Associative Composition Algebra

A depiction of the experience of discovery:
: Flammarion engraving

This book is on associative composition algebras, structures that have been used in kinematics and mathematical physics.

Table of Contents

1. Introduction

Definitions and list of algebras

2. Transcendental paradigm

Transcendental functions, Infinite series

Sample, Alternativity

3. Binarions

Möbius transformations

4. Bibinarions

5. Split-binarions

Rapidity and Simultaneity

6. Quaternions

Screw displacement

Biquaternions, representation of Lorentz transformations of spacetime

7. Split-quaternions

9. Homographies

Representation of the Poincaré and Conformal transformation groups

Readers

This text expands the repertoire of algebra beyond real numbers R and complex numbers C to just five more algebras. The prospective reader will be well-acquainted with the utility of R and C in science, and might like to know (more) about quaternions H and related algebras, and what have been the historical invocations of these algebras. Some group theory and matrix multiplication are prerequisites from linear and abstract algebra. Attention to this text will show some concrete instances of mathematical objects, thus nailing down the abstruse nature of abstract algebra. Whereas linear algebra characteristically is concerned with n-dimensional space and n × n matrices, for this text n = 2 is the limit.

Some of the content of this text was summarized in 1914 by Leonard Dickson when he noted that the complex quaternion and complex matrix algebras are equivalent, but their real subalgebras are not ! The category of composition algebras (which includes octonions and bioctonions) is addressed through exercises in chapter "Transcendental paradigm".

For the history of these algebras see Abstract Algebra/Hypercomplex numbers,  w:Composition algebra#History and w:History of quaternions.