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\section*{E7}

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

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

[[!include exceptional structures -- contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\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{representation}{Representation}\dotfill \pageref*{representation} \linebreak
\noindent\hyperlink{SmallestFundamentalRepresentation}{$\mathbf{56}$ -- The smallest fundamental representation}\dotfill \pageref*{SmallestFundamentalRepresentation} \linebreak
\noindent\hyperlink{AdjointRepresentation}{$\mathbf{133}$ -- The adjoint representation}\dotfill \pageref*{AdjointRepresentation} \linebreak
\noindent\hyperlink{as_part_of_the_ade_pattern}{As part of the ADE pattern}\dotfill \pageref*{as_part_of_the_ade_pattern} \linebreak
\noindent\hyperlink{AsUDualityGroup}{As U-Duality group of 4d SuGra}\dotfill \pageref*{AsUDualityGroup} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_view_of_uduality}{In view of U-duality}\dotfill \pageref*{in_view_of_uduality} \linebreak


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

One of the [[exceptional Lie groups]].

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

Consider the [[vector space]]

\begin{displaymath}
W \coloneqq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast
\end{displaymath}
of [[dimension]] $56$. This is naturally a [[symplectic vector space]] with [[symplectic form]] $\omega$ given by the natural pairing between [[linear 2-forms]] and [[bivectors]].

In addition, consider on this space the quartic form $q \colon W \to \mathbb{R}$ which sends an element $v = (\{v^{a b}, w_{a b}\}) \in W$ to

\begin{displaymath}
q(v) 
   \coloneqq 
   v^{a b } w_{b c} v^{c d} w_{d a}
   - 
   \tfrac{1}{4} v^{a b} w_{a b} v^{c d} w_{c d}
   + 
   \tfrac{1}{96}
   \left(
     \epsilon_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} 
     v^{a_1 a_2}  v^{a_3 a_4}  v^{a_5 a_6}  v^{a_7 a_8} 
     +
     \epsilon^{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} 
     w_{a_1 a_2}  w_{a_3 a_4}  w_{a_5 a_6}  w_{a_7 a_8} 
   \right)
  \,.
\end{displaymath}
Now $E_{7(7)} \subset GL(56,\mathbb{R})$ is the [[subgroup]] of the [[general linear group]] acting on $W$ which preserves both the symplectic form $\omega$ as well as the quartic form $q$. See also \hyperlink{SmallestFundamentalRepresentation}{below}.

This presentation is due to \hyperlink{Cartan}{Cartan}, for review see \hyperlink{CremmerJulia79}{Cremmer-Julia 79, appendix B}, \hyperlink{PachecoWaldram08}{Pacheco-Waldram 08, B.1}. A construction via [[octonions]] is due to (\hyperlink{Freudenthal54}{Freudenthal 54}), one via [[quaternions]] is due to (\hyperlink{Wilson14}{Wilson 2014}).

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

\hypertarget{representation}{}\subsubsection*{{Representation}}\label{representation}

\hypertarget{SmallestFundamentalRepresentation}{}\paragraph*{{$\mathbf{56}$ -- The smallest fundamental representation}}\label{SmallestFundamentalRepresentation}

The smallest [[fundamental representation]] of $E_7$ is the defining one (from the definition \hyperlink{Definition}{above}), of [[dimension]] $56$. Under the [[special linear group|special linear]] [[subgroup]] $SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes as (e.g. \hyperlink{CacciatoriEtAl10}{Cacciatori et al. 10, section 4}, also \hyperlink{PachecoWaldram08}{Pacheco-Waldram 08, appendix B})

\begin{displaymath}
\mathbf{56} 
    \simeq 
  \mathbf{28} \oplus \mathbf{28}^\ast 
    \simeq 
   \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast
  \,.
\end{displaymath}
Under the further subgroup inclusion $SL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes further as

\begin{displaymath}
\mathbf{56}
    \simeq
  \underset{\simeq \wedge^2 \mathbb{R}^8}{\underbrace{\mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast}} 
   \oplus 
  \underset{\simeq \wedge^2 (\mathbb{R}^8)^\ast}{\underbrace{\wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7}}
  \,,
\end{displaymath}
where $\wedge^2 (\mathbb{R}^7) \subset \wedge^2 (\mathbb{R}^8)^\ast$ is regarded as the subspace of 2-forms with vanishing 8-components, and where $\wedge^6 \mathbb{R}^7$ is the [[Poincaré duality|Poincaré dual]] to the complementary subspace of $\wedge^2 (\mathbb{R}^8^\ast)$ of 2-forms with non-trivial 8-component.

This is due to \hyperlink{Cartan}{Cartan}, for review see \hyperlink{CremmerJulia79}{Cremmer-Julia 79, appendix B}, \hyperlink{PachecoWaldram08}{Pacheco-Waldram 08, B.1}.

