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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="group_theory">Group Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a> <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> Classical groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <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> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> </li> </ul> Finite groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a> <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> Group schemes <ul> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a> </li> </ul> Topological groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> </li> </ul> Lie groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a> </li> </ul> Super-Lie groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> </li> </ul> Higher groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> <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> <a class="existingWikiWord" href="/nlab/show/n-group">n-group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> </li> </ul> </li> <li> <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> </li> </ul> Cohomology and Extensions <ul> <li> <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, </li> <li> <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> </li> </ul> Related concepts <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> Introductions <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a> Differentials <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a> </li> </ul> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a> </li> <li> <a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> Models <ul> <li> <a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring) </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a> </li> </ul> </div></div> <h4 id="lie_theory">Lie theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>) Background Smooth structure <ul> <li> <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a> </li> </ul> Higher groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> Lie theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a> </li> </ul> </li> </ul> ∞-Lie groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <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> <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a> </li> </ul> </li> </ul> ∞-Lie algebroids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a> <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <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> <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> Formal Lie groupoids <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> Cohomology <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a> </li> </ul> Homotopy <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> Related topics <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> Examples <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a> </li> </ul> <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a> </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='#lies_three_theorems'>Lie’s three theorems</a></li> <li><a href='#lie_subgroups'>Lie subgroups</a></li> <li><a href='#classification'>Classification</a></li> <li><a href='#different_lie_group_structures_on_a_group'>Different Lie group structures on a group</a></li> <li><a href='#different_topologies_on_a_lie_group'>Different topologies on a Lie group</a></li> <li><a href='#which_topological_groups_admit_lie_group_structure'>Which topological groups admit Lie group structure?</a></li> <li><a href='#relation_to_topological_groups'>Relation to topological groups</a></li> <li><a href='#homotopy_groups'>Homotopy groups</a></li> </ul> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#in_differential_geometry'>In differential geometry</a></li> <li><a href='#in_gauge_theory'>In gauge theory</a></li> </ul> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#basic_examples'>Basic examples</a></li> <li><a href='#classical_lie_groups'>Classical Lie groups</a></li> <li><a href='#exceptional_lie_groups'>Exceptional Lie groups</a></li> <li><a href='#infinitedimensional_examples'>Infinite-dimensional examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#homotopy_groups_2'>Homotopy groups</a></li> <li><a href='#ReferencesOnInfiniteDimensionalLieGroups'>On infinite-dimensional Lie groups</a></li> <li><a href='#spaces_of_homomorphisms'>Spaces of homomorphisms</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> A Lie group is a <a class="existingWikiWord" href="/nlab/show/group">group</a> with <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>. Lie groups form a category, <a class="existingWikiWord" href="/nlab/show/LieGrp">LieGrp</a>. <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> A Lie group is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> whose <a class="existingWikiWord" href="/nlab/show/forgetful+functor">underlying</a> <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/elements">elements</a> is equipped with the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> such that the <a class="existingWikiWord" href="/nlab/show/magma">group multiplication</a> and <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>-assigning <a class="existingWikiWord" href="/nlab/show/functions">functions</a> are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>. In other words, a Lie group is a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. </div> Usually the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> is assumed to be defined over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> and to be of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> (f.d.), but extensions of the definition to some other <a class="existingWikiWord" href="/nlab/show/ground+fields">ground fields</a> or to -<a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifolds">infinite-dimensional manifolds</a> are also