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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> G₂ </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4609/#Item_16" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="exceptional_structures">Exceptional structures</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/exceptional+structures">exceptional structures</a></strong>, <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphisms">exceptional isomorphisms</a></p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+finite+groups">exceptional finite groups</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monster+group">monster group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Conway+group">Conway group</a></p> </li> </ul> </li> <li> <p>exceptional <a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tetrahedral+group">tetrahedral group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/octahedral+group">octahedral group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icosahedral+group">icosahedral group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/F%E2%82%84">F₄</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E%E2%82%86">E₆</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%87">E₇</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a></p> </li> </ul> <p>and <a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/E%E2%82%89">E₉</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%80">E₁₀</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%81">E₁₁</a>, …</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dwyer-Wilkerson+H-space">Dwyer-Wilkerson H-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Lie+algebras">exceptional Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Jordan+algebra">exceptional Jordan algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Albert+algebra">Albert algebra</a></li> </ul> </li> <li> <p>exceptional <a class="existingWikiWord" href="/nlab/show/Jordan+superalgebra">Jordan superalgebra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>10</mn></msub></mrow><annotation encoding="application/x-tex">K_10</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E%E2%82%88+lattice">E₈ lattice</a>, <a class="existingWikiWord" href="/nlab/show/Leech+lattice">Leech lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cayley+plane">Cayley plane</a></p> </li> </ul> <h2 id="interrelations">Interrelations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry+and+division+algebras">supersymmetry and division algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+magic+square">Freudenthal magic square</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moonshine">moonshine</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+moonshine">Mathieu moonshine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/umbral+moonshine">umbral moonshine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/O%27Nan+moonshine">O'Nan moonshine</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+field+theory">exceptional field theory</a></p> </li> </ul> <h2 id="philosophy">Philosophy</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/universal+exceptionalism">universal exceptionalism</a></li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> <h4 id="lie_theory">Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#Orientation'>Orientation</a></li> <li><a href='#Dimension'>Dimension</a></li> <li><a href='#cohomology'>Cohomology</a></li> <li><a href='#Subgroups'>Subgroups</a></li> <li><a href='#supgroups'>Supgroups</a></li> <li><a href='#coset_quotients'>Coset quotients</a></li> <li><a href='#structure_and_exceptional_geometry'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure and exceptional geometry</a></li> <li><a href='#relation_to_higher_prequantum_geometry'>Relation to higher prequantum geometry</a></li> <li><a href='#as_zerodivisors_of_the_sedenions'>As zero-divisors of the sedenions</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#applications_in_physics'>Applications in physics</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> is one (or rather: three) of the <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a>. One way to characterize it is as the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a> as a <a class="existingWikiWord" href="/nlab/show/normed+algebra">normed algebra</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>=</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>𝕆</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_2 = Aut(\mathbb{O}) \,. </annotation></semantics></math></div> <p>Another way to characterize it is as the <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> inside the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(7)</annotation></semantics></math> of the canonical <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential 3-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle ,(-)\times (-) \rangle</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>≃</mo><msub><mi>Stab</mi> <mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_2 \simeq Stab_{GL(7)}(\langle -, -\times -\rangle) \,. </annotation></semantics></math></div> <p>As such, the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> is a higher analog of the <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> (which is the group that preserves a canonical 2-form on any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math>), obtained by passing from <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> to <a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a>.</p> <h2 id="Definition">Definition</h2> <div class="num_defn" id="As2PlectomorphismsOnR7"> <h6 id="definition_2">Definition</h6> <p>On the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math> consider the <em><a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a></em>, the constant <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential 3-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>7</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^3(\mathbb{R}^7)</annotation></semantics></math> given on tangent vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">u,v,w \in \mathbb{R}^7</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">⟨</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>×</mo><mi>w</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega(u,v,w) \coloneqq \langle u , v \times w\rangle \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math> is the canonical <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\times(-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cross+product">cross product</a> of vectors.