Composition algebra


In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
for all and in.
A composition algebra includes an involution called a conjugation: The quadratic form is called the norm of the algebra.
A composition algebra is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N = 0, called a null vector. 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 can be obtained by repeated application of the Cayley–Dickson construction starting from or a 2-dimensional composition subalgebra. The possible dimensions of a composition algebra are,,, and.
For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.

Instances and usage

When the field is taken to be complex numbers and the quadratic form, then four composition algebras over are, the bicomplex numbers, the biquaternions, and the bioctonions, which are also called complex octonions.
Matrix ring has long been an object of interest, first as biquaternions by
Hamilton, later in the isomorphic matrix form, and especially as Pauli algebra.
The squaring function on the real number field forms the primordial composition algebra.
When the field is taken to be real numbers, then there are just six other real composition algebras.
In two, four, and eight dimensions there are both a division algebra and a "split algebra":
Every composition algebra has an associated bilinear form B constructed with the norm N and a polarization identity:

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. 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:
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, and for quaternions and writes a Cayley number. Denoting the quaternion conjugate by, the product of two Cayley numbers is
The conjugate of a Cayley number is, and the quadratic form is, 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.
In 1931 Max Zorn introduced a gamma into the multiplication rule in the Dickson construction to generate split-octonions. 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. Nathan Jacobson described the automorphisms of composition algebras in 1958.
The classical composition algebras over and are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson and Susumu Okubo and others.