octonion

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

1. Form the vector space $𝕆=ℍ\oplus ℍ𝐤$; any element of $𝕆$ can be written as $a+b𝐤$, where $a,b\in ℍ$;

2. Define a binary operation on $𝕆$ called the multiplication in $\mathrm{O}$ by

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

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

   $a(d𝐤)=(da)𝐤$ (1)

   $(b𝐤)c=(b\overline{c})𝐤$ (2)

   $(b𝐤)(d𝐤)=-\overline{d}b.$ (3)

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

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

   $$\overline{a+b𝐤}:=\overline{a}-b𝐤,$$

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

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

   $$N(x)=(a+b𝐤)(\overline{a}-b𝐤)=(a\overline{a}+\overline{b}b)+(-ba+b\overline{\overline{a}})𝐤=a\overline{a}+b\overline{b}\ge 0.$$

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

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

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

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

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

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

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

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

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

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

Title	octonion
Canonical name	Octonion
Date of creation	2013-03-22 15:21:42
Last modified on	2013-03-22 15:21:42
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	15
Author	CWoo (3771)
Entry type	Definition
Classification	msc 17A75
Classification	msc 17D05
Synonym	Cayley algebra
Related topic	TheoremsOnSumsOfSquares
Related topic	DivisionAlgebra
Defines	octonion algebra