Cover image for On quaternions and octonions : their geometry, arithmetic, and symmetry
Title:
On quaternions and octonions : their geometry, arithmetic, and symmetry
Author:
Conway, John H. (John Horton)
Publication Information:
Natick, Mass. : AK Peters, [2003]

©2003
Physical Description:
xii, 159 pages : illustrations ; 24 cm
Language:
English
ISBN:
9781568811345
Format :
Book

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Summary

Summary

This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces 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 authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.


Reviews 1

Choice Review

William Hamilton invented the quaternion algebra in 1843 and precipitated an intellectual fad for quaternions as a tool for mechanics that prevailed until matrix algebra pushed quaternions aside early in the 20th century. Remarkable structures, the quaternion algebra and also its eight-dimensional analog, octonian algebra, remain on the scene; each generation finds new virtues in them. Conway and his student Smith show how quaternions help classify the finite groups of isometries in three and four dimensions; they cleanly prove Hurwitz's famous classification of finite-dimensional composition algebras and develop higher-dimensional analogs of unique factorization in these settings. They also touch on Pfister theory (part of quadratic form theory). Readers will find here Conway's celebrated style: fastidious verbal economy married to ingenious graphics. ^BSumming Up: Recommended. General readers; lower-division undergraduates through professionals. D. V. Feldman University of New Hampshire


Table of Contents

Prefacep. xi
I The Complex Numbersp. 1
1 Introductionp. 3
1.1 The Algebra R of Real Numbersp. 3
1.2 Higher Dimensionsp. 5
1.3 The Orthogonal Groupsp. 6
1.4 The History of Quaternions and Octonionsp. 6
2 Complex Numbers and 2-Dimensional Geometryp. 11
2.1 Rotations and Reflectionsp. 11
2.2 Finite Subgroups of GO[subscript 2] and SO[subscript 2]p. 14
2.3 The Gaussian Integersp. 15
2.4 The Kleinian Integersp. 18
2.5 The 2-Dimensional Space Groupsp. 18
II The Quaternionsp. 21
3 Quaternions and 3-Dimensional Groupsp. 23
3.1 The Quaternions and 3-Dimensional Rotationsp. 23
3.2 Some Spherical Geometryp. 26
3.3 The Enumeration of Rotation Groupsp. 29
3.4 Discussion of the Groupsp. 30
3.5 The Finite Groups of Quaternionsp. 33
3.6 Chiral and Achiral, Diploid and Haploidp. 33
3.7 The Projective or Elliptic Groupsp. 34
3.8 The Projective Groups Tell Us Allp. 35
3.9 Geometric Description of the Groupsp. 36
Appendix v to vqv Is a Simple Rotationp. 40
4 Quaternions and 4-Dimensional Groupsp. 41
4.1 Introductionp. 41
4.2 Two 2-to-1 Mapsp. 42
4.3 Naming the Groupsp. 43
4.4 Coxeter's Notations for the Polyhedral Groupsp. 45
4.5 Previous Enumerationsp. 48
4.6 A Note on Chiralityp. 49
Appendix Completeness of the Tablesp. 50
5 The Hurwitz Integral Quaternionsp. 55
5.1 The Hurwitz Integral Quaternionsp. 55
5.2 Primes and Unitsp. 56
5.3 Quaternionic Factorization of Ordinary Primesp. 58
5.4 The Metacommutation Problemp. 61
5.5 Factoring the Lipschitz Integersp. 61
III The Octonionsp. 65
6 The Composition Algebrasp. 67
6.1 The Multiplication Lawsp. 68
6.2 The Conjugation Lawsp. 68
6.3 The Doubling Lawsp. 69
6.4 Completing Hurwitz's Theoremp. 70
6.5 Other Properties of the Algebrasp. 72
6.6 The Maps L[subscript x], R[subscript x], and B[subscript x]p. 73
6.7 Coordinates for the Quaternions and Octonionsp. 75
6.8 Symmetries of the Octonions: Diassociativityp. 76
6.9 The Algebras over Other Fieldsp. 76
6.10 The 1-, 2-, 4-, and 8-Square Identitiesp. 77
6.11 Higher Square Identities: Pfister Theoryp. 78
Appendix What Fixes a Quaternion Subalgebra?p. 80
7 Moufang Loopsp. 83
7.1 Inverse Loopsp. 83
7.2 Isotopiesp. 84
7.3 Monotopies and Their Companionsp. 86
7.4 Different Forms of the Moufang Lawsp. 88
8 Octonions and 8-Dimensional Geometryp. 89
8.1 Isotopies and SO[subscript 8]p. 89
8.2 Orthogonal Isotopies and the Spin Groupp. 91
8.3 Trialityp. 92
8.4 Seven Rights Can Make a Leftp. 92
8.5 Other Multiplication Theoremsp. 94
8.6 Three 7-Dimensional Groups in an 8-Dimensional Onep. 95
8.7 On Companionsp. 97
9 The Octavian Integers Op. 99
9.1 Defining Integralityp. 99
9.2 Toward the Octavian Integersp. 100
9.3 The E[superscript 8] Lattice of Korkine, Zolotarev, and Gossetp. 105
9.4 Division with Remainder, and Idealsp. 109
9.5 Factorization in O[superscript 8]p. 111
9.6 The Number of Prime Factorizationsp. 114
9.7 "Meta-Problems" for Octavian Factorizationp. 116
10 Automorphisms and Subrings of Op. 119
10.1 The 240 Octavian Unitsp. 119
10.2 Two Kinds of Orthogonalityp. 120
10.3 The Automorphism Group of Op. 121
10.4 The Octavian Unit Ringsp. 125
10.5 Stabilizing the Unit Subringsp. 128
Appendix Proof of Theorem 5p. 131
11 Reading O Mod 2p. 133
11.1 Why Read Mod 2?p. 133
11.2 The E[subscript 8] Lattice, Mod 2p. 135
11.3 What Fixes [left angle bracket] lambda [right angle bracket]?p. 138
11.4 The Remaining Subrings Modulo 2p. 140
12 The Octonion Projective Plane OP[superscript 2]p. 143
12.1 The Exceptional Lie Groups and Freudenthal's "Magic Square"p. 143
12.2 The Octonion Projective Planep. 144
12.3 Coordinates for OP[superscript 2]p. 145
Bibliographyp. 149
Indexp. 153