</p> </li> </ul> <p>Then the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_2 \hookrightarrow GL(7)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> acting on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math> which preserves the canonical <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> and preserves this 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>. Equivalently, it is the subgroup preserving the orientation and the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge dual</a> differential 4-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\star \omega</annotation></semantics></math>.</p> </div> <p>See for instance the introduction of <a href="#Joyce96">Joyce 1996</a>.</p> <h2 id="properties">Properties</h2> <h3 id="Orientation">Orientation</h3> <p>The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_2 \hookrightarrow GL(7)</annotation></semantics></math> of def. <a class="maruku-ref" href="#As2PlectomorphismsOnR7"></a> factors through the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>↪</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_2 \hookrightarrow SO(7) \hookrightarrow GL(7) \,. </annotation></semantics></math></div> <h3 id="Dimension">Dimension</h3> <p>The <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of (the <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>14</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim(G_2) = 14 \,. </annotation></semantics></math></div> <p>One way to see this is via <a href="octonion#ElementaryTriples">octonionic basic triples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>𝕆</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">(e_1, e_2, e_3) \in \mathbb{O}^3</annotation></semantics></math> and the fact (<a href="octonion#BasicTriplesFormAutomorphism">this proposition</a>) that these form a <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>, hence that the space of them has the same dimension as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>:</p> <ul> <li> <p>the space of choices for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e_1</annotation></semantics></math> is the 6-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> of imaginary unit octonions;</p> </li> <li> <p>given that, the space of choices for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">e_2</annotation></semantics></math> is a 5-sphere of imaginary unit octonions orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e_1</annotation></semantics></math>;</p> </li> <li> <p>given that, then the space of choices for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">e_3</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> of imaginary unit octonions orthogonal to both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">e_2</annotation></semantics></math>.</p> </li> </ul> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>dim</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>6</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>dim</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>5</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>dim</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>14</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim(G_2) = dim(S^6) + dim(S^5) + dim(S^3) = 14 \,. </annotation></semantics></math></div> <p>(e.g. <a href="#Baez">Baez, 4.1</a>)</p> <h3 id="cohomology">Cohomology</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/Dwyer-Wilkerson+space">Dwyer-Wilkerson space</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">G_3</annotation></semantics></math> (<a href="Dwyer-Wilkerson+H-space#DwyerWilkerson93">Dwyer-Wilkerson 93</a>) is a <a class="existingWikiWord" href="/nlab/show/p-adic+completion">2-complete</a> <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a>, in fact a finite <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>/<a class="existingWikiWord" href="/nlab/show/infinity-group">infinity-group</a>, such that the mod 2 <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> of its <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>/<a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> is the mod 2 <a class="existingWikiWord" href="/nlab/show/Dickson+invariants">Dickson invariants</a> of rank 4. As such, it is the fourth and last space in a series of <a class="existingWikiWord" href="/nlab/show/infinity-groups">infinity-groups</a> that starts with 3 <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a>:</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">n=</annotation></semantics></math></th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DI</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">DI(n)=</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Z%2F2">Z/2</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%283%29">SO(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G2">G2</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G3">G3</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a href="complex+number#AutomorphismsOfComplexNumbersIsZ2">= Aut(C)</a></td><td style="text-align: left;"><a href="quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3">= Aut(H)</a></td><td style="text-align: left;"><a href="octonion#AutomorphismsOfOctonionAlgebraIsG2">= Aut(O)</a></td><td style="text-align: left;"></td></tr> </tbody></table> <h3 id="Subgroups">Subgroups</h3> <p>We discuss various <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>.</p> <div class="num_defn" id="StabilizerOfQuaternions"> <h6 id="definition_3">Definition</h6> <p>Write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>=</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>𝕆</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_2 = Aut(\mathbb{O})</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of the octonions as a normed alegbra,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Stab</mi> <mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab_{G_2}(\mathbb{H})</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> inside the octonions, i.e. of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma\in G_2</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mrow><mo stretchy="false">|</mo><mi>ℍ</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>ℍ</mi><mo>→</mo><mi>ℍ</mi><mo>↪</mo><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\sigma_{|\mathbb{H}}\colon \mathbb{H}\to \mathbb{H} \hookrightarrow\mathbb{O}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fix</mi> <mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fix_{G_2}(\mathbb{H})</annotation></semantics></math> for the further subgroup of elements that <a