Composition algebra

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. 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". 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. 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. Every composition algebra is an alternative algebra. Using the doubled form ( _ : _ ): A × A → K by then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra. When the field K is taken to be complex numbers C and the quadratic form z2, 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.