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\begin{document}

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\section*{Cayley-Dickson construction}

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

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

[[!include higher algebra - contents]]

\hypertarget{cayleydickson_construction}{}\section*{{Cayley--Dickson construction}}\label{cayleydickson_construction}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

Consider how the [[complex numbers]] are formed from the [[real numbers]]. If generalized carefully, this kind of operation may be performed again to yield the [[quaternions]], then the [[octonions]] (hence the four real [[normed division algebra]]), then the [[sedenions]], and so on.

This is a special case of a construction which takes a real [[star-algebra]] $A$ to a new star-algebra whose elements are pairs of elements of $A$. This operation is the \textbf{Cayley--Dickson construction}, named after [[Arthur Cayley]] and [[Leonard Dickson]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $A$ be an [[nonassociative algebra|possibly nonassociative]] [[star-algebra]] over the [[field]] $\mathbb{R}$ of [[real numbers]]: an algebra equipped with an [[involution]] $\overline(-) \colon x \mapsto \overline{x}$ which is an [[antiautomorphism]]. (Actually, $\mathbb{R}$ could be replaced by any [[commutative ring]] in the definitions, although some properties may depend on this ring.)

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Cayley-Dickson construction, first version)}

The \emph{Cayley--Dickson double} $A^2$ of $A$ is the real algebra whose underlying $\mathbb{R}$-[[vector space]] is is the [[direct sum]] $A \oplus A$, and whose multiplication is given by

\begin{displaymath}
(a,b)\cdot(u,v) \coloneqq (a u - \overline{v} b, b \overline{u} + v a),
\end{displaymath}
and the formula

\begin{displaymath}
\widebar{(a,b)} \coloneqq (\overline{a},-b)
\end{displaymath}
defines an involutive antiautomorphism on $A^2$, so the doubling procedure can be iterated.

\end{defn}
The following description is different but equivalent:

\begin{defn}
\label{CayleyDicksonDoubleByAdjoiningFurtherGenerator}\hypertarget{CayleyDicksonDoubleByAdjoiningFurtherGenerator}{}
\textbf{(Cayley-Dickson double by generators and relations)}

The \emph{Cayley-Dickson double} $A^2$ of $A$ is the real algebra obtained by adjoining one generator $\ell$ to $A$ subject to the following [[generators and relations|relations]]:

\begin{displaymath}
\ell^2 = -1
\end{displaymath}
and

\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}
for all $a, b \in A$.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The map $a\mapsto (a,0)$ is a [[monomorphism]] $A\to A^2$. If $A$ is [[unital algebra|unital]] with unit $1$ then $A^2$ is unital with unit $(1,0)$. In the unital case, the element $\mathrm{i} \coloneqq (0,1)$ has the property $\mathrm{i}^2 = -1 \coloneqq (-1,0)$, and we may write $(a,b)$ as $a + b \mathrm{i}$ (while $a + \mathrm{i} b = (a,\overline{b})$). For this reason, we may write $A[\mathrm{i}]$ in place of $A^2$, at least when $A$ is unital.

Generally speaking, the double $A^2$ of an algebra $A$ has a nice property iff $A$ is one level nicer. For simplicity, assume that $A$ is unital (so that $\mathbb{R}$ is a subalgebra). Since $\overline{\mathrm{i}} = -\mathrm{i}$, we see that the involution on $A^2$ is trivial iff the involution on $A$ is trivial and $A$ further has $2 = 0$. Since $\mathrm{i} a = \overline{a} \mathrm{i}$, $A^2$ is [[commutative algebra|commutative]] iff $A$ is commutative and the involution in $A$ is trivial. Since $a (b \mathrm{i}) = (b a) \mathrm{i}$, $A^2$ is [[associative algebra|associative]] iff $A$ is associative and commutative. Finally, $A^2$ is [[alternative algebra|alternative]] iff $A$ is associative (and hence also alternative).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The standard example is the sequence of consecutive doubles starting with $\mathbb{R}$ itself (with the [[identity map]] as involution); these are the \textbf{Cayley--Dickson algebras}: the [[real numbers]] $\mathbb{R}$, the [[complex numbers]] $\mathbb{C}$, the [[quaternions]] $\mathbb{H}$, the [[octonions]] (or Cayley numbers) $\mathbb{O}$, the [[sedenions]] $\mathbb{S}$, etc. These are the [[normed division algebras]] ($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$), followed by further algebras which are not [[division algebras]]. All of these algebras are [[power-associative algebra|power-associative]], [[flexible algebra|flexible]], and unital, and have all [[inverse elements]]; the subalgebra with $\overline{x} = x$ is always just $\mathbb{R}$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[composition algebra]]

