Cayley-Dickson algebra

An alternative -dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. Quaternion and Alternative rings and algebras). The latter starts out from a given algebra to construct a new algebra (of twice the dimension of ) and is a generalization of the doubling process (see Hypercomplex number). Namely, let be an algebra with a unit 1 over a field , let be some non-zero element of , and let be an -linear mapping which is an involution, and such that

The formula

now defines a multiplication operation on the direct sum of linear spaces , relative to which is an algebra. The algebra may be imbedded in as a subalgebra: , and the involution extends to an involution in :

Moreover,

The extension of to can be repeated resulting in an ascending chain of algebras ; the parameter need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra with basis , multiplication table

and involution , , the first application of the process yields an algebra of generalized quaternions (an associative algebra of dimension 4), and the second — an -dimensional algebra, known as a Cayley–Dickson algebra.

Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over ; conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:

This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form (the norm of ) does not represent the zero in . If is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis with the following multiplication table:'

<tbody> </tbody>

where , , and the involution is defined by the conditions

This algebra is denoted by . The algebras and are isomorphic if and only if their quadratic forms are equivalent. If represents zero, the corresponding Cayley–Dickson algebra is isomorphic to , which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices

where , , with a three-dimensional space over with the usual definition of the scalar product and vector product . Matrix multiplication is defined by

If is the real field, then is the algebra of Cayley numbers (a division algebra). Any Cayley–Dickson algebra over is isomorphic to either or .

The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [1], [2]).

Let be an alternative ring whose associative-commutative centre is distinct from zero and does not contain zero divisors; let be the field of fractions of . Then there is a natural imbedding . If is a Cayley–Dickson algebra over , then is known as a Cayley–Dickson ring.

References

[1]	L.E. Dickson, "Linear algebras" , Cambridge Univ. Press (1930)
[2]	R.D. Schafer, "An introduction to nonassociative algebras" , Acad. Press (1966)
[3]	K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian)