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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="higher_spin_geometry">Higher spin geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/fivebrane+geometry">fivebrane geometry</a></strong> …</p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></p> </li> </ul> <h2 id="spin_geometry">Spin geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/central+product+spin+group">central product spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+conjugation+matrix">charge conjugation matrix</a>, <a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+conjugate">Dirac conjugate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+spinor">Dirac spinor</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a>, <a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/rotation+groups">rotation groups</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/classification+of+simple+Lie+groups">Dynkin label</a></th><th><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">sp. orth. group</a></th><th><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></th><th><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%282%29">Pin(2)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B1</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/Spin%283%29">Spin(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%283%29">Pin(3)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%284%29">Pin(4)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</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/Pin%285%29">Pin(5)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%286%29">SO(6)</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;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%287%29">SO(7)</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;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</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;"></td><td style="text-align: left;"><a href="semi-spin+group#SemiSpin8">SO(8)</a></td></tr> <tr><td style="text-align: left;">B4</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%289%29">SO(9)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></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/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B5</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a></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/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D8</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a></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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D16</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2832%29">SO(32)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2832%29">Spin(32)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a></td></tr> </tbody></table> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29.Spin%283%29">Spin(5).Spin(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a></p> </li> </ul> </div> <h2 id="string_geometry">String geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+2-group">string^c 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+structure">string^c structure</a></p> </li> </ul> <h2 id="fivebrane_geometry">Fivebrane geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> </li> </ul> <h2 id="ninebrane_geometry">Ninebrane geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</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="higher_lie_theory">Higher 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='#general'>General</a></li> <li><a href='#relation_to_whitehead_tower_of_orthogonal_group'>Relation to Whitehead tower of orthogonal group</a></li> <li><a href='#ExceptionalIsomorphisms'>Exceptional isomorphisms</a></li> </ul> <li><a href='#ExampleSection'>Examples</a></li> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#spin_geometry'>Spin geometry</a></li> <li><a href='#in_physics'>In physics</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <strong>spin group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal covering space</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><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math>. By the usual arguments it inherits a group structure for which the operations are smooth and so is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math>.</p> <p>For special cases in low dimensions see at: <a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></p> <h2 id="definition">Definition</h2> <div class="num_defn" id="QuadraticVectorSpace"> <h6 id="definition_2">Definition</h6> <p>A <em>quadratic vector space</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, \langle -,-\rangle)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> over finite <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0, and equipped with a symmetric <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> <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><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\langle -,-\rangle \colon V \otimes V \to k</annotation></semantics></math>.</p> </div> <p>Conventions as in (<a href="#Varadarajan04">Varadarajan 04, section 5.3</a>).</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo lspace="verythinmathspace">:</mo><mi>v</mi><mo>↦</mo><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">q\colon v \mapsto \langle v ,v \rangle</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The <em><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CL</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CL(V,q)</annotation></semantics></math> of a quadratic vector space, def. <a class="maruku-ref" href="#QuadraticVectorSpace"></a>, is the <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>T</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl(V,q) \coloneqq T(V)/I(V,q) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> by the ideal generated by the elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>v</mi><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \otimes v - q(v)</annotation></semantics></math>.</p> </div> <p>Since the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(V)</annotation></semantics></math> is naturally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded, the Clifford algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V,q)</annotation></semantics></math> is naturally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-graded.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>q</mi><mo>=</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R}^n, q = {\vert -\vert})</annotation></semantics></math> be the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> with its canonical <a class="existingWikiWord" href="/nlab/show/scalar+product">scalar product</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Cl</mi> <mi>ℂ</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl^\mathbb{C}(\mathbb{R}^n)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/complexification">complexification</a> of its <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There exists a unique <a class="existingWikiWord" href="/nlab/show/complex+number">complex</a> <a class="existingWikiWord" href="/nlab/show/representation">representation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Cl</mi> <mi>ℂ</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>End</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n) </annotation></semantics></math></div> <p>of the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Cl</mi> <mi>ℂ</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl^\mathbb{C}(\mathbb{R}^n)</annotation></semantics></math> of smallest <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>dim</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mn>2</mn> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="SpinGroup"> <h6 id="definition_4">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Pin+group">Pin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pin</mi><mo stretchy="false">(</mo><mi>V</mi><mo>;</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pin(V;q)</annotation></semantics></math> of a quadratic vector space, def. <a class="maruku-ref" href="#QuadraticVectorSpace"></a>, is the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> in the <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V,q)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Pin</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Pin(V,q) \hookrightarrow GL_1(Cl(V,q)) </annotation></semantics></math></div> <p>on those elements which are multiples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">v_1 \cdots v_{n}</annotation></semantics></math> of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_i \in V</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q(v_i) = 1</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(V,q)</annotation></semantics></math> is the further <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pin</mi><mo stretchy="false">(</mo><mi>V</mi><mo>;</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pin(V;q)</annotation></semantics></math> on those elements which are even number multiples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>v</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">v_1 \cdots v_{2k}</annotation></semantics></math> of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_i \in V</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q(v_i) = 1</annotation></semantics></math>.