CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Lie's three theorems in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Lie's three theorems </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/16196/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="lie_theory"><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 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='#lies_three_theorems'>Lie’s three theorems</a></li> <li><a href='#restriction_to_simply_connected_lie_groups'>Restriction to simply connected Lie groups</a></li> <li><a href='#generalization_of_lies_theorems_to_lie_groupoids'>Generalization of Lie’s theorems to Lie groupoids</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_lie_groups_and_lie_algebras'>For Lie groups and Lie algebras</a></li> <li><a href='#for_lie_groupoids_and_lie_algebroids'>For Lie groupoids and Lie algebroids</a></li> <li><a href='#in_higher_lie_theory'>In higher Lie theory</a></li> </ul> </ul> </div> <h2 id="lies_three_theorems">Lie’s three theorems</h2> <p>There is an obvious <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Lie</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Lie</mi><mi>Gp</mi><mo>→</mo><mi>Lie</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> Lie \;\colon\; Lie Gp \to Lie Alg </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/LieGroups">LieGroups</a> to <a class="existingWikiWord" href="/nlab/show/LieAlgebras">LieAlgebras</a> – <em><a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></em> – which sends any <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> to its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> and every <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a> of Lie groups to the corresponding <a class="existingWikiWord" href="/nlab/show/algebra+homomorphism">algebra homomorphism</a> of Lie algebras.</p> <p>Lie’s three theorems can be understood as establishing salient properties of this functor. More exactly, Lie’s theorems provide a foundation establishing an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between <a class="existingWikiWord" href="/nlab/show/local+Lie+groups">local Lie groups</a> and <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>; subsequent work by <a class="existingWikiWord" href="/nlab/show/%C3%89lie+Cartan">Élie Cartan</a> and others extended the theorems to give information on actual (“global”) <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> via the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi></mrow><annotation encoding="application/x-tex">Lie</annotation></semantics></math>.</p> <ol> <li> <p><strong>Lie’s first theorem</strong> is purely local; see the <a href="http://www.encyclopediaofmath.org/index.php/Lie_theorem">Encyclopedia of Math</a> for a statement. (Here one lacks a good notion of <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a> for extending this to a global result.)</p> </li> <li> <p><strong>Lie II</strong> Let <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> and <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> be Lie groups with Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g} = Lie(G)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔥</mi><mo>=</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{h} = Lie(H)</annotation></semantics></math>, such that <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> is <a class="existingWikiWord" href="/nlab/show/simply+connected+space">simply connected</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">f : \mathfrak{g} \to \mathfrak{h}</annotation></semantics></math> is a morphism of Lie algebras, then there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">F : G \to H</annotation></semantics></math> of Lie groups lifting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f = Lie(F)</annotation></semantics></math>.</p> </li> <li> <p><strong>Lie III</strong> (Cartan-Lie theorem) The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi></mrow><annotation encoding="application/x-tex">Lie</annotation></semantics></math> <strong>is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective on objects</a></strong>: for every <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/real+vector+space">real</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> there is a real 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>≅</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g} \cong Lie(G)</annotation></semantics></math>. Moreover, there exists such <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> which is simply connected.