\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*{G2} \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{orientation}{Orientation}\dotfill \pageref*{orientation} \linebreak \noindent\hyperlink{Dimension}{Dimension}\dotfill \pageref*{Dimension} \linebreak \noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak \noindent\hyperlink{Subgroups}{Subgroups}\dotfill \pageref*{Subgroups} \linebreak \noindent\hyperlink{supgroups}{Supgroups}\dotfill \pageref*{supgroups} \linebreak \noindent\hyperlink{coset_quotients}{Coset quotients}\dotfill \pageref*{coset_quotients} \linebreak \noindent\hyperlink{structure_and_exceptional_geometry}{$G$-Structure and exceptional geometry}\dotfill \pageref*{structure_and_exceptional_geometry} \linebreak \noindent\hyperlink{relation_to_higher_prequantum_geometry}{Relation to higher prequantum geometry}\dotfill \pageref*{relation_to_higher_prequantum_geometry} \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{applications_in_physics}{Applications in physics}\dotfill \pageref*{applications_in_physics} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[Lie group]] $G_2$ is one (or rather: three) of the [[exceptional Lie groups]]. One way to characterize it is as the [[automorphism group]] of the [[octonions]] as a [[normed algebra]]: \begin{displaymath} G_2 = Aut(\mathbb{O}) \,. \end{displaymath} Another way to characterize it is as the [[stabilizer subgroup]] inside the [[general linear group]] $GL(7)$ of the canonical [[differential n-form|differential 3-form]] $\langle ,(-)\times (-) \rangle$ on the [[Cartesian space]] $\mathbb{R}^7$ \begin{displaymath} G_2 \simeq Stab_{GL(7)}(\langle -, -\times -\rangle) \,. \end{displaymath} As such, the group $G_2$ is a higher analog of the [[symplectic group]] (which is the group that preserves a canonical 2-form on any $\mathbb{R}^{2n}$), obtained by passing from [[symplectic geometry]] to [[2-plectic geometry]]. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} \begin{defn} \label{As2PlectomorphismsOnR7}\hypertarget{As2PlectomorphismsOnR7}{} On the [[Cartesian space]] $\mathbb{R}^7$ consider the \emph{[[associative 3-form]]}, the constant [[differential n-form|differential 3-form]] $\omega \in \Omega^3(\mathbb{R}^7)$ given on tangent vectors $u,v,w \in \mathbb{R}^7$ by \begin{displaymath} \omega(u,v,w) \coloneqq \langle u , v \times w\rangle \,, \end{displaymath} where \begin{itemize}% \item $\langle -,-\rangle$ is the canonical [[bilinear form]] \item $(-)\times(-)$ is the [[cross product]] of vectors. \end{itemize} Then the group $G_2 \hookrightarrow GL(7)$ is the [[subgroup]] of the [[general linear group]] acting on $\mathbb{R}^7$ which preserves the canonical [[orientation]] and preserves this 3-form $\omega$. Equivalently, it is the subgroup preserving the orientation and the [[Hodge star operator|Hodge dual]] differential 4-form $\star \omega$. \end{defn} See for instance the introduction of (\hyperlink{Joyce}{Joyce}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{orientation}{}\subsubsection*{{Orientation}}\label{orientation} The inclusion $G_2 \hookrightarrow GL(7)$ of def. \ref{As2PlectomorphismsOnR7} factors through the [[special orthogonal group]] \begin{displaymath} G_2 \hookrightarrow SO(7) \hookrightarrow GL(7) \,. \end{displaymath} \hypertarget{Dimension}{}\subsubsection*{{Dimension}}\label{Dimension} The [[dimension]] of (the [[manifold]] underlying) $G_2$ is \begin{displaymath} dim(G_2) = 14 \,. \end{displaymath} One way to see this is via \href{octonion#ElementaryTriples}{octonionic basic triples} $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact (\href{octonion#BasicTriplesFormAutomorphism}{this proposition}) that these form a [[torsor]] over $G_2$, hence that the space of them has the same dimension as $G_2$: \begin{itemize}% \item the space of choices for $e_1$ is the 6-[[sphere]] of imaginary unit octonions; \item given that, the space of choices for $e_2$ is a 5-sphere of imaginary unit octonions orthogonal to $e_1$; \item given that, then the space of choices for $e_3$ is the [[3-sphere]] of imaginary unit octonions orthogonal to both $e_1$ and $e_2$. \end{itemize} Hence \begin{displaymath} dim(G_2) = dim(S^6) + dim(S^5) + dim(S^3) = 14 \,. \end{displaymath} (e.g. \hyperlink{Baez}{Baez, 4.1}) \hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology} The \emph{[[Dwyer-Wilkerson space]]} $G_3$ (\href{Dwyer-Wilkerson+H-space#DwyerWilkerson93}{Dwyer-Wilkerson 