Lie group - Wikipedia

The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups ... Liegroup FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Groupthatisalsoadifferentiablemanifoldwithgroupoperationsthataresmooth Liegroups Classicalgroups GenerallinearGL(n) SpeciallinearSL(n) OrthogonalO(n) SpecialorthogonalSO(n) UnitaryU(n) SpecialunitarySU(n) SymplecticSp(n) SimpleLiegroups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8 OtherLiegroups Circle Lorentz Poincaré Conformalgroup Diffeomorphism Loop Euclidean Liealgebras Liegroup–Liealgebracorrespondence Exponentialmap Adjointrepresentation KillingformIndex SimpleLiealgebra SemisimpleLiealgebra Dynkindiagrams Cartansubalgebra RootsystemWeylgroup RealformComplexification SplitLiealgebra CompactLiealgebra Representationtheory Liegrouprepresentation Liealgebrarepresentation RepresentationtheoryofsemisimpleLiealgebras RepresentationsofclassicalLiegroups Theoremofthehighestweight Borel–Weil–Botttheorem Liegroupsinphysics Particlephysicsandrepresentationtheory Lorentzgrouprepresentations Poincarégrouprepresentations Galileangrouprepresentations Scientists SophusLie HenriPoincaré WilhelmKilling ÉlieCartan HermannWeyl ClaudeChevalley Harish-Chandra ArmandBorel Glossary TableofLiegroupsvte Algebraicstructure→GrouptheoryGrouptheory Basicnotions Subgroup Normalsubgroup Quotientgroup (Semi-)directproduct Grouphomomorphisms kernel image directsum wreathproduct simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossaryofgrouptheory Listofgrouptheorytopics Finitegroups Classificationoffinitesimplegroups cyclic alternating Lietype sporadic Cauchy'stheorem Lagrange'stheorem Sylowtheorems Hall'stheorem p-group Elementaryabeliangroup Frobeniusgroup Schurmultiplier SymmetricgroupSn Kleinfour-groupV DihedralgroupDn QuaterniongroupQ DicyclicgroupDicn DiscretegroupsLattices Integers( Z {\displaystyle\mathbb{Z}} ) Freegroup ModulargroupsPSL(2, Z {\displaystyle\mathbb{Z}} )SL(2, Z {\displaystyle\mathbb{Z}} ) Arithmeticgroup Lattice Hyperbolicgroup TopologicalandLiegroups Solenoid Circle GenerallinearGL(n) SpeciallinearSL(n) OrthogonalO(n) EuclideanE(n) SpecialorthogonalSO(n) UnitaryU(n) SpecialunitarySU(n) SymplecticSp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop InfinitedimensionalLiegroupO(∞)SU(∞)Sp(∞) Algebraicgroups Linearalgebraicgroup Reductivegroup Abelianvariety Ellipticcurve vte NottobeconfusedwithGroupofLietype. Inmathematics,aLiegroup(pronounced/liː/"Lee")isagroupthatisalsoadifferentiablemanifold.AmanifoldisaspacethatlocallyresemblesEuclideanspace,whereasgroupsdefinetheabstractconceptofabinaryoperationalongwiththeadditionalpropertiesitmusthavetobeagroup,forinstancemultiplicationandthetakingofinverses(division),orequivalently,theconceptofadditionandthetakingofinverses(subtraction).Combiningthesetwoideas,oneobtainsacontinuousgroupwheremultiplyingpointsandtheirinversesarecontinuous.If,inaddition,themultiplicationandtakingofinversesaresmooth(differentiable),oneobtainsaLiegroup. Liegroupsprovideanaturalmodelfortheconceptofcontinuoussymmetry,acelebratedexampleofwhichistherotationalsymmetryinthreedimensions(givenbythespecialorthogonalgroup SO ( 3 ) {\displaystyle{\text{SO}}(3)} ).Liegroupsarewidelyusedinmanypartsofmodernmathematicsandphysics. Liegroupswerefirstfoundbystudyingmatrixsubgroups G {\displaystyleG} containedin GL n ( R ) {\displaystyle{\text{GL}}_{n}(\mathbb{R})} or GL n ( C ) {\displaystyle{\text{GL}}_{n}(\mathbb{C})} ,thegroupsof n × n {\displaystylen\timesn} invertiblematricesover R {\displaystyle\mathbb{R}} or C {\displaystyle\mathbb{C}} .Thesearenowcalledtheclassicalgroups,astheconcepthasbeenextendedfarbeyondtheseorigins.LiegroupsarenamedafterNorwegianmathematicianSophusLie(1842–1899),wholaidthefoundationsofthetheoryofcontinuoustransformationgroups.Lie'soriginalmotivationforintroducingLiegroupswastomodelthecontinuoussymmetriesofdifferentialequations,inmuchthesamewaythatfinitegroupsareusedinGaloistheorytomodelthediscretesymmetriesofalgebraicequations. Contents 1History 2Overview 3Definitionsandexamples 3.1Firstexamples 3.2Non-example 3.3MatrixLiegroups 3.4Relatedconcepts 3.5Topologicaldefinition 4MoreexamplesofLiegroups 4.1Dimensionsoneandtwo 4.2Additionalexamples 4.3Constructions 4.4Relatednotions 5Basicconcepts 5.1TheLiealgebraassociatedwithaLiegroup 5.2Homomorphismsandisomorphisms 5.3LiegroupversusLiealgebraisomorphisms 5.4SimplyconnectedLiegroups 5.5Theexponentialmap 5.6Liesubgroup 6Representations 7Classification 8Infinite-dimensionalLiegroups 9Seealso 10Notes 10.1Explanatorynotes 10.2Citations 11References 12Externallinks History[edit] AccordingtothemostauthoritativesourceontheearlyhistoryofLiegroups(Hawkins,p. 1),SophusLiehimselfconsideredthewinterof1873–1874asthebirthdateofhistheoryofcontinuousgroups.Hawkins,however,suggeststhatitwas"Lie'sprodigiousresearchactivityduringthefour-yearperiodfromthefallof1869tothefallof1873"thatledtothetheory'screation(ibid).SomeofLie'searlyideasweredevelopedinclosecollaborationwithFelixKlein.LiemetwithKleineverydayfromOctober1869through1872:inBerlinfromtheendofOctober1869totheendofFebruary1870,andinParis,GöttingenandErlangeninthesubsequenttwoyears(ibid,p. 2).Liestatedthatalloftheprincipalresultswereobtainedby1884.Butduringthe1870sallhispapers(excepttheveryfirstnote)werepublishedinNorwegianjournals,whichimpededrecognitionoftheworkthroughouttherestofEurope(ibid,p. 76).In1884ayoungGermanmathematician,FriedrichEngel,cametoworkwithLieonasystematictreatisetoexposehistheoryofcontinuousgroups.Fromthiseffortresultedthethree-volumeTheoriederTransformationsgruppen,publishedin1888,1890,and1893.ThetermgroupesdeLiefirstappearedinFrenchin1893inthethesisofLie'sstudentArthurTresse.