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\begin{document}

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\section*{Albert algebra}

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

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

[[!include exceptional structures -- contents]]

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

[[!include higher algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{uniqueness}{Uniqueness}\dotfill \pageref*{uniqueness} \linebreak
\noindent\hyperlink{RelationTo10dSuperSpacetime}{Relation to 10d super-Spacetime}\dotfill \pageref*{RelationTo10dSuperSpacetime} \linebreak
\noindent\hyperlink{Automorphisms}{Automorphisms and exceptional Lie groups}\dotfill \pageref*{Automorphisms} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{octonionic Albert algebra} is the [[Jordan algebra]] of $3$-by-$3$ [[hermitian matrices]] over the [[octonions]] $\mathbb{O}$

\begin{equation}
\mathfrak{h}_3(\mathbb{O})
  \;\coloneqq\;
  \left\{
  \left(
    \itexarray{
      (x_0 + x_1) & y & \psi_1
      \\
      y^\ast & (x_0 - x_1) & \psi_2
      \\    
      \psi_1^\ast & \psi_2^\ast & \phi
    }
  \right)
  \;|\;
  \itexarray{
    x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
    \\
    \psi_1, \psi_2 \in \mathbb{O}
  }
  \right\}
\label{3Times3HermitianMatrix}\end{equation}
Similarly the \emph{split-octonionic Albert algebra} is the algebra of $3$-by-$3$ [[hermitian matrices]] over the [[split-octonions]].

The construction is due to (\hyperlink{Albert}{Albert 1934}), originating in an algebraic approach to [[quantum mechanics]].

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

\hypertarget{uniqueness}{}\subsubsection*{{Uniqueness}}\label{uniqueness}

The octonionic and split-octonionic Albert algebras are (up to [[isomorphism]]) the only [[simple algebra|simple]] [[finite-dimensional space|finite-dimensional]] [[formally real algebra|formally real]] [[Jordan algebras]] over the [[real numbers]] that are not [[special Jordan algebra|special]], together comprising the \emph{real Albert algebras}.

Their [[complexifications]] are [[isomorphism|isomorphic]], the \emph{complex-octonionic Albert algebra}, or simply the \emph{complex Albert algebra}. Analogues exist over any [[field]].

An \emph{exceptional Jordan algebra} (over any [[field]]) is any [[Jordan algebra]] in which an Albert algebra appears as a [[direct summand]]. Every formally real Jordan algebra over the real numbers is either special or exceptional (so they all have excellent self-esteem). The exceptional Jordan algebras are related to the \href{exceptional+Lie+group#lie_algebras_2}{exceptional Lie algebras}.

\hypertarget{RelationTo10dSuperSpacetime}{}\subsubsection*{{Relation to 10d super-Spacetime}}\label{RelationTo10dSuperSpacetime}

The form of the $3 \times 3$-hermitian matrix in \eqref{3Times3HermitianMatrix} makes it manifest that the exceptional Jordan algebra is naturally a [[linear map|linear]] [[direct sum]] of the form

\begin{displaymath}
\mathfrak{h}_3(\mathbb{O})
  \;\simeq_{\mathbb{R}}\;
  \mathfrak{h}_2(\mathbb{O})
  \oplus 
  \mathbb{O}^2
  \oplus
  \mathbb{R}
\end{displaymath}
via

\begin{displaymath}
\underset{
    \mathfrak{h}_3(\mathbb{O})
  }{
  \underbrace{
  \left\{
  \left(
    \itexarray{
      (x_0 + x_1) & y & \psi_1
      \\
      y^\ast & (x_0 - x_1) & \psi_2
      \\    
      \psi_1^\ast & \psi_2^\ast & \phi
    }
  \right)
  \right\}
  }}
  \;\simeq\;
  \underset{
    \mathfrak{h}_2(\mathbb{O})
  }{
  \underbrace{
  \left\{
  \left(
    \itexarray{
      (x_0 + x_1) & y & 0
      \\
      y^\ast & (x_0 - x_1) & 0 
      \\    
      0  & 0 & 0
    }
  \right)
  \right\}
  }
  }
  \oplus
  \underset{
    \mathbb{O}^2
  }{
  \underbrace{
  \left\{
  \left(
    \itexarray{
      0 & 0 & \psi_1
      \\
      0 & 0 & \psi_2
      \\    
      \psi_1^\ast  & \psi_2^\ast & 0
    }
  \right)
  \right\}
  }
  }
  \oplus
  \underset{
    \mathbb{R}
  }{
  \underbrace{
  \left\{
  \left(
    \itexarray{
      0 & 0 & 0
      \\
      0 & 0 & 0
      \\    
      0  & 0 & \phi
    }
  \right)
  \right\}
  }
  }
\end{displaymath}
with

