Composition algebra

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

${\displaystyle N(xy)=N(x)N(y)\,}$

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).

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 ${\displaystyle K\oplus K}$ .
• 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 construction may be applied further, in higher dimensions the algebras will have zero divisors, and hence will not be composition algebras.

See also

• Normed division algebra
• Hurwitz's theorem

References

• Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.
• Springer, T. A. (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. ISBN 3-540-66337-1. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)