[1] Lie'sideasdidnotstandinisolationfromtherestofmathematics.Infact,hisinterestinthegeometryofdifferentialequationswasfirstmotivatedbytheworkofCarlGustavJacobi,onthetheoryofpartialdifferentialequationsoffirstorderandontheequationsofclassicalmechanics.MuchofJacobi'sworkwaspublishedposthumouslyinthe1860s,generatingenormousinterestinFranceandGermany(Hawkins,p. 43).Lie'sidéefixewastodevelopatheoryofsymmetriesofdifferentialequationsthatwouldaccomplishforthemwhatÉvaristeGaloishaddoneforalgebraicequations:namely,toclassifythemintermsofgrouptheory.Lieandothermathematiciansshowedthatthemostimportantequationsforspecialfunctionsandorthogonalpolynomialstendtoarisefromgrouptheoreticalsymmetries.InLie'searlywork,theideawastoconstructatheoryofcontinuousgroups,tocomplementthetheoryofdiscretegroupsthathaddevelopedinthetheoryofmodularforms,inthehandsofFelixKleinandHenriPoincaré.TheinitialapplicationthatLiehadinmindwastothetheoryofdifferentialequations.OnthemodelofGaloistheoryandpolynomialequations,thedrivingconceptionwasofatheorycapableofunifying,bythestudyofsymmetry,thewholeareaofordinarydifferentialequations.However,thehopethatLieTheorywouldunifytheentirefieldofordinarydifferentialequationswasnotfulfilled.SymmetrymethodsforODEscontinuetobestudied,butdonotdominatethesubject.ThereisadifferentialGaloistheory,butitwasdevelopedbyothers,suchasPicardandVessiot,anditprovidesatheoryofquadratures,theindefiniteintegralsrequiredtoexpresssolutions. AdditionalimpetustoconsidercontinuousgroupscamefromideasofBernhardRiemann,onthefoundationsofgeometry,andtheirfurtherdevelopmentinthehandsofKlein.Thusthreemajorthemesin19thcenturymathematicswerecombinedbyLieincreatinghisnewtheory:theideaofsymmetry,asexemplifiedbyGaloisthroughthealgebraicnotionofagroup;geometrictheoryandtheexplicitsolutionsofdifferentialequationsofmechanics,workedoutbyPoissonandJacobi;andthenewunderstandingofgeometrythatemergedintheworksofPlücker,Möbius,Grassmannandothers,andculminatedinRiemann'srevolutionaryvisionofthesubject. AlthoughtodaySophusLieisrightfullyrecognizedasthecreatorofthetheoryofcontinuousgroups,amajorstrideinthedevelopmentoftheirstructuretheory,whichwastohaveaprofoundinfluenceonsubsequentdevelopmentofmathematics,wasmadebyWilhelmKilling,whoin1888publishedthefirstpaperinaseriesentitledDieZusammensetzungderstetigenendlichenTransformationsgruppen(Thecompositionofcontinuousfinitetransformationgroups)(Hawkins,p. 100).TheworkofKilling,laterrefinedandgeneralizedbyÉlieCartan,ledtoclassificationofsemisimpleLiealgebras,Cartan'stheoryofsymmetricspaces,andHermannWeyl'sdescriptionofrepresentationsofcompactandsemisimpleLiegroupsusinghighestweights. In1900DavidHilbertchallengedLietheoristswithhisFifthProblempresentedattheInternationalCongressofMathematiciansinParis. WeylbroughttheearlyperiodofthedevelopmentofthetheoryofLiegroupstofruition,fornotonlydidheclassifyirreduciblerepresentationsofsemisimpleLiegroupsandconnectthetheoryofgroupswithquantummechanics,buthealsoputLie'stheoryitselfonfirmerfootingbyclearlyenunciatingthedistinctionbetweenLie'sinfinitesimalgroups(i.e.,Liealgebras)andtheLiegroupsproper,andbeganinvestigationsoftopologyofLiegroups.[2]ThetheoryofLiegroupswassystematicallyreworkedinmodernmathematicallanguageinamonographbyClaudeChevalley. Overview[edit] Thesetofallcomplexnumberswithabsolutevalue1(correspondingtopointsonthecircleofcenter0andradius1inthecomplexplane)isaLiegroupundercomplexmultiplication:thecirclegroup. Liegroupsaresmoothdifferentiablemanifoldsandassuchcanbestudiedusingdifferentialcalculus,incontrastwiththecaseofmoregeneraltopologicalgroups.OneofthekeyideasinthetheoryofLiegroupsistoreplacetheglobalobject,thegroup,withitslocalorlinearizedversion,whichLiehimselfcalledits"infinitesimalgroup"andwhichhassincebecomeknownasitsLiealgebra. Liegroupsplayanenormousroleinmoderngeometry,onseveraldifferentlevels.FelixKleinarguedinhisErlangenprogramthatonecanconsidervarious"geometries"byspecifyinganappropriatetransformationgroupthatleavescertaingeometricpropertiesinvariant.ThusEuclideangeometrycorrespondstothechoiceofthegroupE(3)ofdistance-preservingtransformationsoftheEuclideanspaceR3,conformalgeometrycorrespondstoenlargingthegrouptotheconformalgroup,whereasinprojectivegeometryoneisinterestedinthepropertiesinvariantundertheprojectivegroup.ThisidealaterledtothenotionofaG-structure,whereGisaLiegroupof"local"symmetriesofamanifold. Liegroups(andtheirassociatedLiealgebras)playamajorroleinmodernphysics,withtheLiegrouptypicallyplayingtheroleofasymmetryofaphysicalsystem.Here,therepresentationsoftheLiegroup(orofitsLiealgebra)areespeciallyimportant.Representationtheoryisusedextensivelyinparticlephysics.GroupswhoserepresentationsareofparticularimportanceincludetherotationgroupSO(3)(oritsdoublecoverSU(2)),thespecialunitarygroupSU(3)andthePoincarégroup. Ona"global"level,wheneveraLiegroupactsonageometricobject,suchasaRiemannianorasymplecticmanifold,thisactionprovidesameasureofrigidityandyieldsarichalgebraicstructure.ThepresenceofcontinuoussymmetriesexpressedviaaLiegroupactiononamanifoldplacesstrongconstraintsonitsgeometryandfacilitatesanalysisonthemanifold.LinearactionsofLiegroupsareespeciallyimportant,andarestudiedinrepresentationtheory. Inthe1940s–1950s,EllisKolchin,ArmandBorel,andClaudeChevalleyrealisedthatmanyfoundationalresultsconcerningLiegroupscanbedevelopedcompletelyalgebraically,givingrisetothetheoryofalgebraicgroupsdefinedoveranarbitraryfield.Thisinsightopenednewpossibilitiesinpurealgebra,byprovidingauniformconstructionformostfinitesimplegroups,aswellasinalgebraicgeometry.Thetheoryofautomorphicforms,animportantbranchofmodernnumbertheory,dealsextensivelywithanaloguesofLiegroupsoveradelerings;p-adicLiegroupsplayanimportantrole,viatheirconnectionswithGaloisrepresentationsinnumbertheory. Definitionsandexamples[edit] ArealLiegroupisagroupthatisalsoafinite-dimensionalrealsmoothmanifold,inwhichthegroupoperationsofmultiplicationandinversionaresmoothmaps.Smoothnessofthegroupmultiplication μ : G × G → G μ ( x , y ) = x y {\displaystyle\mu:G\timesG\toG\quad\mu(x,y)=xy} meansthatμisasmoothmappingoftheproductmanifoldG×GintoG.Thetworequirementscanbecombinedtothesinglerequirementthatthemapping ( x , y ) ↦ x − 1 y {\displaystyle(x,y)\mapstox^{-1}y} beasmoothmappingoftheproductmanifoldintoG. Firstexamples[edit] The2×2realinvertiblematricesformagroupundermultiplication,denotedbyGL(2,R)orbyGL2(R): GL ⁡ ( 2 , R ) = { A = ( a b c d ) : det A = a d − b c ≠ 0 } . {\displaystyle\operatorname{GL}(2,\mathbf{R})=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\detA=ad-bc\neq0\right\}.