# Composition algebra

*https://en.wikipedia.org/wiki/Composition_algebra*

Algebraic structures |
---|

In
mathematics, a **composition algebra** A over a
field K is a
not necessarily associative
algebra over K together with a
nondegenerate
quadratic form N that satisfies

for all x and y in A.

A composition algebra includes an
involution called a **conjugation**: The quadratic form is called the **norm** of the algebra.

A composition algebra (*A*, ∗, *N*) is either a
division algebra or a **split algebra**, depending on the existence of a non-zero *v* in *A* such that *N*(*v*) = 0, called a
null vector.^{
[1]} When *x* is *not* a null vector, the
multiplicative inverse of *x* is . When there is a non-zero null vector, *N* is an
isotropic quadratic form, and "the algebra splits".

## Structure theorem

Every
unital 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.^{
[2]}^{
[3]}^{
[4]}

- 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*⊕*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.

For consistent terminology, algebras of dimension 1 have been called *unarion*, and those of dimension 2 *binarion*.^{
[5]}

## Instances and usage

When the field K is taken to be
complex numbers **C** and the quadratic form *z*^{2}, then four composition algebras over **C** are **C** itself, the
bicomplex numbers, the
biquaternions (isomorphic to the 2×2 complex
matrix ring M(2, **C**)), and the
bioctonions **C** ⊗ **O**, which are also called complex octonions.

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.

The
squaring function *N*(*x*) = *x*^{2} on the
real number field forms the primordial composition algebra.
When the field K is taken to be real numbers **R**, then there are just six other real composition algebras.^{
[3]}^{:166}
In two, four, and eight dimensions there are both a
division algebra and a "split algebra":

- binarions: complex numbers with quadratic form
*x*^{2}+*y*^{2}and split-complex numbers with quadratic form*x*^{2}−*y*^{2}, - quaternions and split-quaternions,
- octonions and split-octonions.

Every composition algebra has an associated
bilinear form B(*x,y*) constructed with the norm N and a
polarization identity:

^{ [6]}

## History

The composition of sums of squares was noted by several early authors.
Diophantus was aware of the identity involving the sum of two squares, now called the
Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied.
Leonhard Euler discussed the
four-square identity in 1748, and it led
W. R. Hamilton to construct his four-dimensional algebra of
quaternions.^{
[5]}^{:62} In 1848
tessarines were described giving first light to bicomplex numbers.

About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra:

- Historically, the first non-associative algebra, the
Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras...
^{ [5]}^{:61}

In 1919
Leonard Dickson advanced the study of the
Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain
Cayley numbers. He introduced a new
imaginary unit e, and for quaternions *q* and *Q* writes a Cayley number *q* + *Q*e. Denoting the quaternion conjugate by *q*′, the product of two Cayley numbers is^{
[7]}

The conjugate of a Cayley number is *q'* – *Q*e, and the quadratic form is *qq*′ + *QQ*′, obtained by multiplying the number by its conjugate. The doubling method has come to be called the
Cayley–Dickson construction.

In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras).

In 1931
Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate
split-octonions.^{
[8]}
Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to any
field with the
squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.^{
[9]}
Nathan Jacobson described the
automorphisms of composition algebras in 1958.^{
[2]}

The classical composition algebras over **R** and **C** are
unital algebras. Composition algebras *without* a
multiplicative identity were found by H.P. Petersson (
Petersson algebras) and Susumu Okubo (
Okubo algebras) and others.^{
[10]}^{:463–81}

## See also

## References

Wikibooks has a book on the topic of:
Associative Composition Algebra |

**^**Springer, T. A.; F. D. Veldkamp (2000).*Octonions, Jordan Algebras and Exceptional Groups*. Springer-Verlag. p. 18. ISBN 3-540-66337-1.- ^
^{a}^{b}Jacobson, Nathan (1958). "Composition algebras and their automorphisms".*Rendiconti del Circolo Matematico di Palermo*.**7**: 55–80. doi: 10.1007/bf02854388. Zbl 0083.02702. - ^
^{a}^{b}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 **^**Schafer, Richard D. (1995) [1966].*An introduction to non-associative algebras*. Dover Publications. pp. 72–75. ISBN 0-486-68813-5. Zbl 0145.25601.- ^
^{a}^{b}^{c}Kevin McCrimmon (2004)*A Taste of Jordan Algebras*, Universitext, Springer ISBN 0-387-95447-3 MR 2014924 **^**Arthur A. Sagle & Ralph E. Walde (1973)*Introduction to Lie Groups and Lie Algebras*, pages 194−200, Academic Press**^**Dickson, L. E. (1919), "On Quaternions and Their Generalization and the History of the Eight Square Theorem",*Annals of Mathematics*, Second Series, Annals of Mathematics,**20**(3): 155–171, doi: 10.2307/1967865, ISSN 0003-486X, JSTOR 1967865**^**Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402**^**Albert, Adrian (1942). "Quadratic forms permitting composition".*Annals of Mathematics*.**43**: 161–177. doi: 10.2307/1968887. Zbl 0060.04003.**^**Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in*The Book of Involutions*, pp. 451–511, Colloquium Publications v 44, American Mathematical Society ISBN 0-8218-0904-0

## Further reading

- Faraut, Jacques;
Korányi, Adam (1994).
*Analysis on symmetric cones*. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. pp. 81–86. ISBN 0-19-853477-9. MR 1446489. -
Lam, Tsit-Yuen (2005).
*Introduction to Quadratic Forms over Fields*. Graduate Studies in Mathematics.**67**. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023. - Harvey, F. Reese (1990).
*Spinors and Calibrations*. Perspectives in Mathematics.**9**. San Diego: Academic Press. ISBN 0-12-329650-1. Zbl 0694.53002.