</p> <p>Specifically, “the” Spin group is</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><mi>n</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Spin</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spin(n) \coloneqq Spin(\mathbb{R}^n) \,. </annotation></semantics></math></div></div> <p>A <em><a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a></em> is a <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a> of the spin group, def. <a class="maruku-ref" href="#SpinGroup"></a>.</p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <p>By definition the spin group sits in a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>Spin</mi><mo>→</mo><mi>SO</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}_2 \to Spin \to SO \,. </annotation></semantics></math></div> <h3 id="relation_to_whitehead_tower_of_orthogonal_group">Relation to Whitehead tower of orthogonal group</h3> <p>The spin group is one element in the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>, which starts out like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><mi>Fivebrane</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math> are for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> and for sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></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><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>4</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>5</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>6</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>7</mn></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">co-killing</a> these groups step by step one gets</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><mi>cokill</mi><mspace width="thinmathspace"></mspace><mi>this</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>to</mi><mspace width="thinmathspace"></mspace><mi>get</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>SO</mi></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Spin</mi></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>ℤ</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>String</mi></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>5</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>6</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>ℤ</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Fivebrane</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ cokill\, this &&&& to\,get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,. </annotation></semantics></math></div> <p>Via the <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-homomorphism</a> this is related to the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a>:</p> <div> <table><thead><tr><th></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th><th>16</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+group">ninebrane group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><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;">higher versions</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><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/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/stable+orthogonal+group">stable orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\mathbb{S})</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>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>24</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{24}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>240</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{240}</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>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2</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>ℤ</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_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><msub><mi>ℤ</mi> <mn>504</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{504}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</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>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2</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>ℤ</mi> <mn>480</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{480} \oplus \mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2 </annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/image+of+J-homomorphism">image of J-homomorphism</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(\pi_n(J))</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>24</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{24}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>240</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{240}</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>504</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{504}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>480</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{480}</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>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> </tbody></table> </div> <h3 id="ExceptionalIsomorphisms">Exceptional isomorphisms</h3> <p>In low <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a> the <a class="existingWikiWord" href="/nlab/show/spin+groups">spin groups</a> happens to be <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to various other <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a>. One speaks of <em><a class="existingWikiWord" href="/nlab/show/exceptional+isomorphisms">exceptional isomorphisms</a></em> or <em><a class="existingWikiWord" href="/nlab/show/sporadic+isomorphisms">sporadic isomorphisms</a></em>.</p> <p>See for instance (<a href="#Garrett13">Garrett 13</a>). See also <em><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></em>.</p> <p>In the following <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(n)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/quaternionic+unitary+group">quaternionic unitary group</a> in <a class="existingWikiWord" href="/nlab/show/quaternion">quaternionic</a> <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><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>We have</p> <ul> <li> <p>in <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean</a> signature</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(1) \simeq O(1)</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\simeq U(1) \simeq SO(2) \simeq S^1</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a>, the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>, see also at <em><a class="existingWikiWord" href="/nlab/show/real+Hopf+fibration">real Hopf fibration</a></em>)</p> <p>the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>→</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2)\to SO(2)</annotation></semantics></math> corresponds to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1\stackrel{\cdot 2}{\longrightarrow} S^1</annotation></semantics></math>, see also at <em><a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\simeq Sp(1) \simeq SU(2) \simeq S^3</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a></p> <p>the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2) \hookrightarrow Spin(3)</annotation></semantics></math> corresponds to the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \hookrightarrow S^3</annotation></semantics></math> (see e.g. <a href="#GorbounovRay92">Gorbounov-Ray 92</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3</annotation></semantics></math></p> <p>this is given by identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup><mo>≃</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^4 \simeq \mathbb{H}</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> and <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><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">SU(2) \simeq S^3</annotation></semantics></math> with the group of unit quternions. Then left and right quaternion multiplication gives a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⟶</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SU(2) \times SU(2) \longrightarrow SO(4) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>x</mi><mo>↦</mo><mspace width="thickmathspace"></mspace><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (g,h) \mapsto ( x \mapsto \; g^{-1} x h ) </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a> and hence exhibits the isomorphism.