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> general linear group 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> general linear group </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/7777/#Item_4" 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="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/higher+linear+algebra">higher linear algebra</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-module">(∞,n)-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field">field</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-field">∞-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a>, <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>, <a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> (<a class="existingWikiWord" href="/nlab/show/square+matrix">square</a>, <a class="existingWikiWord" href="/nlab/show/invertible+matrix">invertible</a>, <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal</a>, <a class="existingWikiWord" href="/nlab/show/hermitian+matrix">hermitian</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+matrix">symmetric</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+group">matrix group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenspace">eigenspace</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <p><a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <p>(…)</p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#AsAtopologicalGroup'>As a topological group</a></li> <li><a href='#context_2'>Context</a></li> <ul> <li><a href='#topology'>Topology</a></li> <li><a href='#definition_2'>Definition</a></li> <li><a href='#properties'>Properties</a></li> </ul> <li><a href='#AsALieGroup'>As a Lie group</a></li> <li><a href='#as_an_algebraic_group'>As an algebraic group</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties_2'>Properties</a></li> <ul> <li><a href='#representation_theory'>Representation theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, the <strong>general linear group</strong> <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_n(k)</annotation></semantics></math>) is the <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/invertible+morphism">invertible</a> <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> from the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> to itself. It may canonically be identified with the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n\times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrix">matrices</a> with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> having nonzero <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>.</p> <h3 id="AsAtopologicalGroup">As a topological group</h3> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context_2">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{R}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">= \mathbb{C}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> or the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> equipped with their <a class="existingWikiWord" href="/nlab/show/Euclidean+topology">Euclidean topology</a>.</p> <h4 id="definition_2">Definition</h4> <div class="num_defn" id="GLnAsTopologicalGroup"> <h6 id="definition_3">Definition</h6> <p><strong>(general linear group as a topological group)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, as a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> the <em>general linear group</em> <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n, k)</annotation></semantics></math> is defined as follows.</p> <p>The underlying group is the <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/real+number">real</a> or <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> whose <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> is non-vanishing</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>A</mi><mo>∈</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> GL(n,k) \;\coloneqq\; \left( A \in Mat_{n \times n}(k) \; \vert \; det(A) \neq 0 \right) </annotation></semantics></math></div> <p>with group operation given by <a class="existingWikiWord" href="/nlab/show/matrix+multiplication">matrix multiplication</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> on this set is the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> as a subset of the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> of <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> Mat_{n \times n}(k) \simeq k^{(n^2)} </annotation></semantics></math></div> <p>with its <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> </div> <div class="num_lemma" id="GLnTopologicalWellDefined"> <h6 id="lemma">Lemma</h6> <p><strong>(group operations are continuous)</strong></p> <p>Definition <a class="maruku-ref" href="#GLnAsTopologicalGroup"></a> is indeed well defined in that the group operations on <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> are indeed <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> with respect to the given topology.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Observe that under the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Mat_{n \times n}(k) \simeq k^{(n^2)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrix+multiplication">matrix multiplication</a> is a <a class="existingWikiWord" href="/nlab/show/polynomial+function">polynomial function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup><mo>×</mo><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>2</mn><msup><mi>n</mi> <mn>2</mn></msup></mrow></msup><mo>⟶</mo><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup><mo>≃</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> k^{(n^2)} \times k^{(n^2)} \simeq k^{ 2 n^2 } \longrightarrow k^{(n^2)} \simeq Mat_{n \times n}(k) \,. </annotation></semantics></math></div> <p>Similarly <a class="existingWikiWord" href="/nlab/show/inverse+matrix">matrix inversion</a> is a <a class="existingWikiWord" href="/nlab/show/rational+function">rational function</a>. Now <a class="existingWikiWord" href="/nlab/show/rational+functions+are+continuous">rational functions are continuous</a> on their <a class="existingWikiWord" href="/nlab/show/domain">domain</a> of definition, and since a real matrix is invertible previsely if its determinant is non-vanishing, the domain of definition for matrix inversion is precisely <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>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k) \subset Mat_{n \times n}(k)</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p><strong>(stable general linear group)</strong></p> <p>The evident <a class="existingWikiWord" href="/nlab/show/tower">tower</a> of embeddings</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↪</mo><msup><mi>k</mi> <mn>2</mn></msup><mo>↪</mo><msup><mi>k</mi> <mn>3</mn></msup><mo>↪</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> k \hookrightarrow k^2 \hookrightarrow k^3 \hookrightarrow \cdots </annotation></semantics></math></div> <p>induces a corresponding <a class="existingWikiWord" href="/nlab/show/tower+diagram">tower diagram</a> of embedding of the general linear groups (def. <a class="maruku-ref" href="#GLnAsTopologicalGroup"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> GL(1,k) \hookrightarrow GL(2,k) \hookrightarrow GL(3,k) \hookrightarrow \cdots \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over this <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> is called the <em>stable general linear group</em> denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> GL(k) \;\coloneqq\; \underset{\longrightarrow}{\lim}_n GL(n,k) \,. </annotation></semantics></math></div></div> <h4 id="properties">Properties</h4> <div