# 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

for all Template:Mvar and Template:Mvar in Template:Mvar. Unital 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
*K*⊕*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 nor commutative.

## The case char(*K*) ≠ 2

### Scalar product

If Template:Mvar has characteristic not equal to 2, then a bilinear form (*a*, *b*) = {{ safesubst:#invoke:Unsubst||$B=1/2}}[*N*(*a* + *b*) − *N*(*a*) − *N*(*b*)] is associated with the quadratic form Template:Mvar.

### Involution in Hurwitz algebras

Assuming Template:Mvar has a multiplicative unity, define involution and right and left multiplication operators by

Evidently Template:Overline is an involution and preserves the quadratic form. The overline notation stresses the fact that complex and quaternion conjugation are partial cases of it. These operators have the following properties:

- The involution is an antiautomorphism, i.e. Template:Overline =
*Template:Overline Template:Overline* *a Template:Overline*=*N*(*a*) 1 =*Template:Overline a**L*(Template:Overline) =*L*(*a*)*,*R*(Template:Overline) =*R*(*a*)*, where * denotes the adjoint operator with respect to the form ( , )- Re(
*a b*) = Re(*b a*) where Re*x*= (*x*+ Template:Overline)/2 = (*x*, 1) - Re((
*a b*)*c*) = Re(*a*(*b c*)) *L*(*a*^{2}) =*L*(*a*)^{2},*R*(*a*^{2}) =*R*(*a*)^{2}, so that Template:Mvar is an alternating algebra

These properties are proved starting from polarized version of the identity (*a b*, *a b*) = (*a*, *a*)(*b*, *b*):

Setting *b* = 1 or *d* = 1 yields *L*(Template:Overline) = *L*(*a*)* and *R*(Template:Overline) = *R*(*c*)*. Hence Re(*a b*) = (*a b*, 1) = (*a*, Template:Overline) = (*b a*, 1) = Re(*b a*). Similarly (Template:Overline, *c*) = (*a b*, Template:Overline) = (*b*, *Template:Overline Template:Overline*) = (1, Template:Overline (*Template:Overline Template:Overline*)) = (1, (*Template:Overline Template:Overline*) Template:Overline) = (*Template:Overline Template:Overline*, *c*). Hence Re(*a b*)*c* = ((*a b*)*c*, 1) = (*a b*, Template:Overline) = (*a*, *Template:Overline Template:Overline*) = (*a*(*b c*), 1) = Re(*a*(*b c*)). By the polarized identity *N*(*a*) (*c*, *d*) = (*a c*, *a d*) = (*Template:Overline a c*, *d*) so *L*(Template:Overline) L(*a*) = *N*(*a*). Applied to 1 this gives *Template:Overline a* = *N*(*a*). Replacing Template:Mvar by Template:Overline gives the other identity. Substituting the formula for Template:Overline in *L*(Template:Overline) *L*(*a*) = *L*(*Template:Overline a*) gives *L*(*a*)^{2} = *L*(*a*^{2}).

### Para-Hurwitz algebra

Another operation ∗ may be defined in a Hurwitz algebra as

*x*∗*y*= Template:Overline Template:Overline

The algebra (*A*, ∗) is a composition algebra not generally unital, known as a **para-Hurwitz algebra**.^{[3]} In dimensions 4 and 8 these are **para-quaternion**^{[4]} and **para-octonion** algebras.^{[5]}

A para-Hurwitz algebra satisfies^{[6]}

Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.^{[7]} Similarly, a flexible algebra satisfying

is either a Hurwitz algebra, a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.^{[7]}

### Euclidean Hurwitz algebras

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If the underlying coefficient field of a Hurwitz algebra is the reals and Template:Mvar is positive-definite, so that (*a*, *b*) = {{ safesubst:#invoke:Unsubst||$B=1/2}}[*q*(*a* + *b*) − *q*(*a*) − *q*(*b*)] is an inner product, then Template:Mvar is called a **Euclidean Hurwitz algebra**. The Euclidean Hurwitz algebras are precisely the real numbers, the complex numbers, the quaternions and the octonions.^{[8]}

## 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 **C** ⊕ **C** 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*) = *x*^{2} 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.

## See also

- 2 × 2 real matrices, a non-Euclidean Hurwitz algebra with
*N*(*a*) = det*a* - Hurwitz problem
- Multiplicative quadratic form
- Petersson algebra

## References

- ↑ Okubo (1995) p. 22
- ↑ Jacobson (1958); Roos (2008); Schafer (1995) p. 73
- ↑ Knus et al (1998) p.464
- ↑ The term "para-quaternions" is sometimes applied to unrelated algebras.
- ↑ Okubo (1995) pp. 40–41
- ↑ Okubo (1995) p. 48
- ↑
^{7.0}^{7.1}Okubo (1995) p. 49 - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ J. Koeplinger & V. Dzhunushaliev (2008) "Nonassociative decomposition of angular momentum operator using complex octonions", presentation at a meeting of the American Physical Society
- ↑ Guy Roos (2008) Theorem 1.10 page 166

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- Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in
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