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\begin{document}

%-------------------------------------------------------------------


\section*{octonion}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- contents]]

\hypertarget{octonions}{}\section*{{Octonions}}\label{octonions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{automorphisms}{Automorphisms}\dotfill \pageref*{automorphisms} \linebreak
\noindent\hyperlink{left_multiplication_by_imaginary_octonions}{Left multiplication by imaginary octonions}\dotfill \pageref*{left_multiplication_by_imaginary_octonions} \linebreak
\noindent\hyperlink{ElementaryTriples}{Basic triples}\dotfill \pageref*{ElementaryTriples} \linebreak
\noindent\hyperlink{relation_to_quaternions}{Relation to quaternions}\dotfill \pageref*{relation_to_quaternions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \emph{octonion} or \emph{Cayley number} is a kind of [[number]] similar to a [[quaternion number]] but with seven instead of just three [[square roots]] of unity adjoined, and satisfying certain relations.

The \textbf{octonions}, usually denoted $\mathbb{O}$, form the largest of the four [[normed division algebras]] over the [[real numbers]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{ComponentDefinition}\hypertarget{ComponentDefinition}{}
The \emph{octonions} $\mathbb{O}$ is the [[nonassociative algebra]] over the [[real numbers]] which is [[generators and relations|generated]] from seven generators $\{e_1, \cdots, e_7\}$ subject to the [[generators and relations|relations]]

\begin{enumerate}%
\item for all $i$

$e_i^2 = -1$


\item for $e_i \to e_j \to e_k$ an edge or circle in the following diagram (a labeled version of the [[Fano plane]]) the relations

\begin{enumerate}%
\item $e_i e_j  = e_k$


\item $e_j e_i  = -e_k$



\end{enumerate}


\end{enumerate}


\begin{quote}%
(graphics grabbed from \hyperlink{Baez02}{Baez 02})


\end{quote}
This becomes a [[star-algebra]] with star [[involution]]

\begin{equation}
\overline{(-)} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}
\label{StarInvolution}\end{equation}
which is the [[antihomomorphism]] $\overline{a b} = \overline{b} \overline{a}$ that is given on the above generators by

\begin{displaymath}
\overline{e_i} \coloneqq - e_i 
  \phantom{AAAA}
  i \in \{1, \cdots, 7\}
 \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{ProductOfImaginaryOctonions}\hypertarget{ProductOfImaginaryOctonions}{}
The product of all the generators with each other, bracketed to the right, is

\begin{displaymath}
e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 ))))))
  \;=\;
  + 1
\end{displaymath}
\end{example}
\begin{proof}
By iteratively using the multiplication table in def. \ref{ComponentDefinition} we compute as follows:

\begin{displaymath}
\begin{aligned}
   & e_7 (e_6 (e_5 (e_4 (e_3 (\underset{-e_4}{\underbrace{e_2 e_1}})))))
   \\
   & =
   - e_7 (e_6 (e_5 (e_4 (\underset{e_6}{\underbrace{e_3 e_4}}))))
   \\
   & =
   - e_7 (e_6 (e_5 (\underset{e_3}{\underbrace{e_4 e_6}})))
   \\
   & =
   - e_7 (e_6 (\underset{-e_2}{\underbrace{e_5 e_3}}))
   \\
   & =
   + e_7 (\underset{-e_7}{\underbrace{e_6 e_2}})
   \\
   & =
   - \underset{= -1}{\underbrace{e_7 e_7}}
   \\
   & =
   + 1
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{defn}
\label{RealAndImaginary}\hypertarget{RealAndImaginary}{}
\textbf{(real and imaginary octonions)}

As for the [[complex numbers]] one says that

\begin{itemize}%
\item an \emph{imaginary octonion} is an $a \in \mathbb{O}$ shuch that under the star involution \eqref{StarInvolution} it is sent to its negative:

\begin{displaymath}
\overline{a} = -a
\end{displaymath}

\item a \emph{real octonions} is an $a \in \mathcal{O}$ shuch that under the star involution \eqref{StarInvolution} it is sent to itself

\begin{displaymath}
\overline{a} = a
\end{displaymath}


\end{itemize}
Accordingly every octonion decomposes into a [[real part]] and an [[imaginary part]]:

\begin{displaymath}
Re(a) \coloneqq \tfrac{1}{2}(a + \overline{a})
  \phantom{AA}
  Im(a) \coloneqq \tfrac{1}{2}(a - \overline{a})
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{}\hypertarget{}{}
The octonions are \emph{not} an [[associative algebra]]. But the \emph{non-zero} octonions and the \emph{unit} octonions form [[Moufang loops]].