93}) is a [[p-adic completion|2-complete]] [[H-space]], in fact a finite [[loop space]]/[[infinity-group]], such that the mod 2 [[cohomology ring]] of its [[classifying space]]/[[delooping]] is the mod 2 [[Dickson invariants]] of rank 4. As such, it is the fourth and last space in a series of [[infinity-groups]] that starts with 3 [[compact Lie groups]]: \begin{tabular}{l|l|l|l|l} $n=$&1&2&3&4\\ \hline $DI(n)=$&[[Z/2]]&[[SO(3)]]&[[G2]]&[[G3]]\\ &\href{complex+number#AutomorphismsOfComplexNumbersIsZ2}{= Aut(C)}&\href{quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3}{= Aut(H)}&\href{octonion#AutomorphismsOfOctonionAlgebraIsG2}{= Aut(O)}&\\ \end{tabular} \hypertarget{Subgroups}{}\subsubsection*{{Subgroups}}\label{Subgroups} We discuss various [[subgroups]] of $G_2$. \begin{defn} \label{StabilizerOfQuaternions}\hypertarget{StabilizerOfQuaternions}{} Write \begin{itemize}% \item $G_2 = Aut(\mathbb{O})$, the [[automorphism group]] of the octonions as a normed alegbra, \item $Stab_{G_2}(\mathbb{H})$, the [[stabilizer subgroup]] of [[generalized the|the]] [[quaternions]] inside the octonions, i.e. of elements $\sigma\in G_2$ such that $\sigma_{|\mathbb{H}}\colon \mathbb{H}\to \mathbb{H} \hookrightarrow\mathbb{O}$; \item $Fix_{G_2}(\mathbb{H})$ for the further subgroup of elements that [[fixed point|fix]] each quaternions (the ``elementwise stabilizer group''), i.e. those $\sigma$ with $\sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}$. \end{itemize} \end{defn} \begin{prop} \label{ElementwiseStabilizerOfHIsSU2}\hypertarget{ElementwiseStabilizerOfHIsSU2}{} The elementwise stabilizer group of the [[quaternions]] is [[SU(2)]]: \begin{displaymath} Fix_{G_2}(\mathbb{H}) \simeq SU(2) \,. \end{displaymath} \end{prop} \begin{proof} Consider \href{octonion#ElementaryTriples}{octonionic basic triples} $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact (\href{octonion#BasicTriplesFormAutomorphism}{this proposition}) that these form a [[torsor]] over $G_2$. The choice of $(e_1,e_2)$ is equivalently a choice of inclusion $\mathbb{H} \hookrightarrow \mathbb{O}$. Then the remaining space of choices for $e_3$ is the [[3-sphere]] (the space of unit imaginary octonions orthogonal to both $e_1$ and $e_2$). This carries a unit group structure, and by the torsor property this is the required subgroup of $SU(2)$. \end{proof} \begin{prop} \label{StabilizingAndFixingTheQuaternions}\hypertarget{StabilizingAndFixingTheQuaternions}{} The subgroups in def. \ref{StabilizerOfQuaternions} sit in a [[short exact sequence]] of the form \begin{displaymath} \itexarray{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) & \simeq & SO(4) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 } \end{displaymath} exhibiting [[special orthogonal group|SO(4)]] as a [[group extension]] of the [[special orthogonal group]] $SO(3)$ by the [[special unitary group]] $SU(2)$. \end{prop} (e.g. \hyperlink{Ferolito}{Ferolito, section 4}) Furthermore there is a [[subgroup]] $SU(3) \hookrightarrow G_2$ whose [[intersection]] with $SO(4)$ is $U(2)$. The [[simple Lie group|simple]] part $SU(2)$ of this intersection is a [[normal subgroup]] of $SO(4)$. (see e.g. \hyperlink{Miyaoka93}{Miyaoka 93}) The [[coset space]] [[G2/SU(3) is the 6-sphere]]. See there for more. (from \hyperlink{Kramer02}{Kramer 02}) The [[Weyl group]] of $G_2$ is the [[dihedral group]] of [[order of a group|order]] 12. (see e.g. \hyperlink{Ishiguro}{Ishiguro, p. 3}). $\,$ \hypertarget{supgroups}{}\subsubsection*{{Supgroups}}\label{supgroups} \begin{prop} \label{QuotientOfSpin7ByG2IsS7}\hypertarget{QuotientOfSpin7ByG2IsS7}{} \textbf{([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])} Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]), \begin{enumerate}% \item This action is is [[transitive action|transitive]]; \item the [[stabilizer group]] of any point on $S^7$ is [[G2]]; \item all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other. \end{enumerate} Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]] \begin{displaymath} Spin(7)/G_2 \;\simeq\; S^7 \,. \end{displaymath} \end{prop} (e.g \hyperlink{Varadarajan01}{Varadarajan 01, Theorem 3}) \hypertarget{coset_quotients}{}\subsubsection*{{Coset quotients}}\label{coset_quotients} [[!include coset space structure on n-spheres -- table]] \hypertarget{structure_and_exceptional_geometry}{}\subsubsection*{{$G$-Structure and exceptional geometry}}\label{structure_and_exceptional_geometry} [[!include