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Associative Composition Algebra

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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[edit]

1. Introduction

Definitions and list of algebras

2. Transcendental paradigm

Transcendental functions, Infinite series, Sample

3. Binarions

Division binarions, Split binarions, Complex binarions

4. Quaternions

Division quaternions, representation of spatial rotation
Biquaternions, representation of Lorentz transformations of spacetime
Split quaternions

5. Homographies

Representation of the Poincare and Conformal transformation groups


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