PlanetMath: Cayley-Dickson construction

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Cayley-Dickson construction

(Definition)

In the foregoing discussion, an algebra shall mean a non-associative algebra.

Let be a normed -algebra, an algebra admitting an involution , over a commutative ring with $1\neq0$ . The Cayley-Dickson construction is a way of enlarging to a new algebra, , extending the as well as the norm operations in , such that is a subalgebra of .

Define to be the module (external) direct sum of with itself:

$\displaystyle KD(A):=A\oplus A.$

Therefore, addition in

is defined by addition componentwise in each copy of

. Next, let $\lambda$ be a unit in

and define three additional operations:

(Multiplication) $(a\oplus b)(c\oplus d):=(ac+\lambda d^*b)\oplus(da+bc^*)$ , where is the involution on ,
(Extended involution) $(a\oplus b)^*:=a^*\oplus(-b)$ , and
(Extended Norm) $N(a\oplus b):=(a\oplus b)(a\oplus b)^*$ .

One readily checks that the multiplication is bilinear, since the involution

(on

) is linear. Therefore,

is an algebra.

Furthermore, since the extended involution is clearly bijective and linear, and that

$\displaystyle {(a\oplus b)}^{**}=(a^*\oplus(-b))^*=a^{**}\oplus b=a\oplus b,$

this extended involution is well-defined and so

is in addition a

-algebra.

Finally, to see that is a normed -algebra, we identify as the first component of , then becomes a subalgebra of and elements of the form $a\oplus0$ can now be written simply as . Now, the extended norm

$\displaystyle N(a\oplus b)=(a\oplus b)(a^*\oplus(-b))=(aa^*-\lambda b^*b)\oplus0=N(a)-\lambda N(b)\in A,$

where

in the subsequent terms of the above equation array is the norm on

given by

. The fact that the $N\colon KD(A)\to A$ , together with the equality $N(0\oplus0)=0$ show that the extended norm

is well-defined. Thus,

is a normed

-algebra.

The normed -algebra , together with the invertible element $\lambda\in R$ , is called the Cayley-Dickson algebra, $KD(A,\lambda)$ , obtained from .

If has a unity 1, then so does $KD(A,\lambda)$ and its unity is $1\oplus0$ . Furthermore, write $i=0\oplus1$ , we check that, $ia=(0\oplus1)(a\oplus0)=0\oplus a^*=(a^*\oplus0)(0\oplus1)=a^*i$ . Therefore, and we can identify the second component of $KD(A,\lambda)$ with and write elements of as for $a\in A$ .

It is not hard to see that $A(Ai)=(Ai)A\subseteq Ai$ and $(Ai)(Ai)\subseteq A$ . We are now able to write

$\displaystyle KD(A,\lambda)=A\oplus Ai,$

where each element $x\in KD(A,\lambda)$ has a unique expression

Properties. Let will be general elements of $KD(A,\lambda)$ .

,
$x+x^*\in A$ ,
.

Examples. All examples considered below have ground ring the reals $\mathbb{R}$ .

$KD(\mathbb{R},-1)=\mathbb{C}$ , the complex numbers.
$KD(\mathbb{C},-1)=\mathbb{H}$ , the quaternions.
$KD(\mathbb{H},-1)=\mathbb{O}$ , the octonions.
$KD(\mathbb{O},-1)=\mathbb{S}$ , which are called the sedenions, an algebra of dimension 16 over $\mathbb{R}$ .

Remark. Starting from $\mathbb{R}$ , notice each stage of Cayley-Dickson construction produces a new algebra that loses some intrinsic properties of the previous one: $\mathbb{C}$ is no longer orderable (or formally real); commutativity is lost in $\mathbb{H}$ ; associativity is gone from $\mathbb{O}$ ; and finally, $\mathbb{S}$ is not even a division algebra anymore!