\hypertarget{AdjointRepresentation}{}\paragraph*{{$\mathbf{133}$ -- The adjoint representation}}\label{AdjointRepresentation}

The [[adjoint representation]] $\mathbf{133}$ of $E_7$ decomposes under $SL(8,\mathbb{R})$ as (\hyperlink{PachecoWaldram08}{Pacheco-Waldram 08 (B.7)})

\begin{displaymath}
\mathfrak{e}_7
  =
  \mathbf{133}
  \simeq
  (\mathbb{R}^8 \otimes (\mathbb{R}^8)^\ast)_{traceless}
  \oplus
  \wedge^4 (\mathbb{R}^8)^\ast
  \,.
\end{displaymath}
In this decomposition the subspace corresponding to the subalgebra $\mathfrak{su}(8) \hookrightarrow \mathfrak{e}_8$ is the vector space

\begin{displaymath}
\mathfrak{su}(8)
  \simeq
  \mathfrak{so}(8)
  \oplus
  (\wedge^4 (\mathbb{R}^8)^\ast)_-
  \,,
\end{displaymath}
where the first summand denotes the skew-symmetric matrices, and the second summand the [[Hodge duality|Hodge anti-self dual]] 4-forms (\hyperlink{PachecoWaldram08}{Pacheco-Waldram 08 (B.29) (B.30) and below (2.34)}).

Under $GL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R})$ the full adjoint representation decomposes further into (\hyperlink{PachecoWaldram08}{Pacheco-Waldram 08 (B.21)})

\begin{displaymath}
\mathbf{133}
  \simeq
  \left(\mathbb{R}^7 \otimes (\mathbb{R}^7)^\ast\right)
  \oplus 
  \left(\wedge^6 \mathbb{R}^7 \oplus \wedge^6 (\mathbb{R}^7)^\ast\right)
  \oplus
  \left(
     \wedge^3 \mathbb{R}^7 \oplus \wedge^3 (\mathbb{R}^7)^\ast
  \right)
  \,.
\end{displaymath}
Here $\wedge^6 (\mathbb{R}^7)^\ast \simeq \mathbb{R}^7$ is the $(-,8)$-component of $\mathbb{R}^7 \oplus (\mathbb{R}^7)^\ast$ and dually, while the $(8,8)$-component carries no information by tracelessness; and $\wedge^3 (\mathbb{R}^7)^\ast$ is the $(-,-,-,8)$-component of $\wedge^4 (\mathbb{R}^8)^\ast$, while $\wedge^3 \mathbb{R}^7$ is the 7-dimensional [[Poincaré duality|Poincaré dual]] of the complement of the $(-,-,-,8)$-component (\hyperlink{PachecoWaldram08}{Pacheco-Waldram 08 (B.22)}).

Taken together this means that under $GL(7,\mathbb{R})$ the subspace $\mathbb{su}(8) \hookrightarrow \mathfrak{e}_8$ is that spanned by

\begin{enumerate}%
\item $\mathfrak{so}(7)$-elements;


\item sums of a 3-form with its 8d-Hodge+7d-Poincar\'e{}-dual 3-vector;


\item sums of a 6-form with its dual 6-vector



\end{enumerate}
hence is

\begin{displaymath}
\mathfrak{su}(8) 
  \simeq
  \mathfrak{so}(8) \oplus \wedge^3 \mathbb{R}^7 \oplus \wedge^6 \mathbb{R}^7
  \,.
\end{displaymath}
Hence the [[tangent space]] to the [[coset]] $E_{7(7)}/(SU(8)/\mathbb{Z}_2)$ may be identified as

\begin{displaymath}
\mathfrak{e}_7/\mathfrak{su}(8)
  \simeq
  \odot^2 (\mathbb{R}^7)^\ast  
  \oplus
  \wedge^3 (\mathbb{R}^7)^\ast
  \oplus
  \wedge^6 (\mathbb{R}^7)^\ast
  \,.
\end{displaymath}
\hypertarget{as_part_of_the_ade_pattern}{}\subsubsection*{{As part of the ADE pattern}}\label{as_part_of_the_ade_pattern}

[[!include ADE -- table]]

\hypertarget{AsUDualityGroup}{}\subsubsection*{{As U-Duality group of 4d SuGra}}\label{AsUDualityGroup}

$E_{7(7)}$ is the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] on a 7-dimensional fiber to [[4-dimensional supergravity]] (e.g. [[M-theory on G2-manifolds]]).

Specifically, (\hyperlink{Hull07}{Hull 07, section 4.4}, \hyperlink{PachecoWaldram08}{Pacheco-Waldram 08, section 2.2}) identifies the [[vector space]] underlying the $SL(7,\mathbb{R})$-decomposition of the \hyperlink{SmallestFundamentalRepresentation}{smallest fundamental representation}

\begin{displaymath}
\mathbf{56}
  \simeq
  \mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast \oplus \wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7
  \,.
\end{displaymath}
as the [[exceptional tangent bundle]]-structure to the 7-dimensional fiber space which one obtains as discussed at \emph{\href{M-theory+supersymmetry+algebra#AsAn11DimensionalBoundaryCondition}{M-theory supersymmetry algebra -- As an 11-dimensional boundary condition}}. Here $\mathbb{R}^7$ is the ordinary [[tangent space]] itself, $\wedge^2 (\mathbb{R}^\ast)^7$ is interpreted as the local incarnation of the possible [[M2-brane]] [[charges]], $\wedge^5 (\mathbb{R}^\ast)^7$ the [[M5-brane]] charges and $\wedge^6 \mathbb{R}^7$ as the charges of [[Kaluza-Klein monopoles]].

