Chapter

Composition algebras

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0009
Composition algebras

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Composition algebras and division algebras are defined using the example of complex numbers. Hurwitz’s theorem is stated: there are only four normed composition algebras with unit element. They are the reals, the complexes, the quaternions and the octonions. The theorem is proved using Clifford numbers. Explicit representation is given in terms of real matrices, which are 1x1, 2x2, 4x4 and 8x8 for the reals R, the complexes C, the quaternions H and the octonions O, respectively. It is noted that while multiplication is commutative and associative in R and C, in H multiplication is associative but not commutative, while in O it is neither associative nor commutative. Biographical notes on Hurwitz, Hamilton, Graves, Cayley and Frobenius are given.

Keywords: composition algebras; quaternions; octonions; non-associativity

Chapter.  3593 words. 

Subjects: Mathematical and Statistical Physics

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