general linear group in nLab

Given a field k, the general linear group GL(n,k) (or GLn(k)) is the group of invertible linear maps from the vector space kn to itself. nLab generallineargroup SkiptheNavigationLinks| HomePage| AllPages| LatestRevisions| Discussthispage| Contents Context GroupTheory grouptheory group,∞-group groupobject,groupobjectinan(∞,1)-category abeliangroup,spectrum superabeliangroup groupaction,∞-action representation,∞-representation progroup homogeneousspace Classicalgroups generallineargroup unitarygroup specialunitarygroup.projectiveunitarygroup orthogonalgroup specialorthogonalgroup symplecticgroup Finitegroups finitegroup symmetricgroup,cyclicgroup,braidgroup classificationoffinitesimplegroups sporadicfinitesimplegroups Monstergroup,Mathieugroup Groupschemes algebraicgroup abelianvariety Topologicalgroups topologicalgroup compacttopologicalgroup,locallycompacttopologicalgroup maximalcompactsubgroup stringgroup Liegroups Liegroup compactLiegroup Kac-Moodygroup Super-Liegroups superLiegroup superEuclideangroup Highergroups 2-group crossedmodule,strict2-group n-group ∞-group simplicialgroup crossedcomplex k-tuplygroupaln-groupoid spectrum circlen-group,string2-group,fivebraneLie6-group CohomologyandExtensions groupcohomology groupextension, ∞-groupextension,Ext-group Relatedconcepts quantumgroup Editthissidebar Contents Definition Asatopologicalgroup Context Topology Definition Properties AsaLiegroup Asanalgebraicgroup Examples Properties Representationtheory Relatedconcepts References Definition Givenafieldkk,thegenerallineargroupGL(n,k)GL(n,k)(orGLn(k)GL_n(k))isthegroupofinvertiblelinearmapsfromthevectorspaceknk^ntoitself.Itmaycanonicallybeidentifiedwiththegroupofn×nn\timesnmatriceswithentriesinkkhavingnonzerodeterminant. Asatopologicalgroup Context Topology topology(point-settopology,point-freetopology) seealsodifferentialtopology,algebraictopology,functionalanalysisandtopologicalhomotopytheory Introduction Basicconcepts opensubset,closedsubset,neighbourhood topologicalspace,locale baseforthetopology,neighbourhoodbase finer/coarsertopology closure,interior,boundary separation,sobriety continuousfunction,homeomorphism uniformlycontinuousfunction embedding openmap,closedmap sequence,net,sub-net,filter convergence categoryTop convenientcategoryoftopologicalspaces Universalconstructions initialtopology,finaltopology subspace,quotientspace, fiberspace,spaceattachment productspace,disjointunionspace mappingcylinder,mappingcocylinder mappingcone,mappingcocone mappingtelescope colimitsofnormalspaces Extrastuff,structure,properties nicetopologicalspace metricspace,metrictopology,metrisablespace Kolmogorovspace,Hausdorffspace,regularspace,normalspace soberspace compactspace,propermap sequentiallycompact,countablycompact,locallycompact,sigma-compact,paracompact,countablyparacompact,stronglycompact compactlygeneratedspace second-countablespace,first-countablespace contractiblespace,locallycontractiblespace connectedspace,locallyconnectedspace simply-connectedspace,locallysimply-connectedspace cellcomplex,CW-complex pointedspace topologicalvectorspace,Banachspace,Hilbertspace topologicalgroup topologicalvectorbundle,topologicalK-theory topologicalmanifold Examples emptyspace,pointspace discretespace,codiscretespace Sierpinskispace ordertopology,specializationtopology,Scotttopology Euclideanspace realline,plane cylinder,cone sphere,ball circle,torus,annulus,Moebiusstrip polytope,polyhedron projectivespace(real,complex) classifyingspace configurationspace path,loop mappingspaces:compact-opentopology,topologyofuniformconvergence loopspace,pathspace Zariskitopology Cantorspace,Mandelbrotspace Peanocurve linewithtwoorigins,longline,Sorgenfreyline K-topology,Dowkerspace Warsawcircle,Hawaiianearringspace Basicstatements Hausdorffspacesaresober schemesaresober continuousimagesofcompactspacesarecompact closedsubspacesofcompactHausdorffspacesareequivalentlycompactsubspaces opensubspacesofcompactHausdorffspacesarelocallycompact quotientprojectionsoutofcompactHausdorffspacesareclosedpreciselyifthecodomainisHausdorff compactspacesequivalentlyhaveconvergingsubnetofeverynet Lebesguenumberlemma sequentiallycompactmetricspacesareequivalentlycompactmetricspaces compactspacesequivalentlyhaveconvergingsubnetofeverynet sequentiallycompactmetricspacesaretotallybounded continuousmetricspacevaluedfunctiononcompactmetricspaceisuniformlycontinuous paracompactHausdorffspacesarenormal paracompactHausdorffspacesequivalentlyadmitsubordinatepartitionsofunity closedinjectionsareembeddings propermapstolocallycompactspacesareclosed injectivepropermapstolocallycompactspacesareequivalentlytheclosedembeddings locallycompactandsigma-compactspacesareparacompact locallycompactandsecond-countablespacesaresigma-compact second-countableregularspacesareparacompact CW-complexesareparacompactHausdorffspaces Theorems Urysohn'slemma Tietzeextensiontheorem Tychonofftheorem tubelemma Michael'stheorem Brouwer'sfixedpointtheorem topologicalinvarianceofdimension Jordancurvetheorem AnalysisTheorems Heine-Boreltheorem intermediatevaluetheorem extremevaluetheorem topologicalhomotopytheory lefthomotopy,righthomotopy homotopyequivalence,deformationretract fundamentalgroup,coveringspace fundamentaltheoremofcoveringspaces homotopygroup weakhomotopyequivalence Whitehead'stheorem Freudenthalsuspensiontheorem nervetheorem homotopyextensionproperty,Hurewiczcofibration cofibersequence Strømmodelcategory classicalmodelstructureontopologicalspaces Letk=ℝk=\mathbb{R}or=ℂ=\mathbb{C}betherealnumbersorthecomplexnumbersequippedwiththeirEuclideantopology. Definition Definition (generallineargroupasatopologicalgroup) Forn∈ℕn\in\mathbb{N},asatopologicalgroupthegenerallineargroupGL(n,k)GL(n,k)isdefinedasfollows. Theunderlyinggroupisthegroupofrealorcomplexn×nn\timesnmatriceswhosedeterminantisnon-vanishing GL(n,k)≔(A∈Matn×n(k)|det(A)≠0) GL(n,k) \;\coloneqq\; \left( A\inMat_{n\timesn}(k) \;\vert\; det(A)\neq0 \right) withgroupoperationgivenbymatrixmultiplication. ThetopologyonthissetisthesubspacetopologyasasubsetoftheEuclideanspaceofmatrices Matn×n(k)≃k(n2) Mat_{n\timesn}(k)\simeqk^{(n^2)} withitsmetrictopology. Lemma (groupoperationsarecontinuous) DefinitionisindeedwelldefinedinthatthegroupoperationsonGL(n,k)GL(n,k)areindeedcontinuousfunctionswithrespecttothegiventopology. Proof ObservethatundertheidentificationMatn×n(k)≃k(n2)Mat_{n\timesn}(k)\simeqk^{(n^2)}matrixmultiplicationisapolynomialfunction k(n2)×k(n2)≃k2n2⟶k(n2)≃Matn×n(k). k^{(n^2)} \times k^{(n^2)} \simeq k^{2n^2} \longrightarrow k^{(n^2)} \simeq Mat_{n\timesn}(k) \,. Similarlymatrixinversionisarationalfunction.Nowrationalfunctionsarecontinuousontheirdomainofdefinition,andsincearealmatrixisinvertiblepreviselyifitsdeterminantisnon-vanishing,thedomainofdefinitionformatrixinversionispreciselyGL(n,k)⊂Matn×n(k)GL(n,k)\subsetMat_{n\timesn}(k). Definition (stablegenerallineargroup) Theevidenttowerofembeddings k↪k2↪k3↪⋯ k\hookrightarrowk^2\hookrightarrowk^3\hookrightarrow\cdots inducesacorrespondingtowerdiagramofembeddingofthegenerallineargroups(def.) GL(1,k)↪GL(2,k)↪GL(3,k)↪⋯. GL(1,k) \hookrightarrow GL(2,k) \hookrightarrow GL(3,k) \hookrightarrow \cdots \,. Thecolimitoverthisdiagraminthecategoryoftopologicalgroupiscalledthestablegenerallineargroupdenoted GL(k)≔lim⟶nGL(n,k). GL(k)\;\coloneqq\;\underset{\longrightarrow}{\lim}_nGL(n,k) \,. Properties Proposition (asasubspaceofthemappingspace) ThetopologyinducedontherealgenerallineargroupwhenregardedasatopologicalsubspaceofEuclideanspacewithitsmetrictopology GL(n,ℝ)⊂Matn×n(ℝ)≃ℝ(n2) GL(n,\mathbb{R}) \subset Mat_{n\timesn}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)} (asindef.)coincideswiththetopologyinducedbyregardingthegenerallineargroupasasubspaceofthemappingspaceMaps(kn,kn)Maps(k^n,k^n), GL(n,ℝ)⊂Maps(kn,kn) GL(n,\mathbb{R}) \subset Maps(k^n,k^n) i.e.thesetofallcontinuousfunctionskn→knk^n\tok^nequippedwiththecompact-opentopology. Proof Ontheonehad,theuniversalpropertyofthemappingspace(thisprop.)givesthattheinclusion GL(n,ℝ)→Maps(ℝn,ℝn) GL(n,\mathbb{R})\toMaps(\mathbb{R}^n,\mathbb{R}^n) isacontinuousfunctionforGL(n,ℝ)GL(n,\mathbb{R})equippedwiththeEuclideanmetrictopology,becausethisistheadjunctofthedefiningcontinuousactionmap GL(n,ℝ)×ℝn→ℝn. GL(n,\mathbb{R})\times\mathbb{R}^n\to\mathbb{R}^n \,. ThisimpliesthattheEuclideanmetrictopologyonGL(n,ℝ)GL(n,\mathbb{R})isequaltoorfinerthanthesubspacetopologycomingfromMap(ℝn,ℝn)Map(\mathbb{R}^n,\mathbb{R}^n). Weconcludebyshowingthatitisalsoequaltoorcoarser,togetherthisthenimpliestheclaims. Sincewearespeakingaboutasubspacetopology,wemayconsidertheopensubsetsoftheambientEuclideanspaceMatn×n(ℝ)≃ℝ(n2)Mat_{n\timesn}(\mathbb{R})\simeq\mathbb{R}^{(n^2)}.ObservethataneighborhoodbaseofalinearmapormatrixAAconsistsofsetsoftheform UAϵ≔{B∈Matn×n(ℝ)|∀1≤i≤n|Aei−Bei|