# Composition algebra

In mathematics, a composition algebra Template:Mvar over a field Template:Mvar is a not necessarily associative algebra over Template:Mvar together with a nondegenerate quadratic form Template:Mvar which satisfies

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

for all Template:Mvar and Template:Mvar in Template:MvarUnital composition algebras are called Hurwitz algebras.[1] If the ground field K is the field of real numbers and Template:Mvar is positive-definite, then Template:Mvar is called an Euclidean Hurwitz algebra.

The quadratic form Template:Mvar is often referred to as a norm on Template:Mvar.  Composition algebras are also called normed algebras: these should not be confused with associative normed algebras, which include Banach algebras, although three associative Euclidean Hurwitz algebras R, C, and H in fact are Banach algebras.

## Structure theorem

Every unital composition algebra over a field Template:Mvar can be obtained by repeated application of the Cayley–Dickson construction starting from Template:Mvar (if the characteristic of Template:Mvar 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.[2]

• 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 Template:Mvar or isomorphic to KK.
• 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 nor commutative.

## Instances and usage

When the field Template:Mvar is taken to be complex numbers C, then the four composition algebras over C are C itself, the direct sum CC known first as tessarines (1848), the Template:Gaps complex matrix ring M(2, C), and the complex octonions CO.

Matrix ring M(2, C) has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra. Complex octonions have been used in a model of angular momentum.[9]

The squaring function N(x) = x2 on the real number field forms the primordial composition algebra. When the field Template:Mvar is taken to be real numbers R, then there are just six other real composition algebras.[10] In two, four, and eight dimensions there are both a "split algebra" and a "division algebra": complex numbers and split-complex numbers, quaternions and split-quaternions, octonions and split-octonions.

## References

1. Okubo (1995) p. 22
2. Jacobson (1958); Roos (2008); Schafer (1995) p. 73
3. Knus et al (1998) p.464
4. The term "para-quaternions" is sometimes applied to unrelated algebras.
5. Okubo (1995) pp. 40–41
6. Okubo (1995) p. 48
7. Okubo (1995) p. 49
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9. J. Koeplinger & V. Dzhunushaliev (2008) "Nonassociative decomposition of angular momentum operator using complex octonions", presentation at a meeting of the American Physical Society
10. Guy Roos (2008) Theorem 1.10 page 166
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• Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4459-5 .
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