nLab Cayley-Dickson construction

CayleyDickson construction

Context

Algebra

higher algebra

universal algebra

Cayley–Dickson construction

Idea

Consider how the complex numbers are formed from the real numbers. If generalized carefully, this kind of operation may be performed again to yield the quaternions, then the octonions (hence the four real normed division algebra), then the sedenions, and so on.

This is a special case of a construction which takes a real star-algebra $A$ to a new star-algebra whose elements are pairs of elements of $A$. This operation is the Cayley–Dickson construction, named after Arthur Cayley and Leonard Dickson.

Definitions

Let $A$ be an possibly nonassociative star-algebra over the field $\mathbb{R}$ of real numbers: an algebra equipped with an involution $\overline(-) \colon x \mapsto \overline{x}$ 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.)

Definition

(Cayley-Dickson construction, first version)

The Cayley–Dickson double $A^2$ of $A$ is the real algebra whose underlying $\mathbb{R}$-vector space is is the direct sum $A \oplus A$, and whose multiplication is given by

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

and the formula

$\widebar{(a,b)} \coloneqq (\overline{a},-b)$

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

The following description is different but equivalent:

Definition

(Cayley-Dickson double by generators and relations)

The Cayley-Dickson double $A^2$ of $A$ is the real algebra obtained by adjoining one generator $\ell$ to $A$ subject to the following relations:

$\ell^2 = -1$

and

$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} a(\ell b) = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell^{-1}) = \overline{a b}$

for all $a, b \in A$.

Properties

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 $a + \mathrm{i} b = (a,\overline{b})$). For this reason, we may write $A[\mathrm{i}]$ in place of $A^2$, at least when $A$ is unital.

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 $\overline{\mathrm{i}} = -\mathrm{i}$, we see that the involution on $A^2$ is trivial iff the involution on $A$ is trivial and $A$ further has $2 = 0$. Since $\mathrm{i} a = \overline{a} \mathrm{i}$, $A^2$ 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, flexible, and unital, and have all inverse elements; the subalgebra with $\overline{x} = x$ is always just $\mathbb{R}$.

Named after Arthur Cayley and Leonard Dickson.

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