\begin{displaymath}
\itexarray{
    x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
    \\
    \psi_1, \psi_2 \in \mathbb{O}
}
\end{displaymath}
By the discussion at \emph{[[geometry of physics -- supersymmetry]]} in the section \emph{\href{geometry+of+physics+--+supersymmetry#InTermsOfNormedDivisionAlgebraInDimension3To10}{Real spinors in dimension 3,4,6,10}} these summands may be further identified as follows:

\begin{itemize}%
\item $\mathfrak{h}_2(\mathbb{O}) \simeq \mathbb{R}^{9,1}$ is the incarnation of 10-dimensional [[super-Minkowski spacetime]] via octonionic [[Pauli matrices]];


\item under this identification $\mathbb{O}^2 \simeq \mathbf{16}$ is the [[Majorana-Weyl spinor]] [[real spin representation]] of the [[spin group]] $Spin(9,1)$.



\end{itemize}
\begin{displaymath}
\mathfrak{h}_3(\mathbb{O})
  \;
  \simeq_{\mathbb{R}}
  \;
  \underset{
    dim_{\mathbb{R}} = 26
  }{
  \underbrace{
    \mathbb{R}^{9,1}
      \oplus
    \mathbf{16}
  }}
  \oplus
  \mathbb{R}
  \,.
\end{displaymath}
Under these identifications, $\phi \in \mathbb{R}$ looks like the size of $S^1/(\mathbb{Z}_2)$ in [[Horava-Witten theory]].

This decomposition hence induces an [[action]] of the [[spin group]] $Spin(9,1)$ on the exceptional Jordan algebra. While only the subgroup $Spin(9) \hookrightarrow Spin(9,1)$ of that is an [[isomorphism]] of the [[Jordan algebra]]-[[structure]] itself, the full $Spin(9,1)$-[[action]] does preserve the [[determinant]] on $\mathfrak{h}_3(\mathbb{O})$.

\hypertarget{Automorphisms}{}\subsubsection*{{Automorphisms and exceptional Lie groups}}\label{Automorphisms}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[general linear group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ preserving [[determinant]] is [[E6]])}

The [[group]] of [[determinant]]-preserving [[linear map|linear]] [[isomorphisms]] of the vector space underlying the octonionic Albert algebra is the [[exceptional Lie group]] [[E6]]${}_{(-26)}$.

\end{prop}
(see e.g. (\hyperlink{ManogueDray09}{Manogue-Dray 09})).

\begin{prop}
\label{JordanAutomorphisms}\hypertarget{JordanAutomorphisms}{}
\textbf{([[Jordan algebra]] [[automorphism group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ is [[F4]])}

The [[group]] of [[automorphism]] with respect to the [[Jordan algebra]] structure $\circ$ on the octonionic Albert algebra is the [[exceptional Lie group]] [[F4]]:

\begin{displaymath}
Aut\left(
    Mat_{3\times 3}^{herm}(\mathbb{O}), \circ
  \right)
  \;\simeq\;
  F_4
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Yokota09}{Yokota 09, section 2.2})

\begin{prop}
\label{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}\hypertarget{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}{}
\textbf{([[Jordan algebra]] [[automorphism group]] of $Mat_{3 \times 3}^{herm}(\mathbb{O})$ fixing an [[imaginary number|imaginary]] [[octonion]])}