</p> <p>In particular therefore the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3) \hookrightarrow Spin(4)</annotation></semantics></math> corresponds to the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3 \hookrightarrow S^3 \times S^3</annotation></semantics></math>.</p> <p>At the level of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</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>4</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∧</mo> <mn>2</mn></msup><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4</annotation></semantics></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\pm 1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/eigenspaces">eigenspaces</a> of the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo><mo lspace="verythinmathspace">:</mo><msup><mo lspace="thinmathspace" rspace="thinmathspace">⋀</mo> <mn>2</mn></msup><msup><mi>ℝ</mi> <mn>4</mn></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4</annotation></semantics></math> gives the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> decomposition <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><mo>≃</mo><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⊕</mo><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⊕</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\simeq Sp(2)</annotation></semantics></math> (an indirect consequence of <a class="existingWikiWord" href="/nlab/show/triality">triality</a>, see e.g. <a href="#CadekVanzura97">Čadek-Vanžura 97</a>)</p> </li> <li> <p><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><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\simeq SU(4)</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <a href="special+unitary+group#SU4">SU(4)</a>)</p> </li> </ul> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentzian</a> signature</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(1,1) \simeq GL(1,\mathbb{R})</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2, \mathbb{R})</annotation></semantics></math> – 2d <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2,\mathbb{C})</annotation></semantics></math> – 2d <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a> of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(4,1) \simeq Sp(1,1)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq SL(2,\mathbb{H})</annotation></semantics></math> – 2d <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a> of <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo>≃</mo> <mrow><mi>in</mi><mspace width="thickmathspace"></mspace><mi>some</mi><mspace width="thickmathspace"></mspace><mi>sense</mi></mrow></msub><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>𝕆</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(9,1) \simeq_{in\;some\;sense} SL(2, \mathbb{O})</annotation></semantics></math> – 2d <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a> of <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a> (see <a class="existingWikiWord" href="/nlab/show/SL%282%2CO%29">SL(2,O)</a> for more on this would-be isomorphism)</p> </li> </ul> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter</a> signature</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,2) \simeq Sp(4,\mathbb{R})</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(4,2) \simeq SU(2,2)</annotation></semantics></math></p> </li> </ul> </li> <li> <p>in mixed signature</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,3) \simeq SL(4,\mathbb{R})</annotation></semantics></math> (<a href="#Garrett13">Garrett 13 (2.12)</a>)</li> </ul> </li> </ul> <p>Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.</p> <div> <p><strong><a href="spin+group#ExceptionalIsomorphisms">exceptional</a> <a class="existingWikiWord" href="/nlab/show/spin+representation">spinors</a> and <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lorentzian+spacetime">Lorentzian</a> <br /> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <br /> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> entry</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><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 = 2+1</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>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2,\mathbb{R})</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</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><mn>4</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4 = 3+1</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>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2, \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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</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><mn>6</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6 = 5+1</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>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">little string</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><mn>10</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10 = 9+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%2C1%29">Spin(9,1)</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/SL%282%2CO%29">SL(2,O)</a>”</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a>/<a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a></td></tr> </tbody></table> </div> <h2 id="ExampleSection">Examples</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/rotation+groups">rotation groups</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/classification+of+simple+Lie+groups">Dynkin label</a></th><th><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">sp. orth. group</a></th><th><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></th><th><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%282%29">Pin(2)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B1</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/Spin%283%29">Spin(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%283%29">Pin(3)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%284%29">Pin(4)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</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/Pin%285%29">Pin(5)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%286%29">SO(6)</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;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%287%29">SO(7)</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;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</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;"></td><td style="text-align: left;"><a href="semi-spin+group#SemiSpin8">SO(8)</a></td></tr> <tr><td style="text-align: left;">B4</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%289%29">SO(9)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></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/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B5</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a></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/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D8</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a></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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D16</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2832%29">SO(32)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2832%29">Spin(32)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a></td></tr> </tbody></table> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29.Spin%283%29">Spin(5).Spin(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a></p> </li> </ul> </div> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%2811%2C3%29">Spin(11,3)</a></li> </ul> <h2 