On Quaternions and Octonions

This book investigates the geometry of the quaternion and octonion algebras. Following a comprehensive historical introduction, the special properties of 3- and 4- dimensional Euclidean spaces are illuminated using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The arithmetics of the quaternions and octonions are also described, and the book concludes with a new theory of octonion factorization. Topics covered include: history; the geometry of complex numbers; quaternions and 3- dimensional groups; quaternions and 4-dimensional groups; the Hurwitz integral quaternions; the composition algebras, Moufang loops; octonions and 8-dimensional geometry; integral octonions; and the octonion projective plane

Preface	p. xi
The Complex Numbers	p. 1
Introduction	p. 3
The Algebra R of Real Numbers	p. 3
Higher Dimensions	p. 5
The Orthogonal Groups	p. 6
The History of Quaternions and Octonions	p. 6
Complex Numbers and 2-Dimensional Geometry	p. 11
Rotations and Reflections	p. 11
Finite Subgroups of GO[subscript 2] and SO[subscript 2]	p. 14
The Gaussian Integers	p. 15
The Kleinian Integers	p. 18
The 2-Dimensional Space Groups	p. 18
The Quaternions	p. 21
Quaternions and 3-Dimensional Groups	p. 23
The Quaternions and 3-Dimensional Rotations	p. 23
Some Spherical Geometry	p. 26
The Enumeration of Rotation Groups	p. 29
Discussion of the Groups	p. 30
The Finite Groups of Quaternions	p. 33
Chiral and Achiral, Diploid and Haploid	p. 33
The Projective or Elliptic Groups	p. 34
The Projective Groups Tell Us All	p. 35
Geometric Description of the Groups	p. 36
v to vqv Is a Simple Rotation	p. 40
Quaternions and 4-Dimensional Groups	p. 41
Introduction	p. 41
Two 2-to-1 Maps	p. 42
Naming the Groups	p. 43
Coxeter's Notations for the Polyhedral Groups	p. 45
Previous Enumerations	p. 48
A Note on Chirality	p. 49
Completeness of the Tables	p. 50
The Hurwitz Integral Quaternions	p. 55
The Hurwitz Integral Quaternions	p. 55
Primes and Units	p. 56
Quaternionic Factorization of Ordinary Primes	p. 58
The Metacommutation Problem	p. 61
Factoring the Lipschitz Integers	p. 61
The Octonions	p. 65
The Composition Algebras	p. 67
The Multiplication Laws	p. 68
The Conjugation Laws	p. 68
The Doubling Laws	p. 69
Completing Hurwitz's Theorem	p. 70
Other Properties of the Algebras	p. 72
The Maps L[subscript x], R[subscript x], and B[subscript x]	p. 73
Coordinates for the Quaternions and Octonions	p. 75
Symmetries of the Octonions: Diassociativity	p. 76
The Algebras over Other Fields	p. 76
The 1-, 2-, 4-, and 8-Square Identities	p. 77
Higher Square Identities: Pfister Theory	p. 78
What Fixes a Quaternion Subalgebra?	p. 80
Moufang Loops	p. 83
Inverse Loops	p. 83
Isotopies	p. 84
Monotopies and Their Companions	p. 86
Different Forms of the Moufang Laws	p. 88
Octonions and 8-Dimensional Geometry	p. 89
Isotopies and SO[subscript 8]	p. 89
Orthogonal Isotopies and the Spin Group	p. 91
Triality	p. 92
Seven Rights Can Make a Left	p. 92
Other Multiplication Theorems	p. 94
Three 7-Dimensional Groups in an 8-Dimensional One	p. 95
On Companions	p. 97
The Octavian Integers O	p. 99
Defining Integrality	p. 99
Toward the Octavian Integers	p. 100
The E[superscript 8] Lattice of Korkine, Zolotarev, and Gosset	p. 105
Division with Remainder, and Ideals	p. 109
Factorization in O[superscript 8]	p. 111
The Number of Prime Factorizations	p. 114
"Meta-Problems" for Octavian Factorization	p. 116
Automorphisms and Subrings of O	p. 119
The 240 Octavian Units	p. 119
Two Kinds of Orthogonality	p. 120
The Automorphism Group of O	p. 121
The Octavian Unit Rings	p. 125
Stabilizing the Unit Subrings	p. 128
Proof of Theorem 5	p. 131
Reading O Mod 2	p. 133
Why Read Mod 2?	p. 133
The E[subscript 8] Lattice, Mod 2	p. 135
What Fixes [left angle bracket] lambda [right angle bracket]?	p. 138
The Remaining Subrings Modulo 2	p. 140
The Octonion Projective Plane OP[superscript 2]	p. 143
The Exceptional Lie Groups and Freudenthal's "Magic Square"	p. 143
The Octonion Projective Plane	p. 144
Coordinates for OP[superscript 2]	p. 145
Bibliography	p. 149
Index	p. 153
Table of Contents provided by Syndetics. All Rights Reserved.

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On Quaternions and Octonions

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