Fix an [[imaginary number|imaginary]] [[octonion]] $i \in \mathbb{O}$, hence a $\mathbb{R}$-[[linear map|linear]] [[direct sum]] decomposition

\begin{displaymath}
\mathbb{O}
  \;\simeq_{\mathbb{R}}\;
  \mathbb{C} \oplus V
  \phantom{AA}\text{with}\phantom{AA}
  V \simeq_{\mathbb{R}} \mathbb{C}^3 
  \,,
\end{displaymath}
and let

\begin{equation}
\itexarray{    
    Mat_{3 \times 3}^{herm}(\mathbb{O}) 
     &\overset{w}{\longrightarrow}&
    Mat_{3 \times 3}^{herm}(\mathbb{O}) 
  }
\label{ComponentwiseiFixingAutomorphism}\end{equation}
be given componentwise by the identity on $\mathbb{C}$ and by multiplication with some fixed non-vanishing number on $V$.

Then the [[subgroup]] of the [[Jordan algebra]] [[automorphism]] group $Aut\left(Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4$ (Prop. \ref{JordanAutomorphisms}) of elements that commute with $w$ \eqref{ComponentwiseiFixingAutomorphism}

\begin{displaymath}
F_4^w 
  \;\coloneqq\;
  \left\{
    \alpha \in F_4 \;\vert\;  w \alpha = \alpha w
  \right\}
\end{displaymath}
is

\begin{displaymath}
F_4^w
  \;\simeq\;
  \big( 
    SU(3) \times SU(3)
  \big)/ \mathbb{Z}_3
  \,,
\end{displaymath}
where every element in the [[direct product group]] of [[SU(3)]] with itself

\begin{displaymath}
(A, B) \in SU(3) \times SU(3)
\end{displaymath}
[[action|acts]] on an element

\begin{displaymath}
\underset{
    \in Mat_{3\times 3}^{herm}(\mathbb{O})
  }{\underbrace{\;X\;}}
  \;\simeq\;
  \underset{
    \in Mat_{3\times 3}^{herm}(\mathbb{C})
  }{\underbrace{\;X_{\mathbb{C}}\;}}
  \;+\;
  \underset{
    \in Mat_{3 \times 3}(\mathbb{C})
  }{\underbrace{X_{V}}}
\end{displaymath}
via [[matrix multiplication]] as

\begin{equation}
X_{\mathbb{C}}
  +
  X_{\mathbb{V}}
  \;\mapsto\;
  A  X_{\mathbb{C}} A^\dagger
  \;+\;
  B X_{V} A^\dagger
\label{MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism}\end{equation}
(with $(-)^\dagger$ being the [[conjugate transpose matrix]], hence the [[inverse matrix]] for the [[unitary matrices]] under consideration) and where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]]

\begin{displaymath}
\mathbb{Z}_3
  \;\subset\;
  SU(3) \times SU(3)
\end{displaymath}
which is [[generators and relations|generated]] by the [[pair]] of [[diagonal matrices]]

\begin{equation}
\left(
    e^{2\pi i \tfrac{1}{3}}   \mathbf{1}_3
    \;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_3
  \right)/
  \;\in \;
  SU(3) \times SU(3)
  \,.
\label{Z3Generator}\end{equation}
\end{prop}
(\hyperlink{Yokota09}{Yokota 09, theorem 2.12.2})

$\backslash$begin\{prpn\} $\backslash$label\{StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra\} The further [[subgroup]] of $F_4^w 
  \simeq
  \big(
    SU(3) \times SU(3)
  \big) / \mathbb{Z}_3
  \;\subset\;
  F_4$ (Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}) which fixes a subspace

\begin{displaymath}
Mat_{2 \times 2}^{herm}(\mathbb{C})
  \;\subset\;
  \underset{
    Mat_{3 \times 3}^{herm}( \mathbb{O} )
  }{
  \underbrace{
    Mat_{3 \times 3}^{herm}(\mathbb{C}) 
    \;\oplus\;
    Mat_{3 \times 3}(V)
  }}
\end{displaymath}
(hence, by the \hyperlink{RelationTo10dSuperSpacetime}{above}, a 4d [[Minkowski spacetime]] (incarnated via its [[Pauli matrices]]) inside the 10d [[Minkowski spacetime]] inside the octonionic Albert algebra)