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

Named after [[Arthur Cayley]] and [[Leonard Dickson]].

Introduction:

\begin{itemize}%
\item [[M M Postnikov]], \emph{Lectures on geometry, Semester V: Lie groups and Lie algebras}, Lec. 14 (russian and english editions)


\item [[John Baez]], \emph{\href{http://math.ucr.edu/home/baez/octonions/node5.html}{The Cayley--Dickson construction}}, in \emph{\href{http://math.ucr.edu/home/baez/octonions/}{The octonions}}, Bull. Amer. Math. Soc. 39 (2002), 145-205, \href{http://dx.doi.org/10.1090/S0273-0979-01-00934-X}{doi}


\item [[John Baez]], \emph{\href{http://math.ucr.edu/home/baez/week59.html}{This Week's Finds --- Week 59}}



\end{itemize}
More:

\begin{itemize}%
\item Daniel K. Biss, [[Daniel Dugger]], [[Daniel Isaksen]], \emph{Large annihilators in Cayley-Dickson algebras}, Communications in Algebra 36 (2), 632-664, 2008 (\href{https://arxiv.org/abs/math/0511691}{arxiv:math/0511691})


\item Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], \emph{Large annihilators in Cayley-Dickson algebras II}, Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 (\href{https://arxiv.org/abs/math/0702075}{arxiv:math/0702075})


\item Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], \emph{Eigentheory of Cayley-Dickson algebras}, Forum Mathematicum 21(5) (2009), 833-851 (\href{https://arxiv.org/abs/0905.2987}{arxiv:0905.2987})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction}{Cayley--Dickson construction}}

\end{itemize}
[[!redirects Cayley-Dickson construction]] [[!redirects Cayley–Dickson construction]] [[!redirects Cayley--Dickson construction]]

[[!redirects Cayley-Dickson constructions]] [[!redirects Cayley–Dickson constructions]] [[!redirects Cayley--Dickson constructions]]

[[!redirects Cayley-Dickson double]] [[!redirects Cayley–Dickson double]] [[!redirects Cayley--Dickson double]]

[[!redirects Cayley-Dickson doubles]] [[!redirects Cayley–Dickson doubles]] [[!redirects Cayley--Dickson doubles]]

[[!redirects Cayley-Dickson double algebra]] [[!redirects Cayley–Dickson double algebra]] [[!redirects Cayley--Dickson double algebra]]

[[!redirects Cayley-Dickson double algebras]] [[!redirects Cayley–Dickson double algebras]] [[!redirects Cayley--Dickson double algebras]]

[[!redirects double of an algebra with involution]] [[!redirects double of algebra with involution]] [[!redirects doubles of algebras with involution]]

[[!redirects double of a star-algebra]] [[!redirects doubles of star-algebras]] [[!redirects double of a \emph{-algebra]] [[!redirects doubles of}-algebras]]

[[!redirects Cayley-Dickson algebra]] [[!redirects Cayley-Dickson algebras]] [[!redirects Cayley–Dickson algebra]] [[!redirects Cayley–Dickson algebras]] [[!redirects Cayley--Dickson algebra]] [[!redirects Cayley--Dickson algebras]]



\end{document}