} Thisisafour-dimensionalnoncompactrealLiegroup;itisanopensubsetof R 4 {\displaystyle\mathbb{R}^{4}} .Thisgroupisdisconnected;ithastwoconnectedcomponentscorrespondingtothepositiveandnegativevaluesofthedeterminant. TherotationmatricesformasubgroupofGL(2,R),denotedbySO(2,R).ItisaLiegroupinitsownright:specifically,aone-dimensionalcompactconnectedLiegroupwhichisdiffeomorphictothecircle.Usingtherotationangle φ {\displaystyle\varphi} asaparameter,thisgroupcanbeparametrizedasfollows: SO ⁡ ( 2 , R ) = { ( cos ⁡ φ − sin ⁡ φ sin ⁡ φ cos ⁡ φ ) : φ ∈ R / 2 π Z } . {\displaystyle\operatorname{SO}(2,\mathbf{R})=\left\{{\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}}:\,\varphi\in\mathbf{R}/2\pi\mathbf{Z}\right\}.} AdditionoftheanglescorrespondstomultiplicationoftheelementsofSO(2,R),andtakingtheoppositeanglecorrespondstoinversion.Thusbothmultiplicationandinversionaredifferentiablemaps. Theaffinegroupofonedimensionisatwo-dimensionalmatrixLiegroup,consistingof 2 × 2 {\displaystyle2\times2} real,upper-triangularmatrices,withthefirstdiagonalentrybeingpositiveandtheseconddiagonalentrybeing1.Thus,thegroupconsistsofmatricesoftheform A = ( a b 0 1 ) , a > 0 , b ∈ R . {\displaystyleA=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quada>0,\,b\in\mathbb{R}.} Non-example[edit] WenowpresentanexampleofagroupwithanuncountablenumberofelementsthatisnotaLiegroupunderacertaintopology.Thegroupgivenby H = { ( e 2 π i θ 0 0 e 2 π i a θ ) : θ ∈ R } ⊂ T 2 = { ( e 2 π i θ 0 0 e 2 π i ϕ ) : θ , ϕ ∈ R } , {\displaystyleH=\left\{\left({\begin{matrix}e^{2\pii\theta}&0\\0&e^{2\piia\theta}\end{matrix}}\right):\,\theta\in\mathbb{R}\right\}\subset\mathbb{T}^{2}=\left\{\left({\begin{matrix}e^{2\pii\theta}&0\\0&e^{2\pii\phi}\end{matrix}}\right):\,\theta,\phi\in\mathbb{R}\right\},} with a ∈ R ∖ Q {\displaystylea\in\mathbb{R}\setminus\mathbb{Q}} afixedirrationalnumber,isasubgroupofthetorus T 2 {\displaystyle\mathbb{T}^{2}} thatisnotaLiegroupwhengiventhesubspacetopology.[3]Ifwetakeanysmallneighborhood U {\displaystyleU} ofapoint h {\displaystyleh} in H {\displaystyleH} ,forexample,theportionof H {\displaystyleH} in U {\displaystyleU} isdisconnected.Thegroup H {\displaystyleH} windsrepeatedlyaroundthetoruswithouteverreachingapreviouspointofthespiralandthusformsadensesubgroupof T 2 {\displaystyle\mathbb{T}^{2}} . Aportionofthegroup H {\displaystyleH} inside T 2 {\displaystyle\mathbb{T}^{2}} .Smallneighborhoodsoftheelement h ∈ H {\displaystyleh\inH} aredisconnectedinthesubsettopologyon H {\displaystyleH} Thegroup H {\displaystyleH} can,however,begivenadifferenttopology,inwhichthedistancebetweentwopoints h 1 , h 2 ∈ H {\displaystyleh_{1},h_{2}\inH} isdefinedasthelengthoftheshortestpathinthegroup H {\displaystyleH} joining h 1 {\displaystyleh_{1}} to h 2 {\displaystyleh_{2}} .Inthistopology, H {\displaystyleH} isidentifiedhomeomorphicallywiththereallinebyidentifyingeachelementwiththenumber θ {\displaystyle\theta} inthedefinitionof H {\displaystyleH} .Withthistopology, H {\displaystyleH} isjustthegroupofrealnumbersunderadditionandisthereforeaLiegroup. Thegroup H {\displaystyleH} isanexampleofa"Liesubgroup"ofaLiegroupthatisnotclosed.SeethediscussionbelowofLiesubgroupsinthesectiononbasicconcepts. MatrixLiegroups[edit] Let GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} denotethegroupof n × n {\displaystylen\timesn} invertiblematriceswithentriesin C {\displaystyle\mathbb{C}} .Anyclosedsubgroupof GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} isaLiegroup;[4]LiegroupsofthissortarecalledmatrixLiegroups.SincemostoftheinterestingexamplesofLiegroupscanberealizedasmatrixLiegroups,sometextbooksrestrictattentiontothisclass,includingthoseofHall,[5]Rossmann,[6]andStillwell.[7] RestrictingattentiontomatrixLiegroupssimplifiesthedefinitionoftheLiealgebraandtheexponentialmap.ThefollowingarestandardexamplesofmatrixLiegroups. Thespeciallineargroupsover R {\displaystyle\mathbb{R}} and C {\displaystyle\mathbb{C}} , SL ⁡ ( n , R ) {\displaystyle\operatorname{SL}(n,\mathbb{R})} and SL ⁡ ( n , C ) {\displaystyle\operatorname{SL}(n,\mathbb{C})} ,consistingof n × n {\displaystylen\timesn} matriceswithdeterminantoneandentriesin R {\displaystyle\mathbb{R}} or C {\displaystyle\mathbb{C}} Theunitarygroupsandspecialunitarygroups, U ( n ) {\displaystyle{\text{U}}(n)} and SU ( n ) {\displaystyle{\text{SU}}(n)} ,consistingof n × n {\displaystylen\timesn} complexmatricessatisfying U ∗ = U − 1 {\displaystyleU^{*}=U^{-1}} (andalso det ( U ) = 1 {\displaystyle\det(U)=1} inthecaseof SU ( n ) {\displaystyle{\text{SU}}(n)} ) Theorthogonalgroupsandspecialorthogonalgroups, O ( n ) {\displaystyle{\text{O}}(n)} and SO ( n ) {\displaystyle{\text{SO}}(n)} ,consistingof n × n {\displaystylen\timesn} realmatricessatisfying R T = R − 1 {\displaystyleR^{\mathrm{T}}=R^{-1}} (andalso det ( R ) = 1 {\displaystyle\det(R)=1} inthecaseof SO ( n ) {\displaystyle{\text{SO}}(n)} ) Alloftheprecedingexamplesfallundertheheadingoftheclassicalgroups. Relatedconcepts[edit] AcomplexLiegroupisdefinedinthesamewayusingcomplexmanifoldsratherthanrealones(example: SL ⁡ ( 2 , C ) {\displaystyle\operatorname{SL}(2,\mathbb{C})} ),andholomorphicmaps.Similarly,usinganalternatemetriccompletionof Q {\displaystyle\mathbb{Q}} ,onecandefineap-adicLiegroupoverthep-adicnumbers,atopologicalgroupwhichisalsoananalyticp-adicmanifold,suchthatthegroupoperationsareanalytic.Inparticular,eachpointhasap-adicneighborhood. Hilbert'sfifthproblemaskedwhetherreplacingdifferentiablemanifoldswithtopologicaloranalyticonescanyieldnewexamples.Theanswertothisquestionturnedouttobenegative:in1952,Gleason,MontgomeryandZippinshowedthatifGisatopologicalmanifoldwithcontinuousgroupoperations,thenthereexistsexactlyoneanalyticstructureonGwhichturnsitintoaLiegroup(seealsoHilbert–Smithconjecture).Iftheunderlyingmanifoldisallowedtobeinfinite-dimensional(forexample,aHilbertmanifold),thenonearrivesatthenotionofaninfinite-dimensionalLiegroup.ItispossibletodefineanaloguesofmanyLiegroupsoverfinitefields,andthesegivemostoftheexamplesoffinitesimplegroups. ThelanguageofcategorytheoryprovidesaconcisedefinitionforLiegroups:aLiegroupisagroupobjectinthecategoryofsmoothmanifolds.Thisisimportant,becauseitallowsgeneralizationofthenotionofaLiegrouptoLiesupergroups.ThiscategoricalpointofviewleadsalsotoadifferentgeneralizationofLiegroups,namelyLiegroupoids,whicharegroupoidobjectsinthecategoryofsmoothmanifoldswithafurtherrequirement. Topologicaldefinition[edit] ALiegroupcanbedefinedasa(Hausdorff)topologicalgroupthat,neartheidentityelement,lookslikeatransformationgroup,withnoreferencetodifferentiablemanifolds.[8]First,wedefineanimmerselylinearLiegrouptobeasubgroupGofthegenerallineargroup GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} suchthat forsomeneighborhoodVoftheidentityelementeinG,thetopologyonVisthesubspacetopologyof GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} andVisclosedin GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} . Ghasatmostcountablymanyconnectedcomponents. (Forexample,aclosedsubgroupof GL ⁡ ( n , C ) {\displaystyle\operatorname{GL}(n,\mathbb{C})} ;thatis,amatrixLiegroupsatisfiestheaboveconditions.) ThenaLiegroupisdefinedasatopologicalgroupthat(1)islocallyisomorphicneartheidentitiestoanimmerselylinearLiegroupand(2)hasatmostcountablymanyconnectedcomponents.Showingthetopologicaldefinitionisequivalenttotheusualoneistechnical(andthebeginningreadersshouldskipthefollowing)butisdoneroughlyasfollows: GivenaLiegroupGintheusualmanifoldsense,theLiegroup–Liealgebracorrespondence(oraversionofLie'sthirdtheorem)constructsanimmersedLiesubgroup G ′ ⊂ GL ⁡ ( n , C ) {\displaystyleG'\subset\operatorname{GL}(n,\mathbb{C})} suchthat G , G ′ {\displaystyleG,G'} sharethesameLiealgebra;thus,theyarelocallyisomorphic.Hence,Gsatisfiestheabovetopologicaldefinition. Conversely,letGbeatopologicalgroupthatisaLiegroupintheabovetopologicalsenseandchooseanimmerselylinearLiegroup G ′ {\displaystyleG'} thatislocallyisomorphictoG.Then,byaversionoftheclosedsubgrouptheorem, G ′ {\displaystyleG'} isareal-analyticmanifoldandthen,throughthelocalisomorphism,Gacquiresastructureofamanifoldneartheidentityelement.OnethenshowsthatthegrouplawonGcanbegivenbyformalpowerseries;[9]sothegroupoperationsarereal-analyticandGitselfisareal-analyticmanifold. ThetopologicaldefinitionimpliesthestatementthatiftwoLiegroupsareisomorphicastopologicalgroups,thentheyareisomorphicasLiegroups.Infact,itstatesthegeneralprinciplethat,toalargeextent,thetopologyofaLiegrouptogetherwiththegrouplawdeterminesthegeometryofthegroup. MoreexamplesofLiegroups[edit] Seealso:TableofLiegroupsandListofsimpleLiegroups Liegroupsoccurinabundancethroughoutmathematicsandphysics.Matrixgroupsoralgebraicgroupsare(roughly)groupsofmatrices(forexample,orthogonalandsymplecticgroups),andthesegivemostofthemorecommonexamplesofLiegroups. Dimensionsoneandtwo[edit] TheonlyconnectedLiegroupswithdimensiononearetherealline R {\displaystyle\mathbb{R}} (withthegroupoperationbeingaddition)andthecirclegroup S 1 {\displaystyleS^{1}} ofcomplexnumberswithabsolutevalueone(withthegroupoperationbeingmultiplication).The S 1 {\displaystyleS^{1}} groupisoftendenotedas U ( 1 ) {\displaystyleU(1)} ,thegroupof 1 × 1 {\displaystyle1\times1} unitarymatrices. Intwodimensions,ifwerestrictattentiontosimplyconnectedgroups,thentheyareclassifiedbytheirLiealgebras.Thereare(uptoisomorphism)onlytwoLiealgebrasofdimensiontwo.TheassociatedsimplyconnectedLiegroupsare R 2 {\displaystyle\mathbb{R}^{2}} (withthegroupoperationbeingvectoraddition)andtheaffinegroupindimensionone,describedintheprevioussubsectionunder"firstexamples". Additionalexamples[edit] ThegroupSU(2)isthegroupof 2 × 2 {\displaystyle2\times2} unitarymatriceswithdeterminant 1 {\displaystyle1} .Topologically, SU ( 2 ) {\displaystyle{\text{SU}}(2)} isthe 3 {\displaystyle3} -sphere S 3 {\displaystyleS^{3}} ;asagroup,itmaybeidentifiedwiththegroupofunitquaternions. TheHeisenberggroupisaconnectednilpotentLiegroupofdimension 3 {\displaystyle3} ,playingakeyroleinquantummechanics. TheLorentzgroupisa6-dimensionalLiegroupoflinearisometriesoftheMinkowskispace. ThePoincarégroupisa10-dimensionalLiegroupofaffineisometriesoftheMinkowskispace. TheexceptionalLiegroupsoftypesG2,F4,E6,E7,E8havedimensions14,52,78,133,and248.AlongwiththeA-B-C-DseriesofsimpleLiegroups,theexceptionalgroupscompletethelistofsimpleLiegroups. Thesymplecticgroup Sp ( 2 n , R ) {\displaystyle{\text{Sp}}(2n,\mathbb{R})} consistsofall 2 n × 2 n {\displaystyle2n\times2n} matricespreservingasymplecticformon R 2 n {\displaystyle\mathbb{R}^{2n}} .ItisaconnectedLiegroupofdimension 2 n 2 + n {\displaystyle2n^{2}+n} . Constructions[edit] ThereareseveralstandardwaystoformnewLiegroupsfromoldones: TheproductoftwoLiegroupsisaLiegroup. AnytopologicallyclosedsubgroupofaLiegroupisaLiegroup.ThisisknownastheClosedsubgrouptheoremorCartan'stheorem. ThequotientofaLiegroupbyaclosednormalsubgroupisaLiegroup. TheuniversalcoverofaconnectedLiegroupisaLiegroup.Forexample,thegroup R {\displaystyle\mathbb{R}} istheuniversalcoverofthecirclegroup S 1 {\displaystyleS^{1}} .Infactanycoveringofadifferentiablemanifoldisalsoadifferentiablemanifold,butbyspecifyinguniversalcover,oneguaranteesagroupstructure(compatiblewithitsotherstructures). Relatednotions[edit] SomeexamplesofgroupsthatarenotLiegroups(exceptinthetrivialsensethatanygrouphavingatmostcountablymanyelementscanbeviewedasa0-dimensionalLiegroup,withthediscretetopology),are: Infinite-dimensionalgroups,suchastheadditivegroupofaninfinite-dimensionalrealvectorspace,orthespaceofsmoothfunctionsfromamanifold X {\displaystyleX} toaLiegroup G {\displaystyleG} , C ∞ ( X , G ) {\displaystyleC^{\infty}(X,G)} .ThesearenotLiegroupsastheyarenotfinite-dimensionalmanifolds. Sometotallydisconnectedgroups,suchastheGaloisgroupofaninfiniteextensionoffields,ortheadditivegroupofthep-adicnumbers.ThesearenotLiegroupsbecausetheirunderlyingspacesarenotrealmanifolds.(Someofthesegroupsare"p-adicLiegroups".)Ingeneral,onlytopologicalgroupshavingsimilarlocalpropertiestoRnforsomepositiveintegerncanbeLiegroups(ofcoursetheymustalsohaveadifferentiablestructure). Basicconcepts[edit] TheLiealgebraassociatedwithaLiegroup[edit] Mainarticle:Liegroup–Liealgebracorrespondence ToeveryLiegroupwecanassociateaLiealgebrawhoseunderlyingvectorspaceisthetangentspaceoftheLiegroupattheidentityelementandwhichcompletelycapturesthelocalstructureofthegroup.InformallywecanthinkofelementsoftheLiealgebraaselementsofthegroupthatare"infinitesimallyclose"totheidentity,andtheLiebracketoftheLiealgebraisrelatedtothecommutatoroftwosuchinfinitesimalelements.Beforegivingtheabstractdefinitionwegiveafewexamples: TheLiealgebraofthevectorspaceRnisjustRnwiththeLiebracketgivenby   [A, B]=0.(IngeneraltheLiebracketofaconnectedLiegroupisalways0ifandonlyiftheLiegroupisabelian.) TheLiealgebraofthegenerallineargroupGL(n,C)ofinvertiblematricesisthevectorspaceM(n,C)ofsquarematriceswiththeLiebracketgivenby   [A, B]=AB − BA. IfGisaclosedsubgroupofGL(n,C)thentheLiealgebraofGcanbethoughtofinformallyasthematricesmofM(n,C)suchthat1 + εmisinG,whereεisaninfinitesimalpositivenumberwithε2 = 0(ofcourse,nosuchrealnumberεexists).Forexample,theorthogonalgroupO(n,R)consistsofmatricesAwithAAT = 1,sotheLiealgebraconsistsofthematricesmwith(1 + εm)(1 + εm)T = 1,whichisequivalenttom + mT = 0becauseε2 = 0. Theprecedingdescriptioncanbemademorerigorousasfollows.TheLiealgebraofaclosedsubgroupGofGL(n,C),maybecomputedas Lie ⁡ ( G ) = { X ∈ M ( n ; C ) | exp ⁡ ( t X ) ∈ G  forall  t  in  R } , {\displaystyle\operatorname{Lie}(G)=\{X\inM(n;\mathbb{C})|\operatorname{exp}(tX)\inG{\text{forall}}t{\text{in}}\mathbb{\mathbb{R}}\},} [10][5]whereexp(tX)isdefinedusingthematrixexponential.ItcanthenbeshownthattheLiealgebraofGisarealvectorspacethatisclosedunderthebracketoperation, [ X , Y ] = X Y − Y X {\displaystyle[X,Y]=XY-YX} .[11] Theconcretedefinitiongivenaboveformatrixgroupsiseasytoworkwith,buthassomeminorproblems:touseitwefirstneedtorepresentaLiegroupasagroupofmatrices,butnotallLiegroupscanberepresentedinthisway,anditisnotevenobviousthattheLiealgebraisindependentoftherepresentationweuse.[12]Togetaroundtheseproblemswegive thegeneraldefinitionoftheLiealgebraofaLiegroup(in4steps): VectorfieldsonanysmoothmanifoldMcanbethoughtofasderivationsXoftheringofsmoothfunctionsonthemanifold,andthereforeformaLiealgebraundertheLiebracket[X, Y] = XY − YX,becausetheLiebracketofanytwoderivationsisaderivation. IfGisanygroupactingsmoothlyonthemanifoldM,thenitactsonthevectorfields,andthevectorspaceofvectorfieldsfixedbythegroupisclosedundertheLiebracketandthereforealsoformsaLiealgebra. WeapplythisconstructiontothecasewhenthemanifoldMistheunderlyingspaceofaLiegroup G,withGactingonG = MbylefttranslationsLg(h) = gh.Thisshowsthatthespaceofleftinvariantvectorfields(vectorfieldssatisfyingLg*Xh= XghforeveryhinG,whereLg*denotesthedifferentialofLg)onaLiegroupisaLiealgebraundertheLiebracketofvectorfields. AnytangentvectorattheidentityofaLiegroupcanbeextendedtoaleftinvariantvectorfieldbylefttranslatingthetangentvectortootherpointsofthemanifold.Specifically,theleftinvariantextensionofanelementvofthetangentspaceattheidentityisthevectorfielddefinedbyv^g = Lg*v.ThisidentifiesthetangentspaceTeGattheidentitywiththespaceofleftinvariantvectorfields,andthereforemakesthetangentspaceattheidentityintoaLiealgebra,calledtheLiealgebraofG,usuallydenotedbyaFraktur g . {\displaystyle{\mathfrak{g}}.} ThustheLiebracketon g {\displaystyle{\mathfrak{g}}} isgivenexplicitlyby[v, w] = [v^, w^]e. ThisLiealgebra g {\displaystyle{\mathfrak{g}}} isfinite-dimensionalandithasthesamedimensionasthemanifoldG.TheLiealgebraofGdeterminesGupto"localisomorphism",wheretwoLiegroupsarecalledlocallyisomorphiciftheylookthesameneartheidentityelement. ProblemsaboutLiegroupsareoftensolvedbyfirstsolvingthecorrespondingproblemfortheLiealgebras,andtheresultforgroupsthenusuallyfollowseasily. Forexample,simpleLiegroupsareusuallyclassifiedbyfirstclassifyingthecorrespondingLiealgebras. WecouldalsodefineaLiealgebrastructureonTeusingrightinvariantvectorfieldsinsteadofleftinvariantvectorfields.ThisleadstothesameLiealgebra,becausetheinversemaponGcanbeusedtoidentifyleftinvariantvectorfieldswithrightinvariantvectorfields,andactsas−1onthetangentspaceTe. TheLiealgebrastructureonTecanalsobedescribedasfollows: thecommutatoroperation (x,y)→xyx−1y−1 onG×Gsends(e, e)toe,soitsderivativeyieldsabilinearoperationonTeG.Thisbilinearoperationisactuallythezeromap,butthesecondderivative,undertheproperidentificationoftangentspaces,yieldsanoperationthatsatisfiestheaxiomsofaLiebracket,anditisequaltotwicetheonedefinedthroughleft-invariantvectorfields. Homomorphismsandisomorphisms[edit] IfGandHareLiegroups,thenaLiegrouphomomorphismf :G→Hisasmoothgrouphomomorphism.InthecaseofcomplexLiegroups,suchahomomorphismisrequiredtobeaholomorphicmap.However,theserequirementsareabitstringent;everycontinuoushomomorphismbetweenrealLiegroupsturnsouttobe(real)analytic.