id="applications">Applications</h2> <h3 id="spin_geometry">Spin geometry</h3> <p>See <a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></p> <h3 id="in_physics">In physics</h3> <p>The name arises due to the requirement that the structure group of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> lifts to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)</p> <p>See <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>, <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>, <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> looks like</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\cdots \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</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">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/string+group">string group</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">\to</annotation></semantics></math> <strong>spin group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <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><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Eh+group">spin^h group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metaplectic+group">metaplectic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/group">group</a></th><th>symbol</th><th><a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a></th><th>symbol</th><th>higher cover</th><th>symbol</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin+group">Pin group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pin(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tring+group">Tring group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tring</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Tring(n)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</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><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/String+group">String group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String(n)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n,1)</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></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><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n,1)</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></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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n,2)</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></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><mi>n</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n,2)</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></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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n+1,t+1)</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></td><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/Narain+group">Narain group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n,n)</annotation></semantics></math></td><td style="text-align: left;"></td><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/Poincar%C3%A9+group">Poincaré group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ISO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ISO(n,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+spin+group">Poincaré spin group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ISO</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat {ISO}(n,1)</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></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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sISO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sISO(n,1)</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></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></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></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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superconformal+group">superconformal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/H.+Blaine+Lawson">H. Blaine Lawson</a>, <a class="existingWikiWord" href="/nlab/show/Marie-Louise+Michelsohn">Marie-Louise Michelsohn</a>, chapter I, section 2 of <em><a class="existingWikiWord" href="/nlab/show/Spin+geometry">Spin geometry</a></em>, Princeton University Press (1989)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckhard+Meinrenken">Eckhard Meinrenken</a>: <em>Clifford algebras and Lie groups</em>, Ergebn. Mathem. & Grenzgeb., Springer (2013) [<a href="https://doi.org/10.1007/978-3-642-36216-3">doi:10.1007/978-3-642-36216-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, §21 & 22 in: <em>Lie Algebras In Particle Physics</em>, Westview Press (1999), CRC Press (2019) [<a href="https://doi.org/10.1201/9780429499210">doi:10.1201/9780429499210</a>]</p> <blockquote> <p>(with an eye towards application to <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a> in (the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a> of) <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a>)</p> </blockquote> </li> </ul> <p>See also</p> <ul> <li id="Varadarajan04"> <p><a class="existingWikiWord" href="/nlab/show/Veeravalli+Varadarajan">Veeravalli Varadarajan</a>, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Supersymmetry+for+mathematicians">Supersymmetry for mathematicians</a>: An introduction</em>, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)</p> </li> <li> <p>wikipedia <em><a href="http://en.wikipedia.org/wiki/Spin_group">Spin group</a></em></p> </li> </ul> <p>Examples of sporadic (exceptional) spin group isomorphisms incarnated as <a class="existingWikiWord" href="/nlab/show/isogenies">isogenies</a> onto <a class="existingWikiWord" href="/nlab/show/orthogonal+groups">orthogonal groups</a> are discussed in</p> <ul> <li id="Garrett13"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Garrett">Paul Garrett</a>, <em>Sporadic isogenies to orthogonal groups</em> (July 2013) [<a href="http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Garrett-SporadicIsogenies.pdf" title="pdf">pdf</a> ]</p> </li> <li id="GorbounovRay92"> <p><a class="existingWikiWord" href="/nlab/show/Vassily+Gorbounov">Vassily Gorbounov</a>, <a class="existingWikiWord" href="/nlab/show/Nigel+Ray">Nigel Ray</a>, <em>Orientations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> Bundles and Symplectic Cobordism</em>, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (<a class="existingWikiWord" href="/nlab/files/GorbunovRaySpinBundles.pdf" title="pdf">pdf</a>, <a href="https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4">doi: 10.2977/prims/1195168855</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphism">exceptional isomorphism</a> <a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</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/Sp%282%29">Sp(2)</a> is discussed via <a class="existingWikiWord" href="/nlab/show/triality">triality</a> in</p> <ul> <li id="CadekVanzura97"><a class="existingWikiWord" href="/nlab/show/Martin+%C4%8Cadek">Martin Čadek</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Van%C5%BEura">Jiří Vanžura</a>, <em>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2) \cdot Sp(1)</annotation></semantics></math>-structures in 8-dimensional vector bundles</em>, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (<a href="https://www.jstor.org/stable/43737249">jstor:43737249</a>)</li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>Spin</mi></mrow><annotation encoding="application/x-tex">B Spin</annotation></semantics></math> includes</p> <ul> <li> <p>E. Thomas, <em>On the cohomology groups of the classifying space for the stable spinor groups</em>, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.</p> </li> <li id="Pittie91"> <p><a class="existingWikiWord" href="/nlab/show/Harsh+Pittie">Harsh Pittie</a>, <em>The integral homology and cohomology rings of SO(n) and Spin(n)</em>, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (<a href="https://doi.org/10.1016/0022-4049(91)90108-E">doi:10.1016/0022-4049(91)90108-E</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 11, 2024 at 09:45:16. 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