Composition algebra

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In mathematics, a composition algebra A over a field K is a unital (but not necessarily associative) algebra over K together with a nondegenerate quadratic form N which satisfies

$N(xy) = N(x)N(y)\,$

for all x and y in A. The quadratic form N is often referred to as a (square) norm on A, although it is not a norm in the usual sense. Composition algebras are also called normed algebras (not to be confused with normed algebras in the sense of functional analysis).

[edit] Structure theorem

Every composition algebra over a field K can be obtained by repeated application of the Cayley-Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.

1-dimensional composition algebras only exist when char(K) ≠ 2.
Composition algebras of dimension 1 and 2 are commutative and associative.
Composition algebras of dimension 2 are either quadratic field extensions of K or isomorphic to $K \oplus K$ .
Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative.
Composition algebras of dimension 8 are called octonion algebras. They are neither associative or commutative.

Although the Cayley-Dickson contruction may be applied further, in higher dimensions the algebras will have zero divisors, and hence will not be composition algebras.

[edit] See also

[edit] References

Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.
Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. ISBN 3-540-66337-1.