This book is on associative composition algebras, structures that have been used in kinematics and mathematical physics.
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This text expands the repertoire of algebra beyond real numbersR and complex numbersC to just five more algebras. The prospective reader will be well-acquainted with the utility ofR andC in science, and might like to know (more) about quaternionsH 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 andn ×n matrices, for this textn = 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 seeAbstract Algebra/Hypercomplex numbers, w:Composition algebra#History andw:History of quaternions.