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Let $\Q$ be the quaternions over the reals $\R$ Apply the Cayley-Dickson construction to $\Q$ 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:
- Form the vector space $\Oc=\Q\oplus\Q\mathbf{k}$ any element of $\Oc$ can be written as $a+b\mathbf{k}$ where $a,b\in\Q$
- Define a binary operation on $\Oc$ called <</SPAN>#156#>the multiplication in $\Oc$ by $$(a+b\mathbf{k})(c+d\mathbf{k}):=(ac-\conj{d}b)+(da+b\conj{c})\mathbf{k},$$ where $a,b,c,d\in\Q$ and $\conj{c}$ is the quaternionic conjugation of $c\in\Q$ When $b=d=0$ the multiplication is reduced the multiplication in $\Q$ In addition, the multiplication rule above imply the following: \begin{eqnarray} a(d\mathbf{k})=(da)\mathbf{k} \\ (b\mathbf{k})c=(b\conj{c})\mathbf{k} \\ (b\mathbf{k})(d\mathbf{k})=-\conj{d}b. \end{eqnarray}In particular, in the last equation, if $b=d=1$ $\mathbf{k}^2=-1$
- Define a unary operation on $\Oc$ called <</SPAN>#157#>the octonionic conjugation in $\Oc$ by $$\overline{a+b\mathbf{k}}:=\conj{a}-b\mathbf{k},$$ where $a,b\in\Q$ Clearly, the octonionic conjugation is an involution ($\overline{\conj{x}}=x$ .
- Finally, define a unary operation $N$ on $\Oc$ called <</SPAN>#158#>the norm in $\Oc$ by $N(x):=x\conj{x}$ where $x\in\Oc$ Write $x=a+b\mathbf{k}$ then $$N(x)=(a+b\mathbf{k})(\conj{a}-b\mathbf{k}) =(a\conj{a}+\conj{b}b)+(-ba+b\conj{\conj{a}})\mathbf{k}= a\conj{a}+b\conj{b}\ge0.$$ It is not hard to see that $N(x)=0$ iff $x=0$
The above four (actually, only the first two suffice) steps makes $\Oc$ into an $8$ dimensional algebra over $\R$ such that $\Q$ is embedded as a subalgebra.
With the last two steps, one can define the inverse of a non-zero element $x\in\Oc$ by $$x^{-1}:=\frac{\conj{x}}{N(x)}$$ so that $xx^{-1}=x^{-1}x=1$ Since $x$ is arbitrary, $\Oc$ has no zero divisors. Upon checking that $x^{-1}(xy)=y=(yx)x^{-1}$ the non-associative algebra $\Oc$ is turned into a division algebra.
Since $N(x)\ge0$ for any $x\in\Oc$ we can define a non-negative real-valued function $\norm{\cdot}$ on $\Oc$ by $\norm{x}= \sqrt{N(x)}$ This is clearly well-defined and $\norm{x}=0$ iff $x=0$ In addition, it is not hard to see that, for any $r\in\R$ and $x\in\Oc$ $\norm{rx}= \abs{r}\norm{x}$ and that $\norm{\cdot}$ satisfies the triangular inequality. This makes $\Oc$ into a
normed division algebra.
Since the multiplication in $\Q$ is noncommutative, $\Oc$ is noncommutative. In fact, if we write $\Q=\C\oplus\C\mathbf{j}$ where $\C$ are the complex numbers and $\mathbf{j}^2=-1$ then $B=\lbrace 1,\mathbf{i},\mathbf{j},\mathbf{ij}\rbrace$ is a basis for the vector space $\Q$ over $\mathbb{R}$ With the introduction of $\mathbf{k}\in\Oc$ 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 $\Oc$ is not associative.
