Cayley-Dickson Construction

The idea of adding additional roots of unity to an algebra can be generalized, and applied recursively. The Cayley-Dickson construction is a the most common generalization, and is designed to preserve the following properties:

  • Addition is a commutative group
  • Multiplication is left and right distributive
  • Conjugation is preserved.
    • x^*{}^*=x
    • (x+y)^*=x^*+y^*
    • (xy)^*=y^*x^*

The construction is defined by its addition, multiplication and conjugation rules, where (a,b) and (c,d) are pairs of the previous level in the algebra, and \gamma is \pm1, the square of the new root.

(a,b)+(c,d)=(a+c,b+d)
(a,b)(c,d)=(ac+\gamma db^*, cb+a^*d)
(a,b)^*=(a^*,-b)
N(x)=xx^*

It should be noted that \gamma is usually presented in the opposite case, so that the complex numbers have a value of 1. Further, multiplication is often defined using a different sign convention in the direction of the new root.

(a,b)(c,d)=(ac+\gamma d^*b, da+bc^*)

The values used here are more convenient for split roots, which is the focus of this study.

Either way, we have the following results, staring with some field.

On the first step we lose the property that numbers are their own conjugate.
On the second step we lose the multiplicative commutative property.
On the third step we lose the multiplicative associative property.
On the third step we lose the property that N(xy)=N(x)N(y).

These properties can not be preserved, without damage to the properties outlined at the top of this page.

This gives us the following normed algebras over the reals, with the gamma’s that produce them.

Real: N/A
Complex: -1
Split-Complex: 1
Quaternions: -1,-1
Split-Quaternions: 1,1; 1,-1; -1,1
Octonions: -1,-1,-1
Split-Octonions: 1,1,1; 1,1,-1; 1,-1,1; 1,-1,-1; -1,1,1; -1,1,-1; -1,-1,1

Essentially, if we only add circular roots, we get a circular algebra, but as soon as we add a single hyperbolic root we get a unique hyperbolic algebra. Up to sign convention.

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