is

\begin{displaymath}
\big(
    U(1)
    \times
    SU(2)
    \times
    SU(3)
  \big) /  \mathbb{Z}_6
  \,,
\end{displaymath}
where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]] which is [[generators and relations|generated]] by the element

\begin{displaymath}
\left(
    \exp\left(2 \pi i \tfrac{1}{6}\right)\;,\;
    \exp\left(2 \pi i \tfrac{1}{2}\right) \mathbf{1}_2\;,\;
    \exp\left(2 \pi i \tfrac{1}{3}\right) \mathbf{1}_3   
  \right)
  \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,.
\end{displaymath}
(Hence this group happens to coincide with the \emph{exact} [[gauge group]] of the [[standard model of particle physics]], see \href{standard+model+of+particle+physics#GaugeGroup}{there}).

$\backslash$end\{prpn\}

This was claimed without proof in \hyperlink{DuboisVioletteTodorov18}{Dubois-Violette \& Todorov 18}.

$\backslash$begin\{proof\} By Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion} \eqref{MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism} it is clear that the subgroup in question is that represented by those [[pairs]] $(A,B) \in SU(3) \times SU(3)$ for which $A$ is $(1 + 2)$-block diagonal. Such matrices $A$ form the [[subgroup]] of [[SU(3)]] of [[matrices]] that may be written in the form

\begin{displaymath}
diag\left(
    c^2, c^{-1} \mathbf{\sigma}
  \right)
\end{displaymath}
for $c \in U(1)$ and $\mathbf{\sigma} \in$ [[SU(2)]]. The [[kernel]] of the [[group homomorphism]]

\begin{equation}
\itexarray{
    U(1) \times SU(2) &\longrightarrow& SU(3)
    \\
    (c,\mathbf{\sigma}) &\mapsto& diag\left(
    c^{2}, c^{-1} \mathbf{\sigma}
  \right)
  }
\label{IdentifyingU1SU2inSU3}\end{equation}
is clearly the [[cyclic group]]

\begin{equation}
\left\{
    (1,\mathbf{1}_2)\;,\; 
    \left(
      e^{2\pi i \tfrac{1}{2}},e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2
    \right) 
  \right\}
  \;\simeq\;
  \mathbb{Z}_2
  \,.
\label{Z2Generator}\end{equation}
Hence the subgroup in question is

\begin{displaymath}
\begin{aligned}
    \Big(
      \big( U(1) \times SU(2) \big)/ \mathbb{Z}_2
      \;\times\;
      SU(3)
    \Big)/ \mathbb{Z}_3
    & \simeq
    \Big(
    \Big(
      \big( U(1) \times SU(2) \big)
      \;\times\;
      SU(3)
    \Big) / \mathbb{Z}_2 
    \Big) / \mathbb{Z}_3
    \\
    &\simeq
    \Big(
      \big( U(1) \times SU(2) \big)
      \;\times\;
      SU(3)
    \Big) / \mathbb{Z}_6
    \,,
  \end{aligned}
\end{displaymath}
where in the first step we extended the $\mathbb{Z}_2$-[[action]] as the [[trivial action]] on the $SU(3)$-factor, and in the second step we used the evident [[isomorphism]] $\mathbb{Z}_2 \times \mathbb{Z}_3 \simeq \mathbb{Z}_6$ (an application of the ``[[fundamental theorem of cyclic groups]]'', if you wish).

It remains to see that the [[action]] of $\mathbb{Z}_6$ is as claimed. By the above identification $\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_3$, it is generated by the \emph{joint} action of that of the generators of $\mathbb{Z}_3$ and of $\mathbb{Z}_2$, which, by \eqref{Z3Generator} and \eqref{Z2Generator}, is

\begin{displaymath}
\underset{
    \text{generator of}\, \mathbb{Z}_3
  }{
  \underbrace{
  \Big(
    e^{2\pi i \tfrac{1}{3}}  \;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_2\;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_3
  \Big)
  }
  }
  \underset{
    \text{generator of} \, \mathbb{Z}_2
  }{
  \underbrace{
    \Big(
      1 , (e^{2 \pi i \tfrac{1}{2}}) (e^{2 \pi i \tfrac{1}{2}}\mathbf{1}_2), \mathbf{1}_3
    \Big)
  }
  }
  \;=\;
  \left(
    e^{2\pi i \tfrac{1}{3}} \;,\;
    \underset{ 
      = e^{2\pi i \tfrac{-1}{6}} 
    }{
    \underbrace{
      e^{2\pi i \tfrac{1}{2}} e^{2\pi i \tfrac{1}{3}}
    }} 
   \;
   ( e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2)
   \;,\;
    e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
  \right)
\end{displaymath}
as an element in $SU(3) \times SU(3)$, hence is

\begin{displaymath}
\Big(
    e^{2\pi i \tfrac{1}{6}} \;,\;
    e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2 \;,\;
    e^{2\pi i \tfrac{1}{3}}\mathbf{1}_3 \;,\;    
  \Big)
  \;\in\;
  U(1) \times SU(2) \times SU(3)
\end{displaymath}
under the lift through \eqref{IdentifyingU1SU2inSU3}.

$\backslash$end\{proof\}

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

The original article is

\begin{itemize}%
\item [[Abraham Adrian Albert]], \emph{On a Certain Algebra of Quantum Mechanics}, Annals of Mathematics, Second Series 35 (1): 65--73, (1934)(doi:\href{http://dx.doi.org/10.2307/1968118}{10.2307/1968118}, \href{https://www.jstor.org/stable/1968118}{JSTOR}).

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 5 of \emph{Octonions, Jordan Algebras, and Exceptional Groups}, Springer Monographs in Mathematics, 2000

\end{itemize}
Further discussion:

\begin{itemize}%
\item [[John Baez]], \href{http://math.ucr.edu/home/baez/octonions/node12.html}{section 3.4} \emph{$\mathbb{O}P^2$ and the Exceptional Jordan Algebra} of \emph{The Octonions}, Bull. Amer. Math. Soc. 39 (2002), 145-205. (\href{http://math.ucr.edu/home/baez/octonions/octonions.html}{web})


\item Ichiro Yokota, \emph{Exceptional Lie groups} (\href{https://arxiv.org/abs/0902.0431}{arXiv:0902.0431})


\item [[Corinne Manogue]], [[Tevian Dray]], \emph{Octonions, $E_6$, and Particle Physics}, J.Phys.Conf.Ser.254:012005,2010 (\href{http://arxiv.org/abs/0911.2253}{arXiv:0911.2253})


\item Michel Dubois-Violette, Ivan Todorov, \emph{Exceptional quantum geometry and particle physics II} (\href{https://arxiv.org/abs/1808.08110}{arXiv:1808.08110})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Albert_algebra}{Albert algebra}}

\end{itemize}
[[!redirects exceptional Jordan algebra]] [[!redirects exceptional Jordan algebras]]

[[!redirects Albert algebra]] [[!redirects Albert algebras]]

[[!redirects real Albert algebra]] [[!redirects real Albert algebras]]

[[!redirects octonionic Albert algebra]] [[!redirects octonionic Albert algebras]]

[[!redirects split-octonionic Albert algebra]] [[!redirects split-octonionic Albert algebras]] [[!redirects split octonionic Albert algebra]] [[!redirects split octonionic Albert algebras]]

[[!redirects complex Albert algebra]] [[!redirects complex Albert algebras]] [[!redirects complexified Albert algebra]] [[!redirects complexified Albert algebras]] [[!redirects complex-octonionic Albert algebra]] [[!redirects complex-octonionic Albert algebras]] [[!redirects complex octonionic Albert algebra]] [[!redirects complex octonionic Albert algebras]]



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