On quaternions and octonions: their geometry, arithmetic and symmetry

Short title......Page 1
Title page......Page 3
Date-line......Page 4
Contents......Page 5
Preface......Page 9
I The Complex Numbers......Page 11
1.1 The Algebra $\\\\mathbb{R}$ of Real Numbers......Page 13
1.2 Higher Dimensions......Page 15
1.4 The History of Quaternions and Octonions......Page 16
2.1 Rotations and Reflections......Page 21
2.2 Finite Subgroups of $GO_2$ and $SO_2$......Page 24
2.3 The Gaussian Integers......Page 25
2.5 The 2-Dimensional Space Groups......Page 28
II The Quaternions......Page 31
3.1 The Quaternions and 3-Dimensional Rotations......Page 33
3.2 Some Spherical Geometry......Page 36
3.3 The Enumeration of Rotation Groups......Page 39
3.4 Discussion of the Groups......Page 40
3.6 Chiral and Achiral, Diploid and Haploid......Page 43
3.7 The Projective or Elliptic Groups......Page 44
3.8 The Projective Groups Tell Us All......Page 45
3.9 Geometric Description of the Groups......Page 46
Appendix: $v\\\\to \\\\bar{v}qv$ Is a Simple Rotation......Page 50
4.1 Introduction......Page 51
4.2 Two 2-to-l Maps......Page 52
4.3 Naming the Groups......Page 53
4.4 Coxeter\'s Notations for the Polyhedral Groups......Page 55
4.5 Previous Enumerations......Page 58
4.6 A Note on Chirality......Page 59
Appendix: Completeness of the Tables......Page 60
5.1 The Hurwitz Integral Quaternions......Page 65
5.2 Primes and Units......Page 66
5.3 Quaternionic Factorization of Ordinary Primes......Page 68
5.5 Factoring the Lipschitz Integers......Page 71
III The Octonions......Page 75
6 The Composition Algebras......Page 77
6.2 The Conjugation Laws......Page 78
6.3 The Doubling Laws......Page 79
6.4 Completing Hurwitz\'s Theorem......Page 80
6.5 Other Properties of the Algebras......Page 82
6.6 The Maps $L_x$, $R_x$ and $B_x$......Page 83
6.7 Coordinates for the Quaternions and Octonions......Page 85
6.9 The Algebras over Other Fields......Page 86
6.10 The 1-, 2-, 4-, and 8-Square Identities......Page 87
6.11 Higher Square Identities: Pfister Theory......Page 88
Appendix: What Fixes a Quaternion Subalgebra?......Page 90
7.1 Inverse Loops......Page 93
7.2 Isotopies......Page 94
7.3 Monotopies and Their Companions......Page 96
7.4 Different Forms of the Moufang Laws......Page 98
8.1 Isotopies and $SO_8$......Page 99
8.2 Orthogonal Isotopies and the Spin Group......Page 101
8.4 Seven Rights Can Make a Left......Page 102
8.5 Other Multiplication Theorems......Page 104
8.6 Three 7-Dimensional Groups in an 8-Dimensional One......Page 105
8.7 On Companions......Page 107
9.1 Defining Integrality......Page 109
9.2 Toward the Octavian Integers......Page 110
9.3 The $E_8$ Lattice of Korkine, Zolotarev, and Gosset......Page 115
9.4 Division with Remainder, and Ideals......Page 119
9.5 Factorization in $O^8$......Page 121
9.6 The Number of Prime Factorizations......Page 124
9.7 \\\"Meta-Problems\\\" for Octavian Factorization......Page 126
10.1 The 240 Octavian Units......Page 129
10.2 Two Kinds of Orthogonality......Page 130
10.3 The Automorphism Group of $O$......Page 131
10.4 The Octavian Unit Rings......Page 135
10.5 Stabilizing the Unit Subrings......Page 138
Appendix: Proof of Theorem 5......Page 141
11.1 Why Read Mod 2?......Page 143
11.2 The $E_8$ Lattice, Mod 2......Page 145
11.3 What Fixes $\\\\langle\\\\lambda\\\\rangle$?......Page 148
11.4 The Remaining Subrings Modulo 2......Page 150
12.1 The Exceptional Lie Groups and FreudenthaFs \\\"Magic Square\\......Page 153
12.2 The Octonion Projective Plane......Page 154
12.3 Coordinates for $\\\\mathbb{O}P^2$......Page 155
Bibliography......Page 159
Index......Page 163