\end{prop}
\begin{prop}
\label{Alternativity}\hypertarget{Alternativity}{}
\textbf{(octonions are alternative)}

The octonions form an [[alternative algebra]]

\end{prop}
\begin{proof}
By linearity it is sufficient to check this on generators. So let $e_i \to e_j \to e_k$ be a circle or a cyclic permutation of an edge in the [[Fano plane]] as in Def. \ref{ComponentDefinition}. Then by definition of the octonion multiplication we have

\begin{displaymath}
\begin{aligned}
    (e_i e_j) e_j
    &=
    e_k e_j
    \\
    &=
    - e_j e_k
    \\
    & =
    -e_i
    \\
    & =
    e_i (e_j e_j)
  \end{aligned}
\end{displaymath}
and similarly

\begin{displaymath}
\begin{aligned}
    (e_i e_i ) e_j
    &=
    - e_j
    \\
    &=
    - e_k e_i
    \\
    &=
    e_i e_k
    \\
    &=
    e_i (e_i e_j)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{automorphisms}{}\subsubsection*{{Automorphisms}}\label{automorphisms}

\begin{prop}
\label{AutomorphismsOfOctonionAlgebraIsG2}\hypertarget{AutomorphismsOfOctonionAlgebraIsG2}{}
The [[automorphism group]] of the octonions, as a real algebra, is the [[exceptional Lie group]] [[G2]].

\end{prop}
See also at \emph{\href{normed+division+algebra#Automorphisms}{normed division algebra -- automorphism}}

\hypertarget{left_multiplication_by_imaginary_octonions}{}\subsubsection*{{Left multiplication by imaginary octonions}}\label{left_multiplication_by_imaginary_octonions}

\begin{defn}
\label{LeftMultiplicationAction}\hypertarget{LeftMultiplicationAction}{}
Given any octonion $o$, then the operation of left multiplication by $o$

\begin{displaymath}
\itexarray{
    \mathbb{O} &\overset{L_o}{\longrightarrow}& \mathbb{O}
    \\
    a &\mapsto& o a
  }
\end{displaymath}
is an $\mathbb{R}-$[[linear map]]. Under [[composition]] of linear maps, this defines an \emph{[[associativity|associative]]} [[monoid]] [[action|acting]] linearly on $\mathbb{O}$.

\end{defn}
\begin{prop}
\label{CliffordActionOfImaginaryOctonions}\hypertarget{CliffordActionOfImaginaryOctonions}{}
\textbf{(Clifford action of imaginary octonions)}

Consider the [[Clifford algebra]]

\begin{displaymath}
Cl(Im(\mathbb{O}), -{\vert -\vert}^2)
\end{displaymath}
on the underlying [[real vector space]] of that of the imaginary octonions (Def. \ref{RealAndImaginary}) regarded as an [[inner product space]] via the [[quadratic form]] given by the \emph{negative} square norm.

Then the operation of left multiplication on $\mathbb{O}$ (def. \ref{LeftMultiplicationAction}) induces a [[representation]] of this Clifford algebra on $\mathbb{R}^8 \simeq_{\mathbb{R}} \mathbb{O}$.

\end{prop}
\begin{proof}
By [[alternative algebra|alternativity]] (Prop. \ref{Alternativity}) we have for every $v \in Im(\mathbb{O})$ and every $a \in \mathbb{O}$

\begin{displaymath}
\begin{aligned}
    L_v L_v (a)
    & \coloneqq
    v (v a)
    \\
    & =
    (v v) a 
    \\
    & =
    - {\vert v\vert}^2 a
    \\
    & =
    L_{- {\vert v\vert}^2} (a)
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{ConsecutiveLeftActionByImaginaryGenerators}\hypertarget{ConsecutiveLeftActionByImaginaryGenerators}{}
\textbf{(consecutive left action by imaginary generators is unity)}