Spin(8)-subgroups and reductions -- table]] \hypertarget{relation_to_higher_prequantum_geometry}{}\subsubsection*{{Relation to higher prequantum geometry}}\label{relation_to_higher_prequantum_geometry} The 3-form $\omega$ from def. \ref{As2PlectomorphismsOnR7} we may regard as equipping $\mathbb{R}^7$ with [[n-plectic geometry|2-plectic structure]]. From this point of view $G_2$ is the linear subgroup of the [[2-plectomorphism group]], hence (up to the translations) the image of the [[Heisenberg group]] of $(\mathbb{R}^7, \omega)$ in the symplectomorphism group. Or, dually, we may regard the 4-form $\star \omega$ of def. \ref{As2PlectomorphismsOnR7} as being a [[n-plectic geometry|3-plectic structure]] and $G_2$ correspondingly as the linear part in the [[3-plectomorphism group]] of $\mathbb{R}^7$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[G2 manifold]], [[generalized G2-manifold]] \item [[M-theory on G2-manifolds]], [[G2-MSSM]] \item \textbf{G2}, [[F4]], [[E6]], [[E7]], [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$ \end{itemize} [[!include special holonomy table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 2 of \emph{Octonions, Jordan Algebras, and Exceptional Groups}, Springer Monographs in Mathematics, 2000 \end{itemize} Surveys are in \begin{itemize}% \item Spiro Karigiannis, \emph{What is\ldots{} a $G_2$-manifold} (\href{http://www.ams.org/notices/201104/rtx110400580p.pdf}{pdf}) \item [[Simon Salamon]], \emph{A tour of exceptional geometry}, (\href{http://calvino.polito.it/~salamon/G/COR/tour.pdf}{pdf}) \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/G2_%28mathematics%29}{G2}} . \end{itemize} The definitions are reviewed for instance in \begin{itemize}% \item [[Dominic Joyce]], \emph{Compact Riemannian 7-manifolds with holonomy $G_2$}, Journal of Differential Geometry vol 43, no 2 (\href{http://www.intlpress.com/JDG/archive/1996/43-2-291.pdf}{pdf}) \item Ferolito \emph{The octonions and $G_2$} (\href{https://www2.bc.edu/~reederma/Ferolito.pdf}{pdf}) \item [[John Baez]], section 4.1 \emph{\href{http://math.ucr.edu/home/baez/octonions/node14.html}{G2}}, of \emph{The Octonions} (\href{http://arxiv.org/abs/math/0105155}{arXiv:math/0105155}) \item Ruben Arenas, \emph{Constructing a Matrix Representation of the Lie Group $G_2$}, 2005 (\href{https://www.math.hmc.edu/seniorthesis/archives/2005/rarenas/rarenas-2005-thesis.pdf}{pdf}) \end{itemize} Discussion in terms of the [[Heisenberg group]] in [[2-plectic geometry]] is in \begin{itemize}% \item Alberto Ibort, \emph{Multisymplectic geometry: generic and exceptional}, \emph{\href{http://rsme.es/public/publi3.htm}{Proceedings of the IX Fall workshop on geometry and physics}} ([[IbortMultisymplectic.pdf:file]]) \end{itemize} A description of the root space decomposition of the [[Lie algebra]] $\mathfrak{g}_2$ is in \begin{itemize}% \item Tathagata Basak, \emph{Root space decomposition of $\mathfrak{g}_2$ from octonions}, arXiv:\href{https://arxiv.org/abs/1708.02367}{1708.02367} \end{itemize} Cohomological properties are discussed in \begin{itemize}% \item Younggi Choi, \emph{Homology of the gauge group of exceptional Lie group $G_2$}, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699--709 \end{itemize} Discussion of [[subgroups]] includes \begin{itemize}% \item [[Reiko Miyaoka]], \emph{The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces}, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (\href{http://projecteuclid.org/euclid.ojm/1200784357}{Euclid}) \item Kenshi Ishiguro, \emph{Classifying spaces and a subgroup of the exceptional Lie group $G_2$} \href{http://hopf.math.purdue.edu/Ishiguro/G2.pdf}{pdf} \item Linus Kramer, 4.27 of \emph{Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces}, AMS 2002 \end{itemize} Discussion of $G_2$ as a subgroup of [[Spin(7)]]: \begin{itemize}% \item [[Veeravalli Varadarajan]], \emph{Spin(7)-subgroups of SO(8) and Spin(8)}, Expositiones Mathematicae, 19 (2001): 163-177 (\href{https://core.ac.uk/download/pdf/81114499.pdf}{pdf}) \end{itemize} \hypertarget{applications_in_physics}{}\subsubsection*{{Applications in physics}}\label{applications_in_physics} Discussion of [[Yang-Mills theory]] with $G_2$ as [[gauge group]] is in \begin{itemize}% \item Ernst-Michael Ilgenfritz, Axel Maas, \emph{Topological aspects of $G_2$ Yang-Mills theory} (\href{http://arxiv.org/abs/1210.5963}{arXiv:1210.5963}) \end{itemize} \end{document}