\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

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

\begin{document}

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


\section*{E11}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_uduality_group_of_0d_supergravity}{As U-duality group of 0d supergravity}\dotfill \pageref*{as_uduality_group_of_0d_supergravity} \linebreak
\noindent\hyperlink{FundamentalRepresentationAndBraneCharges}{Fundamental representation and brane charges}\dotfill \pageref*{FundamentalRepresentationAndBraneCharges} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesRelationToSupergravity}{Relation to supergravity}\dotfill \pageref*{ReferencesRelationToSupergravity} \linebreak


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

A hyperbolic [[Kac-Moody Lie algebra]] in the E-series

\ldots{} [[E6]], [[E7]], [[E8]], [[E9]], [[E10]], $E_{11}$, \ldots{}

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

\hypertarget{as_uduality_group_of_0d_supergravity}{}\subsubsection*{{As U-duality group of 0d supergravity}}\label{as_uduality_group_of_0d_supergravity}

$E_{11}$ is conjectured (\hyperlink{West01}{West 01}) to be the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 0 dimensions.

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

\hypertarget{FundamentalRepresentationAndBraneCharges}{}\subsubsection*{{Fundamental representation and brane charges}}\label{FundamentalRepresentationAndBraneCharges}

The first [[fundamental representation]] of $E_{11}$, usually denoted $l_1$, is argued, since (\hyperlink{West04}{West 04}), to contain in its decomposition into [[representations]] of $GL(11)$ the representations in which the [[charges]] of the [[M-branes]] and other [[BPS states]] transform.

According to (\hyperlink{NicolaiFischbacher03}{Nicolai-Fischbacher 03, first three rows of table 2 on p. 72}, \hyperlink{West04}{West 04}, \hyperlink{KleinschmidtWest04}{Kleinschmidt-West 04}) and concisely stated for instance in (\hyperlink{West11}{West 11, (2.17)}), the level decomposition of $l_1$ under $GL(11)$ starts out as so:

\begin{displaymath}
l_1 
  \simeq
  \underset{level\,0}{
  \underbrace{
    \mathbb{R}^{10,1}
  }}
  \oplus
  \underset{level\,1}{
  \underbrace{
    \wedge^2 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus
  \underset{level\,2}{
  \underbrace{
    \wedge^5 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus
  \underset{level\,3}{\underbrace{
    \wedge^7 (\mathbb{R}^{10,1})^\ast \otimes_s (\mathbb{R}^{10,1})^\ast
    \oplus
    \wedge^8 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus 
  \cdots
\end{displaymath}
Here the $level \leq 2$-truncation happens to coincide with the bosonic [[body]] underlying the [[M-theory super Lie algebra]] and via the relation of that to [[BPS charges]] in [[11-dimensional supergravity]]/[[M-theory]], the [[direct sums|direct summands]] here have been argued to naturally correspond to

\begin{itemize}%
\item level 0: [[momentum]]


\item level 1: [[M2-brane]] charge


\item level 2: [[M5-brane]] charge


\item level 3: [[dual graviton]] charge (\hyperlink{West11}{West 11, section N}) (has two components \footnote{private communication with [[Axel Kleinschmidt]]} )



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

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

\begin{itemize}%
\item [[Peter West]], section 16.7 of \emph{[[Introduction to Strings and Branes]]}


\item [[Hermann Nicolai]], Thomas Fischbacher, \emph{Low Level Representations for E10 and E11} (\href{http://arxiv.org/abs/hep-th/0301017}{arXiv:hep-th/0301017})


\item H. Mkrtchyan, R. Mkrtchyan, \emph{$E_{11},K_{11}$ and $EE_{11}$} (\href{http://arxiv.org/abs/hep-th/0407148}{arXiv:hep-th/0407148})



\end{itemize}
\hypertarget{ReferencesRelationToSupergravity}{}\subsubsection*{{Relation to supergravity}}\label{ReferencesRelationToSupergravity}

Literature discussing $E_{11}$ [[U-duality]] and in the context of [[exceptional generalized geometry]] of [[11-dimensional supergravity]].

Review includes

\begin{itemize}%
\item [[Peter West]], section 17.5 of \emph{[[Introduction to Strings and Branes]]}


\item [[Fabio Riccioni]], \emph{$E_{11}$ and M-theory}, talk at \href{http://www.ift.uam.es/strings07/}{Strings07} (\href{http://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2007/riccioni.pdf}{pdf slides})


\item [[Fabio Riccioni]], [[Peter West]], \emph{The $E_{11}$ origin of all maximal supergravities}, JHEP 0707:063,2007 (\href{http://arxiv.org/abs/0705.0752}{arXiv:0705.0752}, \href{http://inspirehep.net/record/749966/}{spire})


\item [[Paul Cook]], \emph{Connections between Kac-Moody algebras and M-theory} PhD thesis (\href{http://arxiv.org/abs/0711.3498}{arXiv:0711.3498})


\item [[Peter West]], \emph{A brief review of E theory} (\href{http://arxiv.org/abs/1609.06863}{arXiv:1609.06863})



\end{itemize}
Original articles include the following:

The observation that $E_{11}$ seems to neatly organize the structures in [[11-dimensional supergravity]]/[[M-theory]] is due to

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$ and M theory}, Class. Quant. Grav., 18:4443--4460, 2001. (\href{http://arxiv.org/abs/hep-th/0104081}{arXiv:hep-th/0104081})

\end{itemize}
A precursor to (\hyperlink{West01}{West 01}) is

\begin{itemize}%
\item [[Bernard Julia]], \emph{Dualities in the classical supergravity limits} (\href{http://arxiv.org/abs/hep-th/9805083}{arXiv:hep-th/9805083})

\end{itemize}
as explained in (\hyperlink{HenneauxJuliaLevie10}{Henneaux-Julia-Levie 10}).