The consecutive left multiplication action (Def. \ref{LeftMultiplicationAction}) by all the imaginary octonion generators $e_i$ (Def. \ref{ComponentDefinition}) is $\pm$ the [[identity function]] on the octonions. Specifically, if one acts in increasing order of the labels in Def. \ref{ComponentDefinition}, then it is +1:

\begin{displaymath}
L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1}
  \;=\;
  + Id_{\mathbb{O}}
\end{displaymath}
\end{prop}
\begin{proof}
All the generators $e_i$ are imaginary octonions (Def. \ref{RealAndImaginary}). By Prop. \ref{CliffordActionOfImaginaryOctonions} their left action on $\mathbb{O}$ [[representation|represents]] a [[Clifford algebra]]-action of $Cl(Im(\mathbb{O}), -{\vert-\vert}^2) \simeq Cl_{0,7}$ on $\mathbb{R}^{8} \simeq_{\mathbb{R}} \mathbb{O}$.

By the \href{Clifford+algebra#ClassificationOverTheRealNumbers}{classification of real Clifford algebras}, $Cl_{0,7}$ has, up to [[isomorphism]], two different [[irreducible representation|irreducible]] [[modules]]. Their underlying vector space is $\mathbb{R}^8$ in both cases, and so the left action of imaginary octonions we have must be one of the two. The two irreps may be distinguished by the action of the ``volume element'' $\Gamma_7 \Gamma_6 \cdots \Gamma_1$: On one of the two it acts as the identity, on the other as minus the identity.

Hence we may check the remaining sign by acting on any one octonion, for instance on the unit $1 \in \mathbb{O}$. Then the claim follows with the computation in Example \ref{ProductOfImaginaryOctonions}:

\begin{displaymath}
\begin{aligned}
    L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} (1)
    & =
    e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 ))))))
    \\
    & =
    + 1
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{ElementaryTriples}{}\subsubsection*{{Basic triples}}\label{ElementaryTriples}

\begin{defn}
\label{BasicTriple}\hypertarget{BasicTriple}{}
A \emph{special triple} or \emph{basic triple} is a [[triple]] $(e_1, e_2, e_3) \in \mathbb{O}^3$ of three octonions such that

\begin{itemize}%
\item $e_i^2 = -1$


\item $e_i e_j = - e_j e_i$.



\end{itemize}
\end{defn}
(\hyperlink{Whitehead71}{Whitehead 71, p. 691})

\begin{remark}
\label{}\hypertarget{}{}
The choice of $e_1$ identifies an inclusion of the [[complex numbers]] $\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{O}$.

Then the choice of $e_2$ on top of that identifies a compatible inclusion of the [[quaternions]] $\mathbb{R} \hookrightarrow \mathbb{C}\hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}$.

Finally the choice of $e_3$ on top of that induces a basis for all of $\mathbb{O}$.

\end{remark}
\begin{prop}
\label{BasicTriplesFormAutomorphism}\hypertarget{BasicTriplesFormAutomorphism}{}
The set of basic triples, def. \ref{BasicTriple}, forms a [[torsor]] over the [[automorphism group]] [[G2]] $= Aut(\mathbb{O})$.

\end{prop}
(e.g. \hyperlink{Baez02}{Baez 02, 4.1})

\hypertarget{relation_to_quaternions}{}\subsubsection*{{Relation to quaternions}}\label{relation_to_quaternions}

\begin{prop}
\label{InvolutionProjectingOutH}\hypertarget{InvolutionProjectingOutH}{}
Let

\begin{displaymath}
\mathbb{H} = \langle 1, i, j, k\rangle
\end{displaymath}
be the [[quaternions]] equipped with canonical [[linear basis|basis]] elements, and let

\begin{equation}
\mathbb{O} = \mathbb{H} \oplus \ell \mathbb{H}
\label{ForProjectionStatementOcotonionsDirectSumUnderCayleyDickson}\end{equation}
be the octonions equipped with the linear basis induced by the [[Cayley-Dickson construction]] (via \href{Cayley-Dickson+construction#CayleyDicksonDoubleByAdjoiningFurtherGenerator}{this def.}).

Then the linear map

\begin{displaymath}
L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k}
  \;\colon\;
  \mathbb{O} \longrightarrow \mathbb{O}
\end{displaymath}
is an [[involution]] whose +1 [[eigenspace]] is $\ell \mathbb{H}$ and whose -1 eigenspace is $\mathbb{H}$, under the above identification \eqref{ForProjectionStatementOcotonionsDirectSumUnderCayleyDickson}.