relevant, sometimes under other names (such as <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+Lie+group">Fréchet Lie group</a> when the underlying manifold is an infinite-dimensional <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifold">Fréchet manifold</a>). A real Lie group is called a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> (or connected, simply connected Lie group, etc) if its underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> (or <a class="existingWikiWord" href="/nlab/show/connected+space">connected</a>, <a class="existingWikiWord" href="/nlab/show/simply+connected+space">simply connected</a>, etc). Every connected finite dimensional real Lie group is <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/product">product</a> of a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> (its <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>) and a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>. Every <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> connected compact finite dimensional real Lie group is a <a class="existingWikiWord" href="/nlab/show/torus">torus</a> (a product of <a class="existingWikiWord" href="/nlab/show/circles">circles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>n</mi></msup><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>…</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">T^n = S^1\times S^1 \times \ldots \times S^1</annotation></semantics></math>). There is an <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> version of a Lie group, a so-called <a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a>, where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected and simply connected Lie groups. The first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a> approximation to a Lie group is its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. <h2 id="properties">Properties</h2> <h3 id="lies_three_theorems">Lie’s three theorems</h3> <a class="existingWikiWord" href="/nlab/show/Sophus+Lie">Sophus Lie</a> proved several theorems, known as <a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a>, on the relationship between <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> and Lie groups. Lie’s third theorem is about the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> of finite-dimensional real Lie algebras and local Lie groups. Because <a class="existingWikiWord" href="/nlab/show/%C3%89lie+Cartan">Élie Cartan</a> extended this to a global integrability theorem, Lie’s third theorem is also called the Cartan-Lie theorem. <h3 id="lie_subgroups">Lie subgroups</h3> <div class='num_prop' id='CartanClosedSubgroupTheorem'> <h6>Proposition</h6> (<a class="existingWikiWord" href="/nlab/show/Cartan%27s+closed+subgroup+theorem">Cartan's closed subgroup theorem</a>) If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subgroup">closed subgroup</a> of a (<a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimensional</a>) <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, then <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">H</annotation></semantics></math> is a sub-Lie group, hence a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a> such that its group operations are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> with respect to the the <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a> <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>. </div> <h3 id="classification">Classification</h3> <div class='num_prop'> <h6>Proposition</h6> Every connected finite-dimensional real Lie group is <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/product">product</a> of a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> and a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Euclidean space</a>. Every abelian connected compact f.d. real Lie group is a <a class="existingWikiWord" href="/nlab/show/torus">torus</a> (a product of circles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>n</mi></msup><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>…</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">T^n = S^1\times S^1 \times \ldots \times S^1</annotation></semantics></math>). </div> The <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>s have a classification into infinite series of <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+Lie+group">classical Lie group</a>s</li> </ul> and a finite snumber o <ul> <li><a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a>s</li> </ul> <h3 id="different_lie_group_structures_on_a_group">Different Lie group structures on a group</h3> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a bare <a class="existingWikiWord" href="/nlab/show/group">group</a> (without smooth structure) there may be more than one way to equip it with the <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a> of a Lie group. <div class="num_example"> <h6 id="example">Example</h6> As bare <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s, the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> are, for all <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>, <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s over the <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> is the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of the <a class="existingWikiWord" href="/nlab/show/continuum">continuum</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">2^{\aleph_0}</annotation></semantics></math>. Therefore these are all <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> as bare group. But equipped with their canonical Lie group structure (as in the <a href="#Examples">Examples</a>) they are of course not isomorphic. </div> <h3 id="different_topologies_on_a_lie_group">Different topologies on a Lie group</h3> <ul> <li>Linus Kramer, The topology of a simple Lie group is essentially unique, (<a href="http://arxiv.org/abs/1009.5457">arXiv</a>)</li> </ul> <blockquote> Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is very rigid: every ‘abstract’ isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and a locally compact and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-compact group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is automatically a homeomorphism, provided that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is absolutely simple. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all. </blockquote> <h3 id="which_topological_groups_admit_lie_group_structure">Which topological groups admit Lie group structure?</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hilbert%27s+fifth+problem">Hilbert's fifth problem</a></li> </ul> <h3 id="relation_to_topological_groups">Relation to topological groups</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/continuous+homomorphisms+of+Lie+groups+are+smooth">continuous homomorphisms of Lie groups are smooth</a></li> </ul> <h3 id="homotopy_groups">Homotopy groups</h3> List of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of the manifolds underlying the classical <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> are for instance in (<a href="#Abanov09">Abanov 09</a>). <h2 id="applications">Applications</h2> <h3 id="in_differential_geometry">In differential geometry</h3> A central concept of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is that of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a Lie group. <h3 id="in_gauge_theory">In gauge theory</h3> In the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> of <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>s – <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> – Lie groups appear as local <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>s parameterizing <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s: notably the <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a> is modeled by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> for some Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. For models that describe experimental observations the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in question is a quotient of a product of <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>s and the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>. For details see <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> <h2 id="in_higher_category_theory">In higher category theory</h2> The notion of <a class="existingWikiWord" href="/nlab/show/group">group</a> generalizes in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> to that of <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>, … <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>. Accordingly, so does the notion of Lie group generalize to <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, … <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a>. For details see <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a>. <h2 id="Examples">Examples</h2> <h3 id="basic_examples">Basic examples</h3> <ul> <li> The <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <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> with its standard <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> and the group operation being addition is a Lie group. So is every <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> with the componentwise addition of real numbers. </li> <li> The <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of <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> by the subgroup of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \hookrightarrow \mathbb{R}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <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><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">S^1 = \mathbb{R}/\mathbb{Z}</annotation></semantics></math>. The quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>ℤ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n/\mathbb{Z}^n</annotation></semantics></math> is 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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/torus">torus</a>. </li> <li> The <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of any Lie group is canonically itself a Lie group: the <a class="existingWikiWord" href="/nlab/show/automorphism+Lie+group">automorphism Lie group</a>. </li> </ul> <h3 id="classical_lie_groups">Classical Lie groups</h3> The <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a> include <ul> <li> the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> </li> <li> the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <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> and <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>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> <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><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2n)</annotation></semantics></math>. </li> </ul> <h3 id="exceptional_lie_groups">Exceptional Lie groups</h3> The <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a> incude <ul> <li><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a>, <a class="existingWikiWord" href="/nlab/show/F%E2%82%84">F₄</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%86">E₆</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%87">E₇</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a></li> </ul> <h3 id="infinitedimensional_examples">Infinite-dimensional examples</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/stable+unitary+group">stable unitary group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/diffeomorphism+group">diffeomorphism group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></li> </ul> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a> </li> <li> Lie monoid<a href="/nlab/new/Lie+monoid">?