Cayley-Dickson construction

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

Let $A$ be a normed $*$-algebra, an algebra admitting an involution (http://planetmath.org/Involution2) $*$, over a commutative ring $R$ with $1\ne 0$. The Cayley-Dickson construction is a way of enlarging $A$ to a new algebra, $KD(A)$, extending the $*$ as well as the norm operations in $A$, such that $A$ is a subalgebra of $KD(A)$.

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

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

Therefore, addition in $KD(A)$ is defined by addition componentwise in each copy of $A$. Next, let $\lambda$ be a unit in $R$ and define three additional operations:

1. (Multiplication) $(a\oplus b)(c\oplus d):=(ac+\lambda d^{*}b)\oplus (da+b{c}^{*})$, where $*$ is the involution on $A$,

2. (Extended involution) ${(a\oplus b)}^{*}:=a^{*}\oplus (-b)$, and

3. (Extended Norm) $N(a\oplus b):=(a\oplus b){(a\oplus b)}^{*}$.

One readily checks that the multiplication is bilinear, since the involution $*$ (on $A$) is linear. Therefore, $KD(A)$ is an algebra.

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

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

this extended involution is well-defined and so $KD(A)$ is in addition a $*$-algebra.

Finally, to see that $KD(A)$ is a normed $*$-algebra, we identify $A$ as the first component of $KD(A)$, then $A$ becomes a subalgebra of $KD(A)$ and elements of the form $a\oplus 0$ can now be written simply as $a$. Now, the extended norm

$$N(a\oplus b)=(a\oplus b)(a^{*}\oplus (-b))=(a{a}^{*}-\lambda b^{*}b)\oplus 0=N(a)-\lambda N(b)\in A,$$

where $N$ in the subsequent terms of the above equation array is the norm on $A$ given by $N(a)=a{a}^{*}$. The fact that the $N:KD(A)\to A$, together with the equality $N(0\oplus 0)=0$ show that the extended norm $N$ on $KD(A)$ is well-defined. Thus, $KD(A)$ is a normed $*$-algebra.

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

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

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

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

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

1. ${(xy)}^{*}=y^{*}{x}^{*}$,

2. $x+x^{*}\in A$,

3. $N(xy)=N(x)N(y)$.

Examples. All examples considered below have ground ring the reals $ℝ$.

• $KD(ℝ,-1)=ℂ$, the complex numbers.

• $KD(ℂ,-1)=ℍ$, the quaternions.

• $KD(ℍ,-1)=𝕆$, the octonions.

• $KD(𝕆,-1)=𝕊$, which are called the sedenions, an algebra of dimension 16 over $ℝ$.

Remarks.

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

2. More generally, given any field $k$, any algebra obtained by applying the Cayley-Dickson construction twice to $k$ is called a quaternion algebra over $k$, of which $ℍ$ is an example. In other words, a quaternion algebra has the form

   $$KD(KD(k,{\lambda }_1),{\lambda }_2),$$

   where each ${\lambda }_i\in k^{*}:=k-\{0\}$. Any algebra obtained by applying the Cayley-Dickson construction three times to $k$ is called a Cayley algebra, of which $𝕆$ is an example. In other words, a Cayley algebra has the form

   $$KD(KD(KD(k,{\lambda }_1),{\lambda }_2),{\lambda }_3),$$

   where each ${\lambda }_i\in k^{*}$. A Cayley algebra is an octonion algebra when ${\lambda }_1={\lambda }_2={\lambda }_3=-1$.