class="num_example" id="AsSubspaceOfTheMappingSpace"> <h6 id="proposition">Proposition</h6> <p><strong>(as a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">mapping space</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> induced on the real general linear group when regarded as a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> with its <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> GL(n,\mathbb{R}) \subset Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)} </annotation></semantics></math></div> <p>(as in def. <a class="maruku-ref" href="#GLnAsTopologicalGroup"></a>) coincides with the topology induced by regarding the general linear group as a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps(k^n, k^n)</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Maps</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> GL(n,\mathbb{R}) \subset Maps(k^n, k^n) </annotation></semantics></math></div> <p>i.e. the set of all <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n \to k^n</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>On the one had, the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (<a href="Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace">this prop.</a>) gives that the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Maps</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> GL(n, \mathbb{R}) \to Maps(\mathbb{R}^n, \mathbb{R}^n) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> for <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>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{R})</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, because this is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the defining continuous <a class="existingWikiWord" href="/nlab/show/action">action</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n \,. </annotation></semantics></math></div> <p>This implies that the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> on <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>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{R})</annotation></semantics></math> is equal to or <a class="existingWikiWord" href="/nlab/show/finer+topology">finer</a> than the subspace topology coming from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(\mathbb{R}^n, \mathbb{R}^n)</annotation></semantics></math>.</p> <p>We conclude by showing that it is also equal to or <a class="existingWikiWord" href="/nlab/show/coarser+topology">coarser</a>, together this then implies the claims.</p> <p>Since we are speaking about a subspace topology, we may consider the open subsets of the ambient Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}</annotation></semantics></math>. Observe that a <a class="existingWikiWord" href="/nlab/show/neighborhood+base">neighborhood base</a> of a linear map or matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> consists of sets of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mi>A</mi> <mi>ϵ</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>B</mi><mo>∈</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></munder><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mi>A</mi><msub><mi>e</mi> <mi>i</mi></msub><mo>−</mo><mi>B</mi><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mo><</mo><mi>ϵ</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> U_A^\epsilon \;\coloneqq\; \left\{B \in Mat_{n \times n}(\mathbb{R}) \,\vert\, \underset{{1 \leq i \leq n}}{\forall}\; |A e_i - B e_i| \lt \epsilon \right\} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon \in (0,\infty)</annotation></semantics></math>.</p> <p>But this is also a <a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base</a> element for the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, namely</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mi>A</mi> <mi>ϵ</mi></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msubsup><mi>V</mi> <mi>i</mi> <mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> U_A^\epsilon \;=\; \bigcap_{i = 1}^n V_i^{K_i} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">K_i \coloneqq \{e_i\}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub><mo>≔</mo><msubsup><mi>B</mi> <mrow><mi>A</mi><msup><mi>e</mi> <mi>i</mi></msup></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_i \coloneqq B^\circ_{A e^i}(\epsilon)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> of <a class="existingWikiWord" href="/nlab/show/radius">radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msup><mi>e</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">A e^i</annotation></semantics></math>.</p> </div> <div class="num_prop" id="ConnectednessOfGeneralLinearGroup"> <h6 id="proposition_2">Proposition</h6> <p><strong>(connectedness properties of the general linear group)</strong></p> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></p> <ol> <li> <p>the complex general linear group <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>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{C})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a>;</p> </li> <li> <p>the real general linear group <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>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{R})</annotation></semantics></math> is not <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>First observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">GL(1,k) = k \setminus \{0\}</annotation></semantics></math> has this property:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{C} \setminus \{0\}</annotation></semantics></math> is path-connected,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} \setminus \{0\} = (-\infty,0) \sqcup (0,\infty)</annotation></semantics></math> is not path connected.</p> </li> </ol> <p>Now for the general case:</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math>: every invertible complex matrix is diagonalizable by a sequence of elementary matrix operations (<a href="inverse+matrix#FundamentalTheoremOfLinearAlgebra">this prop.</a>). Each of these is clearly path-connected to the identity. Finally the subspace of invertible <a class="existingWikiWord" href="/nlab/show/diagonal+matrices">diagonal matrices</a> is the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></munder><mo stretchy="false">(</mo><mi>ℂ</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{ \{1, \cdots, n\} }{\prod} (\mathbb{C} \setminus \{0\})</annotation></semantics></math> and hence connected (by <a href="ProductSpaceOfConnectedSpacesIsConnected">this prop.</a>, since each factor space is).</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{R}</annotation></semantics></math>: the <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> function is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</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>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℝ</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">GL(n,k) \to \mathbb{R} \setminus \{0\}</annotation></semantics></math>, and since the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> is not path connected, the domain cannot be either.