[[!include U-duality -- table]]

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

\begin{itemize}%
\item [[exceptional geometry]], [[exceptional generalized geometry]]


\item [[G2]], [[F4]],

[[E6]], \textbf{E7}, [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$



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

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

The description of the defining fundamental $\mathbf{56}$-representation of $E_{7(7)}$ is due to

\begin{itemize}%
\item [[Eli Cartan]], Thesis, in Oeuvres compl\`e{}tes T1, Part I, Gauthier-Villars, Paris 1952

\end{itemize}
and recalled for instance in

\begin{itemize}%
\item [[Eugene Cremmer]], [[Bernard Julia]], appendix B of \emph{The $SO(8)$ Supergravity}, Nucl. Phys. B 159 (1979) 141 (\href{http://inspirehep.net/record/140465?ln=en}{spire})

\end{itemize}
See also

\begin{itemize}%
\item Robert B. Brown, \emph{Groups of type $E _7$}, Jour. Reine Angew. Math. 236 (1969), 79-102.

\end{itemize}
A construction via the [[octonions]] is due to

\begin{itemize}%
\item [[Hans Freudenthal]], \emph{Beziehungen der $\mathfrak{e}_7$ und $\mathfrak{e}_8$ zur Oktavenebene}, I, II, Indag. Math. 16 (1954), 218--230, 363--368. III, IV, Indag. Math. 17 (1955), 151--157, 277--285. V --- IX, Indag. Math. 21 (1959), 165--201, 447--474. X, XI, Indag. Math. 25 (1963) 457--487 (\href{http://dspace.library.uu.nl/handle/1874/7436}{dspace})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[John Baez]], section 4.5 \emph{\href{http://math.ucr.edu/home/baez/octonions/node18.html}{E7}} of \emph{The Octonions} (\href{http://arxiv.org/abs/math/0105155}{arXiv:math/0105155})


\item Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti, \emph{E7 groups from octonionic magic square} (\href{http://arxiv.org/abs/1007.4758}{arXiv:1007.4758})



\end{itemize}
A [[quaternion|quaternionic]] construction is given in

\begin{itemize}%
\item [[Robert Wilson]], \emph{A quaternionic approach to $E_7$}, Proc. Amer. Math. Soc. 142 (2014), 867-880. doi:\href{http://dx.doi.org/10.1090/S0002-9939-2013-11838-1}{10.1090/S0002-9939-2013-11838-1}, (\href{http://www.maths.qmul.ac.uk/~raw/pubs_files/E7quat2.pdf}{pre-publication version}), (\href{http://www.maths.qmul.ac.uk/~raw/talks_files/E7quattalk2.pdf}{talk notes}).

\end{itemize}
See also

\begin{itemize}%
\item wikipedia, \emph{\href{http://en.wikipedia.org/wiki/E%E2%82%87}{E7}}

\end{itemize}
\hypertarget{in_view_of_uduality}{}\subsubsection*{{In view of U-duality}}\label{in_view_of_uduality}

The hidden [[E7]]-[[U-duality]] symmetry of the [[KK-compactification]] of [[11-dimensional supergravity]] on a 7-dimensional fiber to [[4d supergravity]] was first noticed in (\hyperlink{CremmerJulia79}{Cremmer-Julia 79}) and then expanded on in

\begin{itemize}%
\item [[Bernard de Wit]], [[Hermann Nicolai]], \emph{D = 11 Supergravity With Local SU(8) Invariance}, Nucl. Phys. B 274, 363 (1986) (\href{http://inspirehep.net/record/227409?ln=en}{spire}), \emph{Local SU(8) invariance in $d = 11$ supergravity} (\href{http://inspirehep.net/record/218601?ln=en}{spire})

\end{itemize}
The proposal to make this hidden $E_7$-symmetry manifest via [[exceptional generalized geometry]] is due to

\begin{itemize}%
\item [[Chris Hull]], \emph{Generalised Geometry for M-Theory}, JHEP 0707:079 (2007) (\href{http://arxiv.org/abs/hep-th/0701203}{arXiv:hep-th/0701203})


\item Paulo Pires Pacheco, [[Daniel Waldram]], appendix B of \emph{M-theory, exceptional generalised geometry and superpotentials} (\href{http://arxiv.org/abs/0804.1362}{arXiv:0804.1362})



\end{itemize}
Further discussion includes

\begin{itemize}%
\item [[Mariana Graña]], [[Jan Louis]], Aaron Sim, [[Daniel Waldram]], section 3.1 of \emph{$E_{7(7)}$ formulation of $N=2$ backgrounds} (\href{http://arxiv.org/abs/0904.2333}{arXiv:0904.2333})

\end{itemize}


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