# Cayley–Dickson construction

## Idea

Consider how the complex numbers are formed from the real numbers. If you generalise this carefully, then you can perform this operation again to get the quaternions, and so on. This operation is the Cayley–Dickson construction.

## Definitions

Let $A$ be an possibly nonassociative $*$-algebra over the field $\mathbb{R}$ of real numbers: an algebra equipped with an involution $x\mapsto \bar{x}$ which is an antiautomorphism. (Actually, $\mathbb{R}$ could be replaced by any commutative ring in the definitions, although some properties may depend on this ring.)

Then one defines a new algebra, the Cayley–Dickson double $A^2$ of $A$ which is the direct sum $A \oplus A$ as an $\mathbb{R}$-vector space, and with the multiplication rule given by

$(a,b)\cdot(u,v) \coloneqq (a u - \bar{v} b, b \bar{u} + v a),$

and the formula

$\widebar{(a,b)} := (\bar{a},-b)$

defines an involutive antiautomorphism on $A^2$, so the doubling procedure can be iterated.

The map $a\mapsto (a,0)$ is a monomorphism $A\to A^2$. If $A$ is unital with unit $1$ then $A^2$ is unital with unit $(1,0)$. In the unital case, the element $\mathrm{i} \coloneqq (0,1)$ has the property $\mathrm{i}^2 = -1 \coloneqq (-1,0)$, and we may write $(a,b)$ as $a + b \mathrm{i}$ (while $a + \mathrm{i} b = (a,\bar{b})$). For this reason, we may write $A[\mathrm{i}]$ in place of $A^2$, at least when $A$ is unital.

## Properties

Generally speaking, the double $A^2$ of an algebra $A$ has a nice property iff $A$ is one level nicer. For simplicity, assume that $A$ is unital (so that $\mathbb{R}$ is a subalgebra). Since $\bar{\mathrm{i}} = -\mathrm{i}$, we see that the involution on $A^2$ is trivial iff the involution on $A$ is trivial and $A$ further has $2 = 0$. Since $\mathrm{i} a = \bar{a} \mathrm{i}$, $A^2$ is commutative iff $A$ is commutative and the involution in $A$ is trivial. Since $a (b \mathrm{i}) = (b a) \mathrm{i}$, $A^2$ is associative iff $A$ is associative and commutative. Finally, $A^2$ is alternative iff $A$ is associative (and hence also alternative).

## Examples

The standard example is the sequence of consecutive doubles starting with $\mathbb{R}$ itself (with the identity map as involution); these are the Cayley–Dickson algebras: the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$, the octonions or Cayley numbers $\mathbb{O}$, the sedenions $\mathbb{S}$, etc. These are the normed division algebras ($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$), followed by further algebras which are not division algebras. All of these algebras are power-associative and unital and have all inverse elements; the subalgebra with $\bar{x} = x$ is always just $\mathbb{R}$.

## References

Revised on October 22, 2013 17:41:35 by Toby Bartels (98.19.41.40)