Book:John H. Conway/On Quaternions And Octonions

Subject Matter

 * Quaternions
 * Octonions

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