Book:John H. Conway/On Quaternions And Octonions

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John H. Conway and Derek A. Smith: On Quaternions And Octonions

Published $\text {2003}$, A K Peters, Ltd.

ISBN 1-56881-134-9


Subject Matter


Contents

preface
I The Complex Numbers
1 Introduction
1.1 The Algebra $\R$ of Real Numbers
1.2 Higher Dimensions
1.3 The Orthogonal Groups
1.4 The History of Quaternions and Octonions
2 Complex Numbers and 2-Dimensional Geometry
2.1 Rotations and Reflections
2.2 Finite Subgroups of $GO_2$ and $SO_2$
2.3 The Gaussian Integers
2.4 The Kleinian Integers
2.5 The 2-Dimensional Space Groups
II The Quaternions
3 Quaternions and 3-Dimensional Groups
3.1 The Quaternions and 3-Dimensional Rotations
3.2 Some Spherical Geometry
3.3 The Enumeration of Rotation Groups
3.4 Discussion of the Groups
3.5 The Finite Groups of Quaternions
3.6 Chiral and Achiral, Diploid and Haploid
3.7 The Projective or Elliptic Groups
3.8 The Projective Groups Tell Us All
3.9 Geometric Description of the Groups
Appendix: $v \to \overline v q v$ Is a Simple Rotation
4 Quaternions and 4-dimensional Groups
4.1 Introduction
4.2 Two 2-to-1 Maps
4.3 Naming the Groups
4.4 Coxeter's Notations for the Polyhedral Groups
4.5 Previous Enumerations
4.6 A Note on Chirality
Appendix: Completeness of the Tables
5 The Hurwitz Integral Quaternions
5.1 The Hurwitz Integral Quaternions
5.2 Primes and Units
5.3 Quaternionic Factorization of Ordinary Primes
5.4 The Metacommutation Problem
5.5 Factoring the Lipschitz Integers
III The Octonions
6 The Composition Algebras
6.1 The Multiplication Laws
6.2 The Conjugation Laws
6.3 The Doubling Laws
6.4 Completing Hurwitz's Theorem
6.5 Other Properties of the Algebras
6.6 The Maps $L_x$, $R_x$, and $B_x$
6.7 Coordinates for the Quaternions and Octonions
6.8 Symmetries of the Octonions: Diassociativity
6.9 The Algebras over Other Fields
6.10 The 1-, 2-, 4-, and 8-square Identities
6.11 Higher Square Identities: Pfister Theory
Appendix: What Fixes a Quaternion Subalgebra?
7 Moufang Loops
7.1 Inverse Loops
7.2 Isotopies
7.3 Monotopies and Their Companions
7.4 Different Forms of the Moufang Laws
8 Octonions and 8-dimensional Geometry
8.1 Isotopies and $SO_8$
8.2 Orthogonal Isotopies and the Spin Group
8.3 Triality
8.4 Seven Rights Can Make a Left
8.5 Other Multiplication Theorems
8.6 Three 7-Dimensional Groups in an 8-Dimensional One
8.7 On Companions
9 The Octavian Integers O
9.1 Defining Internality
9.2 Toward the Octavian Integers
9.3 The $E_8$ Lattice of Korkine, Zolotarev, and Gosset
9.4 Division with Remainder, and Ideals
9.5 factorisation in $O^8$
9.6 The Number of Prime Factorisations
9.7 "Meta-problems" for Octavian Factorisation
10 Automorphisms and Subrings of O
10.1 The 240 Octavian Units
10.2 Two Kinds of Orthogonality
10.3 The Automorphism Group of O
10.4 The Octavian Unit Rings
10.5 Stabilizing the Unit Subrings
Appendix: Proof of Theorem 5
1 1 Reading O Mod 2
11.1 Why Read Mod 2?
11.2 The $E_8$ Lattice, Mod 2
11.3 What Fixes $\langle \lambda \rangle$?
11.4 The Remaining Subrings Modulo 2
12 The Octonion Projective Plane $\mathbb O P^2$
12.1 The Exceptional Lie Groups and Freudenthal's "Magic Square"
12.2 The Octonion Projective Plane
12.3 Coordinates for $\mathbb O P^2$
Bibliography
Index


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