(Here $L_{(-)}$ denotes the linear map on $\mathbb{O}$ given by left multiplication in $\mathbb{O}$.)

\end{prop}
\begin{proof}
We use the Cayley-Dickson relations (\href{Cayley-Dickson+construction#CayleyDicksonDoubleByAdjoiningFurtherGenerator}{this def.})

\begin{displaymath}
a (\ell b) = \ell (\overline{a} b)
  \,,
  \phantom{AA}
  a(\ell b) = (a \overline{b}) \ell
  \,,
  \phantom{AA}
  (\ell a) (b \ell^{-1}) = \overline{a b}
\end{displaymath}
that hold in $\mathbb{O}$ for all $a,b \in \mathbb{H}$, as well as

\begin{displaymath}
\ell e = - e \ell
\end{displaymath}
for all imaginary elements $e \in \mathbb{H}$.

With this we compute

\begin{displaymath}
\begin{aligned}
    & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( a)
    \\
    & =  \ell( (\ell i) ( (\ell j) ( (\ell k) a ) ) )
    \\
    & = - \ell( (\ell i) ( (\ell j) ( (k \ell) a ) ) )
    \\
    & = - \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell ) ) )
    \\
    & = + \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell^{-1} ) ) )
    \\
    & = + \ell( (\ell i) ( \overline{j (k \overline{a})} ) )
    \\
    & = + \ell( (\ell i) ( \overline{i \overline{a}} ) )
    \\
    & = - \ell( (i \ell) ( \overline{i \overline{a}} ) )
    \\
    & = - \ell( ( i i \overline{a}  ) \ell )
    \\
    & = + \ell( \overline{a}  \ell )
    \\
    & = - \ell( \overline{a} \ell^{-1} )
    \\
    & = - a
  \end{aligned}
\end{displaymath}
and

\begin{displaymath}
\begin{aligned}
    & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( \ell a)
    \\
    & = \ell( (\ell i) ( (\ell j) ( (\ell k) (\ell a) ) ) )
    \\
    & =
      \ell( (\ell i) ( (\ell j) ( (\ell k) ( \overline{a} \ell) ) ) )
    \\
    & =
      -
      \ell( (\ell i) ( (\ell j) ( \overline{k {\overline{a}}} ) ) )
    \\
    & =
      + \ell( (\ell i) ( (j \ell) ( \overline{k {\overline{a}}} ) ) )
    \\
    & =
      + \ell( (\ell i) ( (j k {\overline{a}}) \ell ) )
    \\
    & =
      - \ell( (\ell i) ( (j k {\overline{a}}) \ell^{-1} ) )
    \\
    & =
      - \ell( \overline{ i j k {\overline{a}} } )
    \\
    & =
      + \ell( \overline{ {\overline{a}} } )
    \\
    & =
      + \ell a 
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[associative 3-form]]


\item [[Fano plane]]


\item [[Cayley plane]]


\item [[octonionic Hopf fibration]]


\item [[split octonion]]



\end{itemize}
[[!include exceptional spinors and division algebras -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Textbook account:

\begin{itemize}%
\item [[Tonny Springer]], [[Ferdinand Veldkamp]], \emph{Octonions, Jordan Algebras, and Exceptional Groups}, Springer Monographs in Mathematics, 2000

\end{itemize}
A survey is in

\begin{itemize}%
\item [[John Baez]], \emph{The Octonions}, Bull. Amer. Math. Soc. 39 (2002), 145-205. (\href{http://math.ucr.edu/home/baez/octonions/octonions.html}{web})

\end{itemize}
The concept of ``special triples'' or (``basic triples'') used above seems to go back to

\begin{itemize}%
\item [[George Whitehead]], appendix A in \emph{Homotopy Theory}, MIT press 1971

\end{itemize}
Relation to the [[Leech lattice]]:

\begin{itemize}%
\item [[Robert A. Wilson]], \emph{Octonions and the Leech lattice}, Journal of Algebra Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, (\href{http://www.maths.qmul.ac.uk/~raw/pubs_files/octoLeech1.pdf}{pdf}, \href{http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf}{slides})

\end{itemize}
[[!redirects octonion]] [[!redirects octonions]] [[!redirects octonion algebra]]



\end{document}