</p> </li> </ol> </div> <div class="num_prop" id="TopologicalGeneralLinearGroup"> <h6 id="proposition_3">Proposition</h6> <p><strong>(compactness properties of the general linear group)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/topological+group">topological</a> general linear group <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> (def. <a class="maruku-ref" href="#GLnAsTopologicalGroup"></a>) is</p> <ol> <li> <p>not <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+topological+space">paracompact Hausdorff</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> GL_n(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)} </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/open+subset">open</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, since it is the <a class="existingWikiWord" href="/nlab/show/pre-image">pre-image</a> under the <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> function (which is a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> and hence continuous, as in the proof of lemma <a class="maruku-ref" href="#GLnTopologicalWellDefined"></a>) of the of the open subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k \setminus \{0\} \subset k</annotation></semantics></math>.</p> <p>As an open subspace of Euclidean space, <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> is not compact, by the <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a>.</p> <p>As Euclidean space is Hausdorff, and since every <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of a Hausdorff space is again Hausdorff, so <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(n,k)</annotation></semantics></math> is Hausdorff.</p> <p>Similarly, as Euclidean space is <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> and since an open subspace of a locally compact space is again locally compact, it follows that <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> is locally compact.</p> <p>From this it follows that <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>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math> is paracompact, since locally compact topological groups are paracompact (<a href="topological+group#ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact">this prop.</a>).</p> </div> <h3 id="AsALieGroup">As a Lie group</h3> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>Since the general linear group as a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> (def. <a class="maruku-ref" href="#GLnAsTopologicalGroup"></a>) is an <a class="existingWikiWord" href="/nlab/show/open+subspace">open subspace</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> (proof of prop. <a class="maruku-ref" href="#TopologicalGeneralLinearGroup"></a>) it inherits the structure of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (by <a href="differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds">this prop.</a>). The group operations (being <a class="existingWikiWord" href="/nlab/show/rational+functions">rational functions</a>) are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> with respect to this smooth structure. This is the general linear group <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>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{R})</annotation></semantics></math> as a <em><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></em>.</p> </div> <h3 id="as_an_algebraic_group">As an algebraic group</h3> <p>This group can be considered as a (quasi-affine) sub<a class="existingWikiWord" href="/nlab/show/variety">variety</a> of the <a class="existingWikiWord" href="/nlab/show/affine+scheme">affine space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_{n\times n}(k)</annotation></semantics></math> of square matrices of size <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> defined by the condition that the <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">M_{n \times n}(k) \times k</annotation></semantics></math> defined by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mi>t</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(M)t = 1</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> varies over the factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_{n \times n}(k)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> over the factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>).</p> <p>This <a class="existingWikiWord" href="/nlab/show/variety">variety</a> is an algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-group, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is the field of real or complex numbers it is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>One may in fact consider the set of invertible matrices over an arbitrary unital <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, not necessarily commutative. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo>:</mo><mi>R</mi><mo>↦</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_n: R\mapsto GL_n(R)</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of <a class="existingWikiWord" href="/nlab/show/group">group</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo>=</mo><msup><mi>Ring</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Aff=Ring^{op}</annotation></semantics></math> where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a <a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a>; it is in fact the affine group scheme represented by the commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>11</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mi>n</mi></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>det</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}[x_{11}, \ldots, x_{n n}, t]/(det(X)t - 1)</annotation></semantics></math>.</p> <p>Coordinate rings of general linear groups and of special general linear groups have <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum deformations</a> called <a class="existingWikiWord" href="/nlab/show/quantum+linear+groups">quantum linear groups</a>.</p> <h2 id="examples">Examples</h2> <p>Over <a class="existingWikiWord" href="/nlab/show/finite+fields">finite fields</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/GL%282%2C3%29">GL(2,3)</a></li> </ul> <h2 id="properties_2">Properties</h2> <h3 id="representation_theory">Representation theory</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+general+linear+group">representation theory of the general linear group</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/translation+group">translation group</a>, <a class="existingWikiWord" href="/nlab/show/affine+group">affine group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gauss+decomposition">Gauss decomposition</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a>, <a class="existingWikiWord" href="/nlab/show/projective+linear+group">projective linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>, <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a>, <a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+group">jet group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+supergroup">general linear supergroup</a>, <a class="existingWikiWord" href="/nlab/show/quantum+linear+group">quantum linear group</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p>O.T. O’Meara, <em>Lectures on Linear Groups</em>, Amer. Math. Soc., Providence, RI, 1974.</p> </li> <li> <p>B. Parshall, J.Wang, <em>Quantum linear groups</em>, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 19, 2022 at 07:24:45. See the <a href="/nlab/history/general+linear+group" 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/general+linear+group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7777/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/general+linear+group/30" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/general+linear+group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/general+linear+group" accesskey="S" class="navlink" id="history" rel="nofollow">History (30 revisions)</a> <a href="/nlab/show/general+linear+group/cite" style="color: black">Cite</a> <a href="/nlab/print/general+linear+group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/general+linear+group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>