class="existingWikiWord" href="/nlab/show/fixed+point">fix</a> each quaternions (the “elementwise stabilizer group”), i.e. those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mrow><mo stretchy="false">|</mo><mi>ℍ</mi></mrow></msub><mo>=</mo><msub><mi>id</mi> <mi>ℍ</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_prop" id="ElementwiseStabilizerOfHIsSU2"> <h6 id="proposition">Proposition</h6> <p>The elementwise stabilizer group of the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> is <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Fix</mi> <mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Fix_{G_2}(\mathbb{H}) \simeq SU(2) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Consider <a href="octonion#ElementaryTriples">octonionic basic triples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>𝕆</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">(e_1, e_2, e_3) \in \mathbb{O}^3</annotation></semantics></math> and the fact (<a href="octonion#BasicTriplesFormAutomorphism">this proposition</a>) that these form a <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>. The choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e_1,e_2)</annotation></semantics></math> is equivalently a choice of inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi><mo>↪</mo><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{H} \hookrightarrow \mathbb{O}</annotation></semantics></math>. Then the remaining space of choices for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">e_3</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> (the space of unit imaginary octonions orthogonal to both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">e_2</annotation></semantics></math>). This carries a unit group structure, and by the torsor property this is the required subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="StabilizingAndFixingTheQuaternions"> <h6 id="proposition_2">Proposition</h6> <p>The subgroups in def. <a class="maruku-ref" href="#StabilizerOfQuaternions"></a> sit in a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mo>=</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Fix</mi> <mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Stab</mi> <mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Aut</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mo>=</mo></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 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 } </annotation></semantics></math></div> <p>exhibiting <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO(4)</a> as a <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(3)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#Ferolito">Ferolito, section 4</a>, see also at <em><a href="SO%283%29#irreducible_representations">SO(3) – Irreps</a></em>)</p> <p>Furthermore there is a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">SU(3) \hookrightarrow G_2</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(4)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(2)</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple</a> part <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math> of this intersection is a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(4)</annotation></semantics></math>.</p> <p>(see e.g. <a href="#Miyaoka93">Miyaoka 93</a>)</p> <p>The <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <a class="existingWikiWord" href="/nlab/show/G2%2FSU%283%29+is+the+6-sphere">G2/SU(3) is the 6-sphere</a>. See there for more.</p> <p><img src="https://ncatlab.org/nlab/files/3dSubgroupsOfG2.jpg" width="700" /></p> <p>(from <a href="#Kramer02">Kramer 02</a>)</p> <p>The <a class="existingWikiWord" href="/nlab/show/Weyl+group">Weyl group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/dihedral+group">dihedral group</a> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> 12. (see e.g. <a href="#Ishiguro">Ishiguro, p. 3</a>).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="supgroups">Supgroups</h3> <div class="num_prop" id="QuotientOfSpin7ByG2IsS7"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> by <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a> is <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>)</strong></p> <p>Consider the canonical <a class="existingWikiWord" href="/nlab/show/action">action</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> on the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^8</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>),</p> <ol> <li> <p>This action is is <a class="existingWikiWord" href="/nlab/show/transitive+action">transitive</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/stabilizer+group">stabilizer group</a> of any point on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a>;</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a>-subgroups of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> arise this way, and are all <a class="existingWikiWord" href="/nlab/show/conjugate+subgroup">conjugate</a> to each other.</p> </li> </ol> <p>Hence the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> by <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a> is the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>7</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spin(7)/G_2 \;\simeq\; S^7 \,. </annotation></semantics></math></div></div> <p>(e.g <a href="#Varadarajan01">Varadarajan 01, Theorem 3</a>)</p> <h3 id="coset_quotients">Coset quotients</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>-<a class="existingWikiWord" href="/nlab/show/structures">structures</a> on <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>:</strong></p> <table><thead><tr><th><strong>standard:</strong></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{n-1} \simeq_{diff} SO(n)/SO(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#nSphereAsCosetSpace">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#OddDimSphereAsSpecialUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#SphereAsSymplecticUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional</a>:</strong></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(7)/G_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29%2FG%E2%82%82+is+the+7-sphere">Spin(7)/G₂ is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(6)/SU(3)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SU%284%29">SU(4)</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(5)/SU(2)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Sp%282%29">Sp(2)</a> is <a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a> and <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> is <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>, see <a class="existingWikiWord" href="/nlab/show/Spin%285%29%2FSU%282%29+is+the+7-sphere">Spin(5)/SU(2) is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>6</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^6 \simeq_{diff} G_2/SU(3)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G%E2%82%82%2FSU%283%29+is+the+6-sphere">G₂/SU(3) is the 6-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>15</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^15 \simeq_{diff} Spin(9)/Spin(7)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29%2FSpin%287%29+is+the+15-sphere">Spin(9)/Spin(7) is the 15-sphere</a></td></tr> </tbody></table> <p>see also <em><a class="existingWikiWord" href="/nlab/show/Spin%288%29-subgroups+and+reductions+--+table">Spin(8)-subgroups and reductions</a></em></p> <p id="HomotopyTheoretic"> <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>:</p> <p><img src="/nlab/files/ExceptionalSpheres.jpg" width="730px" /></p> <p>(from <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation">FSS 19, 3.4</a>)</p> </div> <h3 id="structure_and_exceptional_geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure and exceptional geometry</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a>-<a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> and <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+group">reductions</a> to <a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/reduction+of+structure+group">reduction</a></th><th>from <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th>to <a class="existingWikiWord" href="/nlab/show/maximal+subgroup">maximal</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29-structure">Spin(7)-structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-structure">G₂-structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">CY3-structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%283%29">SU(3)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%282%29-structure">SU(2)-structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized</a> <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+group">reduction</a></strong></td><td style="text-align: left;"><strong>from <a class="existingWikiWord" href="/nlab/show/Narain+group">Narain group</a></strong></td><td style="text-align: left;"><strong>to <a class="existingWikiWord" href="/nlab/show/direct+product+group">direct product group</a></strong></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+Spin%287%29-structure">generalized Spin(7)-structure</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>8</mn><mo>,</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(8,8)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(7) \times Spin(7)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+G%E2%82%82-structure">generalized G₂-structure</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo>,</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(7,7)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2 \times G_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized CY3</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(6,6)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3) \times SU(3)</annotation></semantics></math></td></tr> </tbody></table> <p>see also: <em><a class="existingWikiWord" href="/nlab/show/coset+space+structure+on+n-spheres+--+table">coset space structure on n-spheres</a></em></p> </div> <h3 id="relation_to_higher_prequantum_geometry">Relation to higher prequantum geometry</h3> <p>The 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> from def. <a class="maruku-ref" href="#As2PlectomorphismsOnR7"></a> we may regard as equipping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">2-plectic structure</a>. From this point of view <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> is the linear subgroup of the <a class="existingWikiWord" href="/nlab/show/2-plectomorphism+group">2-plectomorphism group</a>, hence (up to the translations) the image of the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>7</mn></msup><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R}^7, \omega)</annotation></semantics></math> in the symplectomorphism group.</p> <p>Or, dually, we may regard the 4-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\star \omega</annotation></semantics></math> of def. <a class="maruku-ref" href="#As2PlectomorphismsOnR7"></a> as being a <a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">3-plectic structure</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> correspondingly as the linear part in the <a class="existingWikiWord" href="/nlab/show/3-plectomorphism+group">3-plectomorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math>.</p> <h3 id="as_zerodivisors_of_the_sedenions">As zero-divisors of the sedenions</h3> <p>It is shown in Corollary 2.14 of <a href="#Moreno97">Moreno (1997)</a> that the unit-norm <a class="existingWikiWord" href="/nlab/show/zero-divisor">zero-divisors</a> of the <a class="existingWikiWord" href="/nlab/show/sedenion">sedenions</a> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>. (More precisely, the smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>p</mi><mi>q</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo><mo>=</mo><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">|</mo><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(p, q) \mid p q = 0, |p| = |q| = 1\}</annotation></semantics></math> is diffeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>. This can also be improved to an isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-torsors.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifold">G₂ manifold</a>, <a class="existingWikiWord" href="/nlab/show/generalized+G%E2%82%82-manifold">generalized G₂-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>, <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> <li> <p><strong>G₂</strong>, <a class="existingWikiWord" href="/nlab/show/F%E2%82%84">F₄</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/E%E2%82%86">E₆</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%87">E₇</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%89">E₉</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%80">E₁₀</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%81">E₁₁</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></p> </li> </ul> <div> <p><strong>classification of <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> by <a class="existingWikiWord" href="/nlab/show/Berger%27s+theorem">Berger's theorem</a>:</strong></p> <table><thead><tr><th></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>preserved <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{C}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unitary+group">U(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+forms">Kähler forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\omega_2\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{H}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternionic+K%C3%A4hler+manifold">quaternionic Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).Sp(1)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/hyper-K%C3%A4hler+manifold">hyper-Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/symplectic+group">Sp(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ω</mi><mo>=</mo><mi>a</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>c</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a^2 + b^2 + c^2 = 1</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{O}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29+manifold">Spin(7) manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>8<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cayley+form">Cayley form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifold">G₂ manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>7</mn><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,7\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="SpringerVeltkamp00"><a class="existingWikiWord" href="/nlab/show/Tonny+Springer">Tonny Springer</a>, <a class="existingWikiWord" href="/nlab/show/Ferdinand+Veldkamp">Ferdinand Veldkamp</a>, chapter 2 of <em>Octonions, Jordan Algebras, and Exceptional Groups</em>, Springer Monographs in Mathematics, 2000</li> </ul> <p>Surveys are in</p> <ul> <li> <p>Spiro Karigiannis, <em>What is… a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifold</em> (<a href="http://www.ams.org/notices/201104/rtx110400580p.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Salamon">Simon Salamon</a>, <em>A tour of exceptional geometry</em>, (<a href="http://calvino.polito.it/~salamon/G/COR/tour.pdf">pdf</a>)</p> </li> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/G2_%28mathematics%29">G₂</a></em> .</p> </li> </ul> <p>The definitions are reviewed for instance in</p> <ul> <li id="Joyce96"> <p><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, <em>Compact Riemannian 7-manifolds with holonomy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math></em>, Journal of Differential Geometry <strong>43</strong> 2 (1996) [<a href="https://projecteuclid.org/journals/journal-of-differential-geometry/volume-43/issue-2/Compact-Riemannian-7-manifolds-with-holonomy-G_2-I/10.4310/jdg/1214458109.full">doi:10.4310/jdg/1214458109</a>, <a href="http://www.intlpress.com/JDG/archive/1996/43-2-291.pdf">pdf</a>]</p> </li> <li id="Ferolito"> <p>Ferolito <em>The octonions and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math></em> (<a href="https://www2.bc.edu/~reederma/Ferolito.pdf">pdf</a>)</p> </li> <li id="Baez"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, section 4.1 <em><a href="http://math.ucr.edu/home/baez/octonions/node14.html">G2</a></em> of: <em>The Octonions</em> (<a href="http://arxiv.org/abs/math/0105155">arXiv:math/0105155</a>)</p> </li> <li> <p>Ruben Arenas, <em>Constructing a Matrix Representation of the Lie Group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math></em>, 2005 (<a href="https://www.math.hmc.edu/seniorthesis/archives/2005/rarenas/rarenas-2005-thesis.pdf">pdf</a>)</p> </li> </ul> <p>Discussion in terms of the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> in <a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a> is in</p> <ul> <li id="Ibort">Alberto Ibort, <em>Multisymplectic geometry: generic and exceptional</em>, <em><a href="http://rsme.es/public/publi3.htm">Proceedings of the IX Fall workshop on geometry and physics</a></em> (<a class="existingWikiWord" href="/nlab/files/IbortMultisymplectic.pdf" title="pdf">pdf</a>)</li> </ul> <p>A description of the root space decomposition of the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_2</annotation></semantics></math> is in</p> <ul> <li id="Basak17">Tathagata Basak, <em>Root space decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_2</annotation></semantics></math> from octonions</em>, arXiv:<a href="https://arxiv.org/abs/1708.02367">1708.02367</a></li> </ul> <p>As the group of <a class="existingWikiWord" href="/nlab/show/zero-divisor">zero divisors</a> of the <a class="existingWikiWord" href="/nlab/show/sedenion">sedenions</a></p> <ul> <li id="Moreno97">Guillermo Moreno. <em>The zero divisors of the Cayley-Dickson algebras over the real numbers</em>. (1997) (<a href="https://arxiv.org/abs/q-alg/9710013">doi</a>)</li> </ul> <p>Cohomological properties are discussed in</p> <ul> <li>Younggi Choi, <em>Homology of the gauge group of exceptional Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math></em>, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> includes</p> <ul> <li id="Miyaoka93"> <p><a class="existingWikiWord" href="/nlab/show/Reiko+Miyaoka">Reiko Miyaoka</a>, <em>The linear isotropy group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_2/SO(4)</annotation></semantics></math>, the Hopf fibering and isoparametric hypersurfaces</em>, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (<a href="http://projecteuclid.org/euclid.ojm/1200784357">Euclid</a>)</p> </li> <li id="Ishiguro"> <p>Kenshi Ishiguro, <em>Classifying spaces and a subgroup of the exceptional Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math></em> <a href="http://hopf.math.purdue.edu/Ishiguro/G2.pdf">pdf</a></p> </li> <li id="Kramer02"> <p>Linus Kramer, 4.27 of <em>Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces</em>, AMS 2002</p> </li> </ul> <p>Discussion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> as a subgroup of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a>:</p> <ul> <li id="Varadarajan01"><a class="existingWikiWord" href="/nlab/show/Veeravalli+Varadarajan">Veeravalli Varadarajan</a>, <em>Spin(7)-subgroups of SO(8) and Spin(8)</em>, Expositiones Mathematicae, 19 (2001): 163-177 (<a href="https://core.ac.uk/download/pdf/81114499.pdf">pdf</a>)</li> </ul> <h3 id="applications_in_physics">Applications in physics</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> is in</p> <ul> <li>Ernst-Michael Ilgenfritz, Axel Maas, <em>Topological aspects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Yang-Mills theory</em> (<a href="http://arxiv.org/abs/1210.5963">arXiv:1210.5963</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 15, 2025 at 07:31:15. 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