</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/Lie+category">Lie category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/real+form">real form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a>, <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal torus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/rank+of+a+Lie+group">rank of a Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conjugacy+class">conjugacy class</a>, <a class="existingWikiWord" href="/nlab/show/Cartan-Dirac+structure+on+a+Lie+group">Cartan-Dirac structure on a Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Weyl+group">Weyl group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/In%C3%B6n%C3%BC-Wigner+group+contraction">Inönü-Wigner group contraction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+Lie+group">infinite-dimensional Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+dynamics+on+Lie+groups">Hamiltonian dynamics on Lie groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+Lie+group">complex Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+group">Poisson Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> </li> </ul> <div> Examples of sequences of local structures <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><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><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integration">integration</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></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/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</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/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</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>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a></td><td style="text-align: left;"></td><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><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><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><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><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>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="Serre64"> <a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [<a href="https://doi.org/10.1007/978-3-540-70634-2">doi:10.1007/978-3-540-70634-2</a>] </li> <li id="Bredon72"> <a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Section 0.5 of: <a class="existingWikiWord" href="/nlab/show/Introduction+to+compact+transformation+groups">Introduction to compact transformation groups</a>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>) <blockquote> (in the broader context of <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a>) </blockquote> </li> <li> Arthur A. Sagle, Ralph E. Walde: Introduction to Lie Groups and Lie Algebras, Pure and Applied Mathematics 51, Elsevier (1973) 215-227 </li> <li> <a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, Lie groups and Lie algebras – Chapters 1-3, Springer (1975, 1989) [<a href="https://link.springer.com/book/9783540642428">ISBN:9783540642428</a>] </li> <li id="Adams82"> <a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, Lectures on Lie groups, University of Chicago Press, 1982 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/L/bo3614673.html">ISBN:9780226005300</a>, <a href="https://www.google.com/books/edition/Lectures_on_Lie_Groups/TC7d3ZcqjfsC?hl=en">gbooks</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/M.+M.+Postnikov">M. M. Postnikov</a>, Lectures on geometry: Semester V, Lie groups and algebras (1986) [<a href="https://archive.org/details/postnikov-lectures-in-geometry-semester-v-lie-group-and-lie-algebras">ark:/13960/t4cp9jn4p</a>] </li> <li id="Onishchik93"> A. L. Onishchik (ed.) Lie Groups and Lie Algebras <ul> <li> I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory, </li> <li> II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups </li> </ul> Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993 </li> <li> <a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, Ch. I of: Representations of compact Lie groups, Springer 1985 (<a href="https://link.springer.com/book/10.1007/978-3-662-12918-0">doi:10.1007/978-3-662-12918-0</a>) <blockquote> (in the context of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, <a class="existingWikiWord" href="/nlab/show/Lie+Groups%2C+Lie+Algebras%2C+Cohomology+and+Some+Applications+in+Physics">Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</a>, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [<a href="https://doi.org/10.1017/CBO9780511599897">doi:10.1017/CBO9780511599897</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [<a href="https://doi.org/10.1201/9780429499210">doi:10.1201/9780429499210</a>] <blockquote> with an eye towards application to (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> </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hans+Duistermaat">Hans Duistermaat</a>, <a class="existingWikiWord" href="/nlab/show/Johan+A.+C.+Kolk">Johan A. C. Kolk</a>, Lie groups, Springer (2000) [<a href="https://doi.org/10.1007/978-3-642-56936-4">doi:10.1007/978-3-642-56936-4</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Sigurdur+Helgason">Sigurdur Helgason</a>, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [<a href="https://bookstore.ams.org/gsm-34">ams:gsm-34</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Mark+Haiman">Mark Haiman</a> (notes by <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>), Lie groups, lecture notes, Berkeley (2008) [<a class="existingWikiWord" href="/nlab/files/Haiman-LieGroups09.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Eckhard+Meinrenken">Eckhard Meinrenken</a>, Lie groups and Lie algebras, Lecture notes 2010 (<a href="http://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf">pdf</a>) </li> <li id="HilgertNeeb12"> <a class="existingWikiWord" href="/nlab/show/Joachim+Hilgert">Joachim Hilgert</a>, <a class="existingWikiWord" href="/nlab/show/Karl-Hermann+Neeb">Karl-Hermann Neeb</a>, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (<a href="https://link.springer.com/book/10.1007/978-0-387-84794-8">doi:10.1007/978-0-387-84794-8</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Daniel+Bump">Daniel Bump</a>, Lie groups, Graduate Texts in Mathematics 225, Springer (2013) [<a href="https://doi.org/10.1007/978-1-4614-8024-2">doi:10.1007/978-1-4614-8024-2</a>, <a href="https://www-fourier.ujf-grenoble.fr/~panchish/ETE%20LAMA%202018-AP/lecturesZETAS2018/BumpD-Lie%20groups.pdf">pdf</a>] </li> <li id="Hall15"> <a class="existingWikiWord" href="/nlab/show/Brian+C.