337-1......Page 6
Preface......Page 7
Contents......Page 9
1.1 Quadratic and Bilinear Forms......Page 11
1.2 Composition Algebras. The Minimum Equation......Page 13
1.3 Conjugation. Inverses......Page 17
1.4 Moufang Identities. Alternative Laws......Page 19
1.5 Subalgebras. Doubling......Page 21
1.6 Structure and Dimension of a Composition Algebra......Page 23
1.7 A Composition Algebra is Determined by its Norm\r......Page 25
1.8 Split Composition Algebras......Page 27
1.9 Center and Associating Elements......Page 29
1.10 Classification over Special Fields......Page 31
1.11 Historical Notes......Page 33
2.2 Connectedness and Dimension of the Automorphism Group......Page 35
2.3 The Automorphism Group is of Type G2......Page 39
2.4 Derivations and the Lie Algebra of the Automorphism Group......Page 43
2.5 Historical Notes......Page 45
3.1 Similarities. Clifford Algebras, Spin Groups and Spinor Norms......Page 47
3.2 The Principle of Triality......Page 51
3.3 Outer Automorphisms Defined by Triality......Page 55
3.4 Automorphism Group and Rotation Group of an Octonion Algebra......Page 57
3.5 Local Triality......Page 59
3.6 The Spin Group of an Octonion Algebra......Page 67
3.8 Historical Notes......Page 75
4.1 Normal Twisted Composition Algebras......Page 79
4.2 Nonnormal Twisted Composition Algebras......Page 89
4.3 Twisted Composition Algebras over Split Cubic Extensions......Page 99
4.4 Automorphism Groups of Twisted Octonion Algebras......Page 101
4.5 Normal Twisted Octonion Algebras with Isotropic Norm......Page 103
4.6 A Construction of Isotropic Normal Twisted Octonion Algebras......Page 109
4.7 A Related Central Simple Associative Algebra......Page 111
4.8 A Criterion for Reduced Twisted Octonion Algebras. Applications......Page 115
4.9 More on Isotropic Normal Twisted Octonion Algebras......Page 117
4.10 Nonnormal Twisted Octonion Algebras with Isotropic Norm......Page 119
4.11 Twisted Composition Algebras with Anisotropic Norm......Page 121
4.12 Historical Notes......Page 126
5.1 J-algebras. Definition and Basic Properties......Page 127
5.2 Cross Product. Idempotents......Page 131
5.3 Reduced J-algebras and Their Decomposition......Page 135
5.4 Classification of Reduced J-algebras......Page 143
5.5 Further Properties of Reduced J-algebras......Page 151
5.6 Uniqueness of the Composition Algebra......Page 155
5.7 Norm Class of a Primitive Idempotent......Page 159
5.8 Isomorphism Criterion. Classification over Some Fields......Page 161
5.9 Isotopes. Orbits of the Invariance Group of the Determinant......Page 163
5.10 Historical Notes......Page 169
6.1 Reducing Fields of J-algebras......Page 171
6.2 From J-algebras to Twisted Composition Algebras......Page 173
6.3 From Twisted Composition Algebras to J-algebras......Page 177
6.4 Historical Notes......Page 181
7.1 The Automorphisms Fixing a Given Primitive Idempotent......Page 183
7.2 The Automorphism Group of an Albert Algebra......Page 187
7.3 The Invariance Group of the Determinant in anAlbert Algebra......Page 189
7.4 Historical Notes......Page 191
8.1 Galois Cohomology......Page 195
8.2 An Invariant of Composition Algebras......Page 199
8.3 An Invariant of Twisted Octonion Algebras......Page 201
8.4 An Invariant of Albert Algebras......Page 205
8.5 The Freudenthal-Tits Construction......Page 209
References......Page 211
Index......Page 215
Back cover\r......Page 220