The derivation of the [[equations of motion]] of [[11-dimensional supergravity]] and maximally supersymmetric [[5d supergravity]] from a [[vielbein]] with values in the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] is due to

\begin{itemize}%
\item [[Peter West]], \emph{Generalised geometry, eleven dimensions and $E_{11}$}, J. High Energ. Phys. (2012) 2012: 18 (\href{http://arxiv.org/abs/1111.1642}{arXiv:1111.1642})


\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ must be a symmetry of strings and branes}, Physics Letters B Volume 759, 10 August 2016, Pages 663--671 (\href{https://arxiv.org/abs/1512.01644}{arXiv:1512.01644})


\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ in $11d$}, Physics Letters B Volume 758, 10 July 2016, Pages 278--285 (\href{https://arxiv.org/abs/1601.03974}{arXiv:1601.03974})



\end{itemize}
This way that elements of [[cosets]] of the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] may encode [[equations of motion]] of [[11-dimensional supergravity]] follows previous considerations for [[Einstein equations]] in

\begin{itemize}%
\item [[Abdus Salam]], J. Strathdee, \emph{Nonlinear realizations. 1: The Role of Goldstone bosons, Phys. Rev. 184 (1969) 1750,}


\item [[Chris Isham]], [[Abdus Salam]], J. Strathdee, \emph{Spontaneous, breakdown of conformal symmetry}, Phys. Lett. 31B (1970) 300.


\item A. Borisov, V. Ogievetsky, \emph{Theory of dynamical affine and conformal symmetries as the theory of the gravitational field}, Theor. Math. Phys. 21 (1973) 1179-1188 (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3902&option_lang=eng}{web})


\item V. Ogievetsky, \emph{Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups}, Nuovo. Cimento, 8 (1973) 988.



\end{itemize}
Further developments of the proposed $E_{11}$ formulation of [[M-theory]] include

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$, ten forms and supergravity}, JHEP0603:072,2006 (\href{http://arxiv.org/abs/hep-th/0511153}{arXiv:hep-th/0511153})


\item [[Fabio Riccioni]], [[Peter West]], \emph{Dual fields and $E_{11}$}, Phys.Lett.B645:286-292,2007 (\href{http://arxiv.org/abs/hep-th/0612001}{arXiv:hep-th/0612001})


\item [[Fabio Riccioni]], [[Peter West]], \emph{E(11)-extended spacetime and gauged supergravities}, JHEP 0802:039,2008 (\href{http://arxiv.org/abs/0712.1795}{arXiv:0712.1795})


\item [[Fabio Riccioni]], Duncan Steele, [[Peter West]], \emph{The E(11) origin of all maximal supergravities - the hierarchy of field-strengths}, JHEP 0909:095 (2009) (\href{http://arxiv.org/abs/0906.1177}{arXiv:0906.1177})


\item [[Eric Bergshoeff]], I. De Baetselier, T. Nutma, \emph{E(11) and the Embedding Tensor} (\href{http://arxiv.org/abs/0705.1304}{arXiv:0705.1304}, \href{http://mms.technologynetworks.net/posters/0364.pdf}{poster})


\item Guillaume Bossard, [[Axel Kleinschmidt]], Jakob Palmkvist, [[Christopher Pope]], [[Ergin Sezgin]], \emph{Beyond $E_{11}$} (\href{https://arxiv.org/abs/1703.01305}{arXiv:1703.01305})



\end{itemize}
Discussion of the [[semidirect product]] of $E_{11}$ with its $l_1$-[[representation]], and arguments that the [[charges]] of the [[M-theory super Lie algebra]] and in fact further brane charges may be identified inside $l_1$ originate in

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$, $SL(32)$ and Central Charges}, Phys.Lett.B575:333- 342,2003 (\href{http://arxiv.org/abs/hep-th/0307098}{arXiv:hep-th/0307098})

\end{itemize}
and was further explored in

\begin{itemize}%
\item [[Axel Kleinschmidt]], [[Peter West]], \emph{Representations of $G^{+++}$ and the role of space-time}, JHEP 0402 (2004) 033 (\href{http://arxiv.org/abs/hep-th/0312247}{arXiv:hep-th/0312247})


\item [[Paul Cook]], [[Peter West]], \emph{Charge multiplets and masses for $E(11)$}, JHEP 11 (2008) 091 (\href{http://arxiv.org/abs/0805.4451}{arXiv:0805.4451})


\item [[Peter West]], \emph{$E_{11}$ origin of Brane charges and U-duality multiplets}, JHEP 0408 (2004) 052 (\href{http://arxiv.org/abs/hep-th/0406150}{arXiv:hep-th/0406150})



\end{itemize}
Relation to [[exceptional field theory]] is discussed in

\begin{itemize}%
\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ and exceptional field theory} (\href{http://arxiv.org/abs/1507.08912}{arXiv:1507.08912})

\end{itemize}
Relation to [[Borcherds superalgebras]] is discussed in

\begin{itemize}%
\item Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, \emph{Borcherds symmetries in M-theory}, JHEP 0204 (2002) 049 (\href{http://arxiv.org/abs/hep-th/0203070}{arXiv:hep-th/0203070})


\item [[Marc Henneaux]], [[Bernard Julia]], J\'e{}r\^o{}me Levie, \emph{$E_{11}$, Borcherds algebras and maximal supergravity} (\href{http://arxiv.org/abs/1007.5241}{arxiv:1007.5241})


\item Jakob Palmkvist, \emph{Tensor hierarchies, Borcherds algebras and $E_{11}$}, JHEP 1202 (2012) 066 (\href{http://arxiv.org/abs/1110.4892}{arXiv:1110.4892})



\end{itemize}


\end{document}