# 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 $1\neq 0$. 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. 1.

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

2. 2.

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

3. 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

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

where $N$ in the subsequent terms of the above equation array is the norm on $A$ given by $N(a)=aa^{*}$. The fact that the $N\colon KD(A)\to A$, 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. 1.

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

2. 2.

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

3. 3.

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

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 $\mathbb{R}$.

Remarks.

1. 1.

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!

2. 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 $\mathbb{H}$ 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 $\mathbb{O}$ 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 $\lambda_{1}=\lambda_{2}=\lambda_{3}=-1$.

## References

 Title Cayley-Dickson construction Canonical name CayleyDicksonConstruction Date of creation 2013-03-22 14:54:11 Last modified on 2013-03-22 14:54:11 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 25 Author CWoo (3771) Entry type Definition Classification msc 17A99 Synonym Cayley-Dickson process Synonym doubling process Synonym octonion algebra Related topic TheoremsOnSumsOfSquares Defines Cayley-Dickson algebra Defines sedenion Defines quaternion algebra Defines Cayley algebra