Inmathematics, acomposition algebraA over afieldK is anot necessarily associativealgebra overK together with anondegeneratequadratic formN that satisfies

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

for allx andy inA.

A composition algebra includes aninvolution called aconjugation:${\displaystyle x\mapsto x^{\ast }.}$ The quadratic form${\displaystyle N(x)=x{x}^{\ast }}$ is called thenorm of the algebra.

A composition algebra (A, ∗,N) is either adivision algebra or asplit algebra, depending on the existence of a non-zerov inA such thatN(v) = 0, called anull vector.^[1] Whenx isnot a null vector, themultiplicative inverse ofx is$\frac{x^{\ast }}{N(x)}$. When there is a non-zero null vector,N is anisotropic quadratic form, and "the algebra splits".

Everyunital composition algebra over a fieldK can be obtained by repeated application of theCayley–Dickson construction starting fromK (if thecharacteristic ofK is different from2) or a 2-dimensional composition subalgebra (ifchar(K) = 2).  The possible dimensions of a composition algebra are1,2,4, and8.^[2][3][4]

• 1-dimensional composition algebras only exist whenchar(K) ≠ 2.
• Composition algebras of dimension 1 and 2 are commutative and associative.
• Composition algebras of dimension 2 are eitherquadratic field extensions ofK or isomorphic toK ⊕K.
• Composition algebras of dimension 4 are calledquaternion algebras.  They are associative but not commutative.
• Composition algebras of dimension 8 are calledoctonion algebras.  They are neither associative nor commutative.

For consistent terminology, algebras of dimension 1 have been calledunarion, and those of dimension 2binarion.^[5]

Every composition algebra is analternative algebra.^[3]

Using the doubled form ( _ : _ ):A ×A →K by${\displaystyle (a:b)=n(a+b)-n(a)-n(b),}$  then the trace ofa is given by (a:1) and the conjugate bya* = (a:1)e –a where e is the basis element for 1. A series of exercises proves that a composition algebra is always an alternative algebra.^[6]

When the fieldK is taken to becomplex numbersC and the quadratic formz², then four composition algebras overC areC itself, thebicomplex numbers, thebiquaternions (isomorphic to the2×2 complexmatrix ringM(2, C)), and thebioctonionsC ⊗O, which are also called complex octonions.

The matrix ringM(2, C) has long been an object of interest, first asbiquaternions byHamilton (1853), later in the isomorphic matrix form, and especially asPauli algebra.

Thesquaring functionN(x) =x² on thereal number field forms the primordial composition algebra.When the fieldK is taken to be real numbersR, then there are just six other real composition algebras.^[3]: 166 In two, four, and eight dimensions there are both adivision algebra and asplit algebra:

binarions: complex numbers with quadratic formx² +y² andsplit-complex numbers with quadratic formx² −y²,

quaternions andsplit-quaternions,

octonions andsplit-octonions.

Every composition algebra has an associatedbilinear form B(x,y) constructed with the norm N and apolarization identity:

${\displaystyle B(x,y)=[N(x+y)-N(x)-N(y)]/2.}$ ^[7]

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 theBrahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied.Leonhard Euler discussed thefour-square identity in 1748, and it ledW. R. Hamilton to construct his four-dimensional algebra ofquaternions.^[5]: 62 In 1848tessarines were described giving first light to bicomplex numbers.

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

Historically, the first non-associative algebra, theCayley 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 1919Leonard Dickson advanced the study of theHurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtainCayley numbers. He introduced a newimaginary unite, and for quaternionsq andQ writes a Cayley numberq +Qe. Denoting the quaternion conjugate byq′, the product of two Cayley numbers is^[8]

${\displaystyle (q+Qe)(r+Re)=(qr-R^{\prime }Q)+(Rq+Q{r}^{\prime })e.}$ 

The conjugate of a Cayley number isq' –Qe, and the quadratic form isqq′ +QQ′, obtained by multiplying the number by its conjugate. The doubling method has come to be called theCayley–Dickson construction.

In 1923 the case of real algebras withpositive definite forms was delimited by theHurwitz's theorem (composition algebras).

In 1931Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generatesplit-octonions.^[9]Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to anyfield with thesquaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.^[10]Nathan Jacobson described theautomorphisms of composition algebras in 1958.^[2]

The classical composition algebras overR andC areunital algebras. Composition algebraswithout amultiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.^{[11]: 463–81}

• Freudenthal magic square
• Pfister form
• Triality

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• Lam, Tsit-Yuen (2005).Introduction to Quadratic Forms over Fields.Graduate Studies in Mathematics. Vol. 67. American Mathematical Society.ISBN 0-8218-1095-2.Zbl 1068.11023.
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