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\begin{document}

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\section*{Cayley plane}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{Cayley plane} is the [[projective plane]] over the [[octonions]], $\mathbb{O} P^2$.

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

It can't be constructed using homogeneous coordinates the way we do for projective planes over [[division rings]], since multiplication of octonions is not associative. However, it can be constructed in several other ways that generalize the approach for division rings:

\begin{itemize}%
\item One can start with $\mathbb{O}^2$ and give it the structure of an [[affine plane]] in the obvious way, and then add ``points at infinity'' in the usual way to obtain a projective plane. This is the most straightforward approach, but as always it has the defect that it makes the line at infinity appear special.


\item One can consider the space of $3\times 3$ matrices over $\mathbb{O}$ that are ``Hermitian'' and idempotent, hence can be imagined as ``projections onto dimension-1 subspaces of $\mathbb{O}^3$''.


\item Writing out the components of such a matrix explicitly, one obtains a \emph{Veronese vector} $(x_1,x_2,x_3;\xi_1,\xi_2,\xi_3)$ where $x_i\in \mathbb{O}$ and $\xi_i\in \mathbb{R}$, such that $\xi_i \overline{x_i} = x_j x_k$ and $\Vert x_i\Vert^2 = \xi_j \xi_k$ for all cyclic permutations $(i,j,k)$ of $(1,2,3)$. The Cayley plane can be identified with the space of nonzero such vectors modulo the scalar action of $\mathbb{R}$.



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

\begin{itemize}%
\item [[F4]] is the [[isometry group]] of $\mathbb{O} P^2$, with the [[stabilizer]] of a point being [[Spin(9)]]. Hence $\mathbb{O} P^2 \cong F_4/Spin(9)$.


\item It is a non-Desarguesian plane, that is, Desargues' theorem does not hold. See [[projective plane]].



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

\begin{itemize}%
\item [[Cayley 4-form]]

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

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Cayley_plane}{Cayley plane}}


\item Salzmann et. al., \emph{Compact Projective Planes, with an introduction to Octonion Geometry}



\end{itemize}
Discussion of the [[Witten genus]] of Cayley plane-[[fiber bundles]] is in

\begin{itemize}%
\item [[Carl McTague]], \emph{The Cayley Plane and the Witten Genus} (\href{http://arxiv.org/abs/1006.0728}{arXiv:1006.0728})

\end{itemize}
[[!redirects Cayley planes]] [[!redirects octonion projective plane]] [[!redirects octonion projective planes]] [[!redirects octonionic projective plane]] [[!redirects octonionic projective planes]]



\end{document}