</p> </li> </ol> <p>For a classical account see:</p> <ul> <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>, Chapter 1 of: <em>Lie groups</em>, Springer (2000) [<a href="https://doi.org/10.1007/978-3-642-56936-4">doi:10.1007/978-3-642-56936-4</a>]</li> </ul> <p> <div class='num_remark'> <h6>Remark</h6> <p></p> <p>In his third theorem, Lie proved only the existence of a <a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a>, but not the global existence (nor the simply connected choice) which were established only a few decades later by <a class="existingWikiWord" href="/nlab/show/%C3%89lie+Cartan">Élie Cartan</a>. Hence the full theorem would properly be called the <em>Cartan-Lie theorem</em>. From an <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>, the third Lie theorem establishes the essential surjectivity of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi></mrow><annotation encoding="application/x-tex">Lie</annotation></semantics></math> from the category of <em>local Lie groups</em> to the category of finite dimensional real Lie algebras, and similarly the second Lie theorem establishes that this functor is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> (so the two together establish that this functor is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a>). The historically incorrect naming of the Cartan-Lie theorem as the “third Lie theorem” is largely due to the influence the lectures of <a href="#Serre64">Serre 1964</a>.</p> <p></p> </div> </p> <h2 id="restriction_to_simply_connected_lie_groups">Restriction to simply connected Lie groups</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LieGroups</mi> <mi>simpl</mi></msub></mrow><annotation encoding="application/x-tex">LieGroups_{simpl}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LieGroups</mi></mrow><annotation encoding="application/x-tex">LieGroups</annotation></semantics></math> consisting of simply connected Lie groups. Then the above implies that restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LieGroups</mi> <mi>simpl</mi></msub></mrow><annotation encoding="application/x-tex">LieGroups_{simpl}</annotation></semantics></math>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi></mrow><annotation encoding="application/x-tex">Lie</annotation></semantics></math> becomes an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> <h2 id="generalization_of_lies_theorems_to_lie_groupoids">Generalization of Lie’s theorems to Lie groupoids</h2> <p>The <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> of Lie’s theorems for Lie groups leads to analogous statements for <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoids</a>. In other words, there are analogous properties for the differentiation functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>diff</mi><mo>:</mo><mi>LieGroupoids</mi><mo>→</mo><mi>LieAlgebroids</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">diff : LieGroupoids \to LieAlgebroids.</annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoids</a> to <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroids</a>.</p> <p>In the case of Lie groupoids, the condition of a group being simply connected which plays an important role in the above statements is generalized to the condition that <em>source fibers</em> of the Lie groupoid (the preimages <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s^{-1}(x)</annotation></semantics></math> of the source map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s : C_1 \to C_0</annotation></semantics></math> at every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in C_0</annotation></semantics></math> of the Lie groupoid <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">C</annotation></semantics></math>) are simply connected. One says</p> <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>C</mi><mi>is</mi><mi>source</mi><mo>−</mo><mi>simply</mi><mi>connected</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mo>∀</mo><mi>x</mi><mo>∈</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (C is source-simply connected) \Leftrightarrow (\forall x \in C_0 : \pi_1(s^{-1}(x)) = 0 ). </annotation></semantics></math></div> <p><strong><em>Lie II</em> for Lie groupoids</strong> now reads exactly as <em>Lie II</em> for Lie groups, with “simply connected” replaced by “source simply connected”.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a href="Leibniz+algebra#LieThirdTheorem">Lie’s third theorem for Leibniz algebras</a></li> </ul> <h2 id="references">References</h2> <h3 id="for_lie_groups_and_lie_algebras">For Lie groups and Lie algebras</h3> <p>Review:</p> <ul> <li id="Serre64"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: <em>Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University</em>, Lecture Notes in Mathematics <strong>1500</strong> (1992) [<a href="https://doi.org/10.1007/978-3-540-70634-2">doi:10.1007/978-3-540-70634-2</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sigurdur+Helgason">Sigurdur Helgason</a>, Thm. 7.5 in: <em>Differential geometry, Lie groups and symmetric spaces</em>, Graduate Studies in Mathematics <strong>34</strong> (2001) [<a href="https://bookstore.ams.org/gsm-34">ams:gsm-34</a>]</p> </li> </ul> <h3 id="for_lie_groupoids_and_lie_algebroids">For