+Hall">Brian C. Hall</a>, Lie Groups, Lie Algebras, and Representations, Springer 2015 (<a href="https://doi.org/10.1007/978-3-319-13467-3">doi:10.1007/978-3-319-13467-3</a>) </li> <li id="GallierQuaintance20a"> <a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyn+Quaintance">Jocelyn Quaintance</a>, Differential Geometry and Lie Groups – A computational perspective, Geometry and Computing 12, Springer (2020) [<a href="https://doi.org/10.1007/978-3-030-46040-2">doi:10.1007/978-3-030-46040-2</a>, <a href="https://www.cis.upenn.edu/~jean/gbooks/manif.html">webpage</a>] </li> <li id="GallierQuaintance20b"> <a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyn+Quaintance">Jocelyn Quaintance</a>, Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [<a href="https://doi.org/10.1007/978-3-030-46047-1">doi:10.1007/978-3-030-46047-1</a>, <a href="https://www.cis.upenn.edu/~jean/gbooks/manif.html">webpage</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, Lie groups and Lie algebras [<a href="https://arxiv.org/abs/2201.09397">arXiv:2201.09397</a>] </li> </ul> In the generality of <a class="existingWikiWord" href="/nlab/show/Lie+semigroups">Lie semigroups</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Joachim+Hilgert">Joachim Hilgert</a>, <a class="existingWikiWord" href="/nlab/show/Karl-Hermann+Neeb">Karl-Hermann Neeb</a>, Lie Semigroups and their Applications. Lecture Notes in Mathematics 1552, Springer 1993 (<a href="https://link.springer.com/book/10.1007/BFb0084640">doi:10.1007/BFb0084640</a>)</li> </ul> In the generality of <a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Borcherds">Richard Borcherds</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Haiman">Mark Haiman</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <a class="existingWikiWord" href="/nlab/show/Nicolai+Reshetikhin">Nicolai Reshetikhin</a>, <a class="existingWikiWord" href="/nlab/show/Vera+Serganova">Vera Serganova</a>, Berkeley Lectures on Lie Groups and Quantum Groups (2020-2024) [<a href="http://categorified.net/LieQuantumGroups.pdf">pdf</a>]</li> </ul> <h3 id="homotopy_groups_2">Homotopy groups</h3> <ul> <li id="Abanov09">Alexander Abanov, Homotopy groups of Lie groups (2009) [<a href="http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf">pdf</a>]</li> </ul> <h3 id="ReferencesOnInfiniteDimensionalLieGroups">On infinite-dimensional Lie groups</h3> References on <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+Lie+groups">infinite-dimensional Lie groups</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andreas+Kriegl">Andreas Kriegl</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Michor">Peter Michor</a>, Regular infinite dimensional Lie groups Journal of Lie Theory Volume 7 (1997) 61-99 (<a href="http://www.heldermann-verlag.de/jlt/jlt07/MICHPL.PDF">pdf</a>) </li> <li> Rudolf Schmid, Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics Advances in Mathematical Physics Volume 2010, (<a href="http://www.emis.de/journals/HOA/AMP/Volume2010/280362.pdf">pdf</a>) </li> <li> Josef Teichmann, Infinite dimensional Lie Theory from the point of view of Functional Analysis (<a href="http://www.math.ethz.ch/~jteichma/diss.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Karl-Hermann+Neeb">Karl-Hermann Neeb</a>, Monastir summer school: Infinite-dimensional Lie groups (<a href="http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf">pdf</a>) </li> </ul> <h3 id="spaces_of_homomorphisms">Spaces of homomorphisms</h3> On <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> of <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> from <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> to <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>: for maps out of <a class="existingWikiWord" href="/nlab/show/finitely+generated+group">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>“: <ul> <li><a class="existingWikiWord" href="/nlab/show/Frederick+R.+Cohen">Frederick R. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Mentor+Stafa">Mentor Stafa</a>, A survey on spaces of homomorphisms to Lie groups, In: Callegaro F., Cohen F., De Concini C., Feichtner E., Gaiffi G., Salvetti M. (eds.) Configuration Spaces, INdAM Series, vol 14, Springer 2016 (<a href="https://arxiv.org/abs/1412.5668">arXiv:1412.5668</a>, <a href="https://doi.org/10.1007/978-3-319-31580-5_15">doi:10.1007/978-3-319-31580-5_15</a>)</li> </ul> for maps out of <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> and the fact that <a class="existingWikiWord" href="/nlab/show/nearby+homomorphisms+from+compact+Lie+groups+are+conjugate">nearby homomorphisms from compact Lie groups are conjugate</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, Ch. III, Lem. 38.1 in: Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete 33, Springer 1964 (<a href="https://link.springer.com/book/10.1007/978-3-662-41633-4">doi:10.1007/978-3-662-41633-4</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, Nearby homomorphisms from compact Lie groups are conjugate (<a href="https://mathoverflow.net/q/123624/381">MO:q/123624</a>) </li> <li id="Rezk14"> <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, Rem. 2.2.1 in: <a class="existingWikiWord" href="/nlab/show/Global+Homotopy+Theory+and+Cohesion">Global Homotopy Theory and Cohesion</a>, 2014 (<a href="http://www.math.uiuc.edu/~rezk/global-cohesion.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rezk14.pdf" title="pdf">pdf</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on September 10, 2024 at 22:02:27. 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