Since $\Oc=\Q\oplus\Q\mathbf{k}$ the set $\lbrace 1,\mathbf{i,j,ij,k,ik,jk,(ij)k}\rbrace$ $=B\cup B\mathbf{k}$ is a basis for $\Oc$ over $\R$ A less messy way to represent these basis elements is done the following assignment:
| basis element |
$1$ |
$\mathbf{i}$ |
$\mathbf{j}$ |
$\mathbf{ij}$ |
$\mathbf{k}$ |
$\mathbf{ik}$ |
$\mathbf{jk}$ |
$\mathbf{(ij)k}$ basis element rewritten |
Any element $x$ of $\Oc$ can thus be expressed uniquely as $\sum_{n=0}^{7}r_n\mathbf{i_n}$ where $r_n\in\R$ Using Equations (1)-(3) above, one can form a multiplication table for these basis elements $i_n$ s:
| row$\times$ |
$\mathbf{i_1}$ |
$\mathbf{i_2}$ |
$\mathbf{i_3}$ |
$\mathbf{i_4}$ |
$\mathbf{i_5}$ |
$\mathbf{i_6}$ |
$\mathbf{i_7}$ |
| $\mathbf{i_1}$ |
-1 |
$\mathbf{i_3}$ |
-$\mathbf{i_2}$ |
$\mathbf{i_5}$ |
-$\mathbf{i_4}$ |
-$\mathbf{i_7}$ |
$\mathbf{i_6}$ |
| $\mathbf{i_2}$ |
-$\mathbf{i_3}$ |
-1 |
$\mathbf{i_1}$ |
$\mathbf{i_6}$ |
$\mathbf{i_7}$ |
-$\mathbf{i_4}$ |
-$\mathbf{i_5}$ |
| $\mathbf{i_3}$ |
$\mathbf{i_2}$ |
-$\mathbf{i_1}$ |
-1 |
$\mathbf{i_7}$ |
-$\mathbf{i_6}$ |
$\mathbf{i_5}$ |
-$\mathbf{i_4}$ |
| $\mathbf{i_4}$ |
-$\mathbf{i_5}$ |
-$\mathbf{i_6}$ |
-$\mathbf{i_7}$ |
-1 |
$\mathbf{i_1}$ |
$\mathbf{i_2}$ |
$\mathbf{i_3}$ |
| $\mathbf{i_5}$ |
$\mathbf{i_4}$ |
-$\mathbf{i_7}$ |
$\mathbf{i_6}$ |
-$\mathbf{i_1}$ |
-1 |
-$\mathbf{i_3}$ |
$\mathbf{i_2}$ |
| $\mathbf{i_6}$ |
$\mathbf{i_7}$ |
$\mathbf{i_4}$ |
-$\mathbf{i_5}$ |
-$\mathbf{i_2}$ |
$\mathbf{i_3}$ |
-1 |
-$\mathbf{i_1}$ |
| $\mathbf{i_7}$ |
-$\mathbf{i_6}$ |
$\mathbf{i_5}$ |
$\mathbf{i_4}$ |
-$\mathbf{i_3}$ |
-$\mathbf{i_2}$ |
$\mathbf{i_1}$ |
-1 |
Other well known properties of the octonions are
- $\conj{xy}=\conj{y}\hspace{2pt}\conj{x}$ for any $x,y\in\Oc$
- $N(xy)=N(x)N(y)$ so that $\Oc$ is a composition algebra. It also proves that the product of sums of eight squares is a sum of eight squares.
- $\Oc$ is an alternative algebra. As a result, any two elements of $\Oc$ generate an associative algebra. If fact, the algebra is isomorphic of one of $\R$ $\C$ and $\Q$ This is the consequence of Artin's Theorem.
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Cross-references: consequence, isomorphic, generate, alternative algebra, squares, sums, product, composition algebra, properties, row, represent, associative, basis, complex numbers, noncommutative, inequality, well-defined, function, division algebra, non-associative algebra, zero divisors, inverse, iff, norm, operation, unary, equation, imply, addition, reduced, conjugation, multiplication, binary operation, vector space, algebra, Cayley-Dickson construction, reals, quaternions
There are 12 references to this entry.
This is version 12 of octonion, born on 2005-06-22, modified 2007-12-18.
Object id is 7185, canonical name is Octonion.
Accessed 4495 times total.
Classification:
| AMS MSC: | 17D05 (Nonassociative rings and algebras :: Other nonassociative rings and algebras :: Alternative rings) | | | 17A75 (Nonassociative rings and algebras :: General nonassociative rings :: Composition algebras) |
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Pending Errata and Addenda
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