# octonion

Let $\mathbb{H}$ be the quaternions over the reals $\mathbb{R}$. Apply the Cayley-Dickson construction to $\mathbb{H}$ once, and we obtain an algebra, variously called Cayley algebra, the octonion algebra, or simply the octonions, over $\mathbb{R}$. Specifically the construction is carried out as follows:

1. 1.

Form the vector space $\mathbb{O}=\mathbb{H}\oplus\mathbb{H}\mathbf{k}$; any element of $\mathbb{O}$ can be written as $a+b\mathbf{k}$, where $a,b\in\mathbb{H}$;

2. 2.

Define a binary operation on $\mathbb{O}$ called the multiplication in $\mathbb{O}$ by

 $(a+b\mathbf{k})(c+d\mathbf{k}):=(ac-\overline{d}b)+(da+b\overline{c})\mathbf{k},$

where $a,b,c,d\in\mathbb{H}$, and $\overline{c}$ is the quaternionic conjugation of $c\in\mathbb{H}$. When $b=d=0$, the multiplication is reduced the multiplication in $\mathbb{H}$. In addition, the multiplication rule above imply the following:

 $\displaystyle a(d\mathbf{k})=(da)\mathbf{k}$ (1) $\displaystyle(b\mathbf{k})c=(b\overline{c})\mathbf{k}$ (2) $\displaystyle(b\mathbf{k})(d\mathbf{k})=-\overline{d}b.$ (3)

In particular, in the last equation, if $b=d=1$, $\mathbf{k}^{2}=-1$.

3. 3.

Define a unary operation on $\mathbb{O}$ called the octonionic conjugation in $\mathbb{O}$ by

 $\overline{a+b\mathbf{k}}:=\overline{a}-b\mathbf{k},$

where $a,b\in\mathbb{H}$. Clearly, the octonionic conjugation is an involution (http://planetmath.org/Involution2) ($\overline{\overline{x}}=x$).

4. 4.

Finally, define a unary operation $N$ on $\mathbb{O}$ called the norm in $\mathbb{O}$ by $N(x):=x\overline{x}$, where $x\in\mathbb{O}$. Write $x=a+b\mathbf{k}$, then

 $N(x)=(a+b\mathbf{k})(\overline{a}-b\mathbf{k})=(a\overline{a}+\overline{b}b)+(% -ba+b\overline{\overline{a}})\mathbf{k}=a\overline{a}+b\overline{b}\geq 0.$

It is not hard to see that $N(x)=0$ iff $x=0$.

The above four (actually, only the first two suffice) steps makes $\mathbb{O}$ into an $8$-dimensional algebra over $\mathbb{R}$ such that $\mathbb{H}$ is embedded as a subalgebra.

With the last two steps, one can define the inverse of a non-zero element $x\in\mathbb{O}$ by

 $x^{-1}:=\frac{\overline{x}}{N(x)}$

so that $xx^{-1}=x^{-1}x=1$. Since $x$ is arbitrary, $\mathbb{O}$ has no zero divisors. Upon checking that $x^{-1}(xy)=y=(yx)x^{-1}$, the non-associative algebra $\mathbb{O}$ is turned into a division algebra.

Since $N(x)\geq 0$ for any $x\in\mathbb{O}$, we can define a non-negative real-valued function $\lVert\cdot\rVert$ on $\mathbb{O}$ by $\lVert x\rVert=\sqrt{N(x)}$. This is clearly well-defined and $\lVert x\rVert=0$ iff $x=0$. In addition, it is not hard to see that, for any $r\in\mathbb{R}$ and $x\in\mathbb{O}$, $\lVert rx\rVert=\lvert r\rvert\lVert x\rVert$, and that $\lVert\cdot\rVert$ satisfies the triangular inequality. This makes $\mathbb{O}$ into a normed division algebra.

Since the multiplication in $\mathbb{H}$ is noncommutative, $\mathbb{O}$ is noncommutative. In fact, if we write $\mathbb{H}=\mathbb{C}\oplus\mathbb{C}\mathbf{j}$, where $\mathbb{C}$ are the complex numbers and $\mathbf{j}^{2}=-1$, then $B=\{1,\mathbf{i},\mathbf{j},\mathbf{ij}\}$ is a basis for the vector space $\mathbb{H}$ over $\mathbb{R}$. With the introduction of $\mathbf{k}\in\mathbb{O}$, we quickly check that $\mathbf{k}$ anti-commute with the non-real basis elements in $B$:

 $\mathbf{ik=-ki},\qquad\mathbf{jk=-kj},\qquad\mathbf{(ij)k=-k(ij)}.$

Furthermore, one checks that $\mathbf{i(jk)=(ji)k=-(ij)k}$, so that $\mathbb{O}$ is not associative.

Since $\mathbb{O}=\mathbb{H}\oplus\mathbb{H}\mathbf{k}$, the set $\{1,\mathbf{i,j,ij,k,ik,jk,(ij)k}\}$($=B\cup B\mathbf{k}$) is a basis for $\mathbb{O}$ over $\mathbb{R}$. A less messy way to represent these basis elements is done the following assignment:

Title octonion Octonion 2013-03-22 15:21:42 2013-03-22 15:21:42 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 17A75 msc 17D05 Cayley algebra TheoremsOnSumsOfSquares DivisionAlgebra octonion algebra