Lie groupoids and Lie algebroids</h3> <p>Lie II for <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> was proven in</p> <ul> <li>K. C. H. Mackenzie and P. Xu, <em>Integration of Lie bialgebroids</em>, Topology, 39(3):445-467</li> </ul> <p>and</p> <ul> <li>I. Moerdijk and J Mrčun, <em>On integrability of infinitesimal actions</em>, Amer. J. Math. 124(3):567-593, 2002</li> </ul> <p><strong><em>Lie III</em> for Lie groupoids</strong> does <em>not</em> hold in direct generalization:</p> <p>by the general mechanism of <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> the space of morphisms of the source simply-connected <em>topological</em> groupoid <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> integrating a Lie algebroid <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">\mathfrak{g}</annotation></semantics></math> is a <em>quotient</em> space. This quotient may fail to be a <em>manifold</em> due to singularities.</p> <p>On the precise conditions under which <a class="existingWikiWord" href="/nlab/show/Lie+algebroids">Lie algebroids</a> do have <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> integrating them:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a>, <a class="existingWikiWord" href="/nlab/show/Rui+Loja+Fernandes">Rui Loja Fernandes</a>, <em>Integrability of Lie brackets</em>, Ann. of Math. <strong>157</strong> 2 (2003) 575-620 [<a href="http://arxiv.org/abs/math.DG/0105033">arXiv:math.DG/0105033</a>, <a href="https://doi.org/10.4007/annals.2003.157.575">doi:10.4007/annals.2003.157.575</a>]</li> </ul> <p>Enlarging <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> to groupoids in the category of <a class="existingWikiWord" href="/nlab/show/etale+stacks">etale stacks</a> and smooth maps results in a Cartan–Lie theorem for <a class="existingWikiWord" href="/nlab/show/Lie+algebroids">Lie algebroids</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hsian-Hua+Tseng">Hsian-Hua Tseng</a>, <a class="existingWikiWord" href="/nlab/show/Chenchang+Zhu">Chenchang Zhu</a>, <em>Integrating Lie algebroids via stacks</em>, Compositio Mathematica <strong>142</strong> (2006) 251–270 [<a href="https://doi.org/10.1112/S0010437X05001752">doi:10.1112/S0010437X05001752</a>]</li> </ul> <p>In particular, a <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> can be integrated to an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> if and only if the integrating groupoid in <a class="existingWikiWord" href="/nlab/show/etale+stacks">etale stacks</a> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> <p>Comprehensive review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rui+Loja+Fernandes">Rui Loja Fernandes</a>, <a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a>: <em>Lectures on Integrability of Lie Brackets</em>, Geometry & Topology Monographs <strong>17</strong> (2011) 1–107 [<a href="https://arxiv.org/abs/math/0611259">arxiv:math.DG/0611259</a>, <a href="http://dx.doi.org/10.2140/gtm.2011.17.1">doi:10.2140/gtm.2011.17.1</a>]</li> </ul> <p>and in the introduction of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chenchang+Zhu">Chenchang Zhu</a>, <em>Lie II theorem for Lie algebroids via stacky Lie groupoids</em> [<a href="http://arxiv.org/abs/math/0701024">arXiv:math/0701024</a>]</li> </ul> <h3 id="in_higher_lie_theory">In higher Lie theory</h3> <p>On <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras">L-infinity algebras</a> related to <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groups">smooth infinity-groups</a> in <a class="existingWikiWord" href="/nlab/show/higher+Lie+theory">higher Lie theory</a>:</p> <blockquote> <p>(see also references at <em><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></em>)</p> </blockquote> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christopher+L.+Rogers">Christopher L. Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Jesse+Wolfson">Jesse Wolfson</a>: <em>Lie’s Third Theorem for Lie <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>-Algebras</em> [<a href="https://arxiv.org/abs/2409.08957">arXiv:2409.08957</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 18, 2024 at 01:11:47. See the <a href="/nlab/history/Lie%27s+three+theorems" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Lie%27s+three+theorems" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/16196/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/Lie%27s+three+theorems/32" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Lie%27s+three+theorems" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Lie%27s+three+theorems" accesskey="S" class="navlink" id="history" rel="nofollow">History (32 revisions)</a> <a href="/nlab/show/Lie%27s+three+theorems/cite" style="color: black">Cite</a> <a href="/nlab/print/Lie%27s+three+theorems" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Lie%27s+three+theorems" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>