[13] ThecompositionoftwoLiehomomorphismsisagainahomomorphism,andtheclassofallLiegroups,togetherwiththesemorphisms,formsacategory.Moreover,everyLiegrouphomomorphisminducesahomomorphismbetweenthecorrespondingLiealgebras.Let ϕ : G → H {\displaystyle\phi\colonG\toH} beaLiegrouphomomorphismandlet ϕ ∗ {\displaystyle\phi_{*}} beitsderivativeattheidentity.IfweidentifytheLiealgebrasofGandHwiththeirtangentspacesattheidentityelements,then ϕ ∗ {\displaystyle\phi_{*}} isamapbetweenthecorrespondingLiealgebras: ϕ ∗ : g → h , {\displaystyle\phi_{*}\colon{\mathfrak{g}}\to{\mathfrak{h}},} whichturnsouttobeaLiealgebrahomomorphism(meaningthatitisalinearmapwhichpreservestheLiebracket).Inthelanguageofcategorytheory,wethenhaveacovariantfunctorfromthecategoryofLiegroupstothecategoryofLiealgebraswhichsendsaLiegrouptoitsLiealgebraandaLiegrouphomomorphismtoitsderivativeattheidentity. TwoLiegroupsarecalledisomorphicifthereexistsabijectivehomomorphismbetweenthemwhoseinverseisalsoaLiegrouphomomorphism.Equivalently,itisadiffeomorphismwhichisalsoagrouphomomorphism.Observethat,bytheabove,acontinuoushomomorphismfromaLiegroup G {\displaystyleG} toaLiegroup H {\displaystyleH} isanisomorphismofLiegroupsifandonlyifitisbijective. LiegroupversusLiealgebraisomorphisms[edit] IsomorphicLiegroupsnecessarilyhaveisomorphicLiealgebras;itisthenreasonabletoaskhowisomorphismclassesofLiegroupsrelatetoisomorphismclassesofLiealgebras. ThefirstresultinthisdirectionisLie'sthirdtheorem,whichstatesthateveryfinite-dimensional,realLiealgebraistheLiealgebraofsome(linear)Liegroup.OnewaytoproveLie'sthirdtheoremistouseAdo'stheorem,whichsayseveryfinite-dimensionalrealLiealgebraisisomorphictoamatrixLiealgebra.Meanwhile,foreveryfinite-dimensionalmatrixLiealgebra,thereisalineargroup(matrixLiegroup)withthisalgebraasitsLiealgebra.[14] Ontheotherhand,LiegroupswithisomorphicLiealgebrasneednotbeisomorphic.Furthermore,thisresultremainstrueevenifweassumethegroupsareconnected.Toputitdifferently,theglobalstructureofaLiegroupisnotdeterminedbyitsLiealgebra;forexample,ifZisanydiscretesubgroupofthecenterofGthenGandG/ZhavethesameLiealgebra(seethetableofLiegroupsforexamples).AnexampleofimportanceinphysicsarethegroupsSU(2)andSO(3).ThesetwogroupshaveisomorphicLiealgebras,[15]butthegroupsthemselvesarenotisomorphic,becauseSU(2)issimplyconnectedbutSO(3)isnot.[16] Ontheotherhand,ifwerequirethattheLiegroupbesimplyconnected,thentheglobalstructureisdeterminedbyitsLiealgebra:twosimplyconnectedLiegroupswithisomorphicLiealgebrasareisomorphic.[17](SeethenextsubsectionformoreinformationaboutsimplyconnectedLiegroups.)InlightofLie'sthirdtheorem,wemaythereforesaythatthereisaone-to-onecorrespondencebetweenisomorphismclassesoffinite-dimensionalrealLiealgebrasandisomorphismclassesofsimplyconnectedLiegroups. SimplyconnectedLiegroups[edit] Seealso:Liegroup–LiealgebracorrespondenceandFundamentalgroup§ Liegroups ALiegroup G {\displaystyleG} issaidtobesimplyconnectedifeveryloopin G {\displaystyleG} canbeshrunkcontinuouslytoapointin G {\displaystyleG} .Thisnotionisimportantbecauseofthefollowingresultthathassimpleconnectednessasahypothesis: Theorem:[18]Suppose G {\displaystyleG} and H {\displaystyleH} areLiegroupswithLiealgebras g {\displaystyle{\mathfrak{g}}} and h {\displaystyle{\mathfrak{h}}} andthat f : g → h {\displaystylef:{\mathfrak{g}}\rightarrow{\mathfrak{h}}} isaLiealgebrahomomorphism.If G {\displaystyleG} issimplyconnected,thenthereisauniqueLiegrouphomomorphism ϕ : G → H {\displaystyle\phi:G\rightarrowH} suchthat ϕ ∗ = f {\displaystyle\phi_{*}=f} ,where ϕ ∗ {\displaystyle\phi_{*}} isthedifferentialof ϕ {\displaystyle\phi} attheidentity. Lie'sthirdtheoremsaysthateveryfinite-dimensionalrealLiealgebraistheLiealgebraofaLiegroup.ItfollowsfromLie'sthirdtheoremandtheprecedingresultthateveryfinite-dimensionalrealLiealgebraistheLiealgebraofauniquesimplyconnectedLiegroup. AnexampleofasimplyconnectedgroupisthespecialunitarygroupSU(2),whichasamanifoldisthe3-sphere.TherotationgroupSO(3),ontheotherhand,isnotsimplyconnected.(SeeTopologyofSO(3).)ThefailureofSO(3)tobesimplyconnectedisintimatelyconnectedtothedistinctionbetweenintegerspinandhalf-integerspininquantummechanics.OtherexamplesofsimplyconnectedLiegroupsincludethespecialunitarygroupSU(n),thespingroup(doublecoverofrotationgroup)Spin(n)for n ≥ 3 {\displaystylen\geq3} ,andthecompactsymplecticgroupSp(n).[19] MethodsfordeterminingwhetheraLiegroupissimplyconnectedornotarediscussedinthearticleonfundamentalgroupsofLiegroups. Theexponentialmap[edit] Mainarticle:Exponentialmap(Lietheory) Seealso:derivativeoftheexponentialmapandnormalcoordinates TheexponentialmapfromtheLiealgebra M ( n ; C ) {\displaystyleM(n;\mathbb{C})} ofthegenerallineargroup G L ( n ; C ) {\displaystyleGL(n;\mathbb{C})} to G L ( n ; C ) {\displaystyleGL(n;\mathbb{C})} isdefinedbythematrixexponential,givenbytheusualpowerseries: exp ⁡ ( X ) = 1 + X + X 2 2 ! + X 3 3 ! + ⋯ {\displaystyle\exp(X)=1+X+{\frac{X^{2}}{2!}}+{\frac{X^{3}}{3!}}+\cdots} formatrices X {\displaystyleX} .If G {\displaystyleG} isaclosedsubgroupof G L ( n ; C ) {\displaystyleGL(n;\mathbb{C})} ,thentheexponentialmaptakestheLiealgebraof G {\displaystyleG} into G {\displaystyleG} ;thus,wehaveanexponentialmapforallmatrixgroups.Everyelementof G {\displaystyleG} thatissufficientlyclosetotheidentityistheexponentialofamatrixintheLiealgebra.[20] Thedefinitionaboveiseasytouse,butitisnotdefinedforLiegroupsthatarenotmatrixgroups,anditisnotclearthattheexponentialmapofaLiegroupdoesnotdependonitsrepresentationasamatrixgroup.WecansolvebothproblemsusingamoreabstractdefinitionoftheexponentialmapthatworksforallLiegroups,asfollows. Foreachvector X {\displaystyleX} intheLiealgebra g {\displaystyle{\mathfrak{g}}} of G {\displaystyleG} (i.e.,thetangentspaceto G {\displaystyleG} attheidentity),oneprovesthatthereisauniqueone-parametersubgroup c : R → G {\displaystylec:\mathbb{R}\rightarrowG} suchthat c ′ ( 0 ) = X {\displaystylec'(0)=X} .Sayingthat c {\displaystylec} isaone-parametersubgroupmeanssimplythat c {\displaystylec} isasmoothmapinto G {\displaystyleG} andthat c ( s + t ) = c ( s ) c ( t )   {\displaystylec(s+t)=c(s)c(t)\} forall s {\displaystyles} and t {\displaystylet} .Theoperationontherighthandsideisthegroupmultiplicationin G {\displaystyleG} .Theformalsimilarityofthisformulawiththeonevalidfortheexponentialfunctionjustifiesthedefinition exp ⁡ ( X ) = c ( 1 ) .   {\displaystyle\exp(X)=c(1).\} Thisiscalledtheexponentialmap,anditmapstheLiealgebra g {\displaystyle{\mathfrak{g}}} intotheLiegroup G {\displaystyleG} .Itprovidesadiffeomorphismbetweenaneighborhoodof0in g {\displaystyle{\mathfrak{g}}} andaneighborhoodof e {\displaystylee} in G {\displaystyleG} .Thisexponentialmapisageneralizationoftheexponentialfunctionforrealnumbers(because R {\displaystyle\mathbb{R}} istheLiealgebraoftheLiegroupofpositiverealnumberswithmultiplication),forcomplexnumbers(because C {\displaystyle\mathbb{C}} istheLiealgebraoftheLiegroupofnon-zerocomplexnumberswithmultiplication)andformatrices(because M ( n , R ) {\displaystyleM(n,\mathbb{R})} withtheregularcommutatoristheLiealgebraoftheLiegroup G L ( n , R ) {\displaystyleGL(n,\mathbb{R})} ofallinvertiblematrices). Becausetheexponentialmapissurjectiveonsomeneighbourhood N {\displaystyleN} of e {\displaystylee} ,itiscommontocallelementsoftheLiealgebrainfinitesimalgeneratorsofthegroup G {\displaystyleG} .Thesubgroupof G {\displaystyleG} generatedby N {\displaystyleN} istheidentitycomponentof G {\displaystyleG} . TheexponentialmapandtheLiealgebradeterminethelocalgroupstructureofeveryconnectedLiegroup,becauseoftheBaker–Campbell–Hausdorffformula:thereexistsaneighborhood U {\displaystyleU} ofthezeroelementof g {\displaystyle{\mathfrak{g}}} ,suchthatfor X , Y ∈ U {\displaystyleX,Y\inU} wehave exp ⁡ ( X ) exp ⁡ ( Y ) = exp ⁡ ( X + Y + 1 2 [ X , Y ] + 1 12 [ [ X , Y ] , Y ] − 1 12 [ [ X , Y ] , X ] − ⋯ ) , {\displaystyle\exp(X)\,\exp(Y)=\exp\left(X+Y+{\tfrac{1}{2}}[X,Y]+{\tfrac{1}{12}}[\,[X,Y],Y]-{\tfrac{1}{12}}[\,[X,Y],X]-\cdots\right),} wheretheomittedtermsareknownandinvolveLiebracketsoffourormoreelements.Incase X {\displaystyleX} and Y {\displaystyleY} commute,thisformulareducestothefamiliarexponentiallaw exp ⁡ ( X ) exp ⁡ ( Y ) = exp ⁡ ( X + Y ) {\displaystyle\exp(X)\exp(Y)=\exp(X+Y)} TheexponentialmaprelatesLiegrouphomomorphisms.Thatis,if ϕ : G → H {\displaystyle\phi:G\toH} isaLiegrouphomomorphismand ϕ ∗ : g → h {\displaystyle\phi_{*}:{\mathfrak{g}}\to{\mathfrak{h}}} theinducedmaponthecorrespondingLiealgebras,thenforall x ∈ g {\displaystylex\in{\mathfrak{g}}} wehave ϕ ( exp ⁡ ( x ) ) = exp ⁡ ( ϕ ∗ ( x ) ) . {\displaystyle\phi(\exp(x))=\exp(\phi_{*}(x)).\,} Inotherwords,thefollowingdiagramcommutes,[Note1] (Inshort,expisanaturaltransformationfromthefunctorLietotheidentityfunctoronthecategoryofLiegroups.) TheexponentialmapfromtheLiealgebratotheLiegroupisnotalwaysonto,evenifthegroupisconnected(thoughitdoesmapontotheLiegroupforconnectedgroupsthatareeithercompactornilpotent).Forexample,theexponentialmapofSL(2,R)isnotsurjective.Also,theexponentialmapisneithersurjectivenorinjectiveforinfinite-dimensional(seebelow)LiegroupsmodelledonC∞Fréchetspace,evenfromarbitrarysmallneighborhoodof0tocorrespondingneighborhoodof1. Liesubgroup[edit] ALiesubgroup H {\displaystyleH} ofaLiegroup G {\displaystyleG} isaLiegroupthatisasubsetof G {\displaystyleG} andsuchthattheinclusionmapfrom H {\displaystyleH} to G {\displaystyleG} isaninjectiveimmersionandgrouphomomorphism.AccordingtoCartan'stheorem,aclosedsubgroupof G {\displaystyleG} admitsauniquesmoothstructurewhichmakesitanembeddedLiesubgroupof G {\displaystyleG} —i.e.aLiesubgroupsuchthattheinclusionmapisasmoothembedding. Examplesofnon-closedsubgroupsareplentiful;forexampletake G {\displaystyleG} tobeatorusofdimension2orgreater,andlet H {\displaystyleH} beaone-parametersubgroupofirrationalslope,i.e.onethatwindsaroundinG.ThenthereisaLiegrouphomomorphism φ : R → G {\displaystyle\varphi:\mathbb{R}\toG} with i m ( φ ) = H {\displaystyle\mathrm{im}(\varphi)=H} .Theclosureof H {\displaystyleH} willbeasub-torusin G {\displaystyleG} . Theexponentialmapgivesaone-to-onecorrespondencebetweentheconnectedLiesubgroupsofaconnectedLiegroup G {\displaystyleG} andthesubalgebrasoftheLiealgebraof G {\displaystyleG} .[21]Typically,thesubgroupcorrespondingtoasubalgebraisnotaclosedsubgroup.Thereisnocriterionsolelybasedonthestructureof G {\displaystyleG} whichdetermineswhichsubalgebrascorrespondtoclosedsubgroups. Representations[edit] Mainarticle:RepresentationofaLiegroup Seealso:Compactgroup§ RepresentationtheoryofaconnectedcompactLiegroup,andLiealgebrarepresentation OneimportantaspectofthestudyofLiegroupsistheirrepresentations,thatis,thewaytheycanact(linearly)onvectorspaces.Inphysics,Liegroupsoftenencodethesymmetriesofaphysicalsystem.Thewayonemakesuseofthissymmetrytohelpanalyzethesystemisoftenthroughrepresentationtheory.Consider,forexample,thetime-independentSchrödingerequationinquantummechanics, H ^ ψ = E ψ {\displaystyle{\hat{H}}\psi=E\psi} .AssumethesysteminquestionhastherotationgroupSO(3)asasymmetry,meaningthattheHamiltonianoperator H ^ {\displaystyle{\hat{H}}} commuteswiththeactionofSO(3)onthewavefunction ψ {\displaystyle\psi} .(OneimportantexampleofsuchasystemistheHydrogenatom,whichhasasinglesphericalorbital.)Thisassumptiondoesnotnecessarilymeanthatthesolutions ψ {\displaystyle\psi} arerotationallyinvariantfunctions.Rather,itmeansthatthespaceofsolutionsto H ^ ψ = E ψ {\displaystyle{\hat{H}}\psi=E\psi} isinvariantunderrotations(foreachfixedvalueof E {\displaystyleE} ).Thisspace,therefore,constitutesarepresentationofSO(3).Theserepresentationshavebeenclassifiedandtheclassificationleadstoasubstantialsimplificationoftheproblem,essentiallyconvertingathree-dimensionalpartialdifferentialequationtoaone-dimensionalordinarydifferentialequation. ThecaseofaconnectedcompactLiegroupK(includingthejust-mentionedcaseofSO(3))isparticularlytractable.[22]Inthatcase,everyfinite-dimensionalrepresentationofKdecomposesasadirectsumofirreduciblerepresentations.Theirreduciblerepresentations,inturn,wereclassifiedbyHermannWeyl.Theclassificationisintermsofthe"highestweight"oftherepresentation.TheclassificationiscloselyrelatedtotheclassificationofrepresentationsofasemisimpleLiealgebra. Onecanalsostudy(ingeneralinfinite-dimensional)unitaryrepresentationsofanarbitraryLiegroup(notnecessarilycompact).Forexample,itispossibletogivearelativelysimpleexplicitdescriptionoftherepresentationsofthegroupSL(2,R)andtherepresentationsofthePoincarégroup. Classification[edit] Liegroupsmaybethoughtofassmoothlyvaryingfamiliesofsymmetries.Examplesofsymmetriesincluderotationaboutanaxis.Whatmustbeunderstoodisthenatureof'small'transformations,forexample,rotationsthroughtinyangles,thatlinknearbytransformations.ThemathematicalobjectcapturingthisstructureiscalledaLiealgebra(Liehimselfcalledthem"infinitesimalgroups").ItcanbedefinedbecauseLiegroupsaresmoothmanifolds,sohavetangentspacesateachpoint. TheLiealgebraofanycompactLiegroup(veryroughly:oneforwhichthesymmetriesformaboundedset)canbedecomposedasadirectsumofanabelianLiealgebraandsomenumberofsimpleones.ThestructureofanabelianLiealgebraismathematicallyuninteresting(sincetheLiebracketisidenticallyzero);theinterestisinthesimplesummands.Hencethequestionarises:whatarethesimpleLiealgebrasofcompactgroups?Itturnsoutthattheymostlyfallintofourinfinitefamilies,the"classicalLiealgebras"An,Bn,CnandDn,whichhavesimpledescriptionsintermsofsymmetriesofEuclideanspace.Buttherearealsojustfive"exceptionalLiealgebras"thatdonotfallintoanyofthesefamilies.E8isthelargestofthese. Liegroupsareclassifiedaccordingtotheiralgebraicproperties(simple,semisimple,solvable,nilpotent,abelian),theirconnectedness(connectedorsimplyconnected)andtheircompactness. AfirstkeyresultistheLevidecomposition,whichsaysthateverysimplyconnectedLiegroupisthesemidirectproductofasolvablenormalsubgroupandasemisimplesubgroup. ConnectedcompactLiegroupsareallknown:theyarefinitecentralquotientsofaproductofcopiesofthecirclegroupS1andsimplecompactLiegroups(whichcorrespondtoconnectedDynkindiagrams). AnysimplyconnectedsolvableLiegroupisisomorphictoaclosedsubgroupofthegroupofinvertibleuppertriangularmatricesofsomerank,andanyfinite-dimensionalirreduciblerepresentationofsuchagroupis1-dimensional.Solvablegroupsaretoomessytoclassifyexceptinafewsmalldimensions. AnysimplyconnectednilpotentLiegroupisisomorphictoaclosedsubgroupofthegroupofinvertibleuppertriangularmatriceswith1'sonthediagonalofsomerank,andanyfinite-dimensionalirreduciblerepresentationofsuchagroupis1-dimensional.Likesolvablegroups,nilpotentgroupsaretoomessytoclassifyexceptinafewsmalldimensions. SimpleLiegroupsaresometimesdefinedtobethosethataresimpleasabstractgroups,andsometimesdefinedtobeconnectedLiegroupswithasimpleLiealgebra.Forexample,SL(2,R)issimpleaccordingtotheseconddefinitionbutnotaccordingtothefirst.Theyhaveallbeenclassified(foreitherdefinition). SemisimpleLiegroupsareLiegroupswhoseLiealgebraisaproductofsimpleLiealgebras.[23]TheyarecentralextensionsofproductsofsimpleLiegroups. TheidentitycomponentofanyLiegroupisanopennormalsubgroup,andthequotientgroupisadiscretegroup.TheuniversalcoverofanyconnectedLiegroupisasimplyconnectedLiegroup,andconverselyanyconnectedLiegroupisaquotientofasimplyconnectedLiegroupbyadiscretenormalsubgroupofthecenter.AnyLiegroupGcanbedecomposedintodiscrete,simple,andabeliangroupsinacanonicalwayasfollows.Write Gconfortheconnectedcomponentoftheidentity Gsolforthelargestconnectednormalsolvablesubgroup Gnilforthelargestconnectednormalnilpotentsubgroup sothatwehaveasequenceofnormalsubgroups 1⊆Gnil⊆Gsol⊆Gcon⊆G. Then G/Gconisdiscrete Gcon/GsolisacentralextensionofaproductofsimpleconnectedLiegroups. Gsol/Gnilisabelian.AconnectedabelianLiegroupisisomorphictoaproductofcopiesofRandthecirclegroupS1. Gnil/1isnilpotent,andthereforeitsascendingcentralserieshasallquotientsabelian. ThiscanbeusedtoreducesomeproblemsaboutLiegroups(suchasfindingtheirunitaryrepresentations)tothesameproblemsforconnectedsimplegroupsandnilpotentandsolvablesubgroupsofsmallerdimension. ThediffeomorphismgroupofaLiegroupactstransitivelyontheLiegroup EveryLiegroupisparallelizable,andhenceanorientablemanifold(thereisabundleisomorphismbetweenitstangentbundleandtheproductofitselfwiththetangentspaceattheidentity) Infinite-dimensionalLiegroups[edit] Liegroupsareoftendefinedtobefinite-dimensional,buttherearemanygroupsthatresembleLiegroups,exceptforbeinginfinite-dimensional.Thesimplestwaytodefineinfinite-dimensionalLiegroupsistomodelthemlocallyonBanachspaces(asopposedtoEuclideanspaceinthefinite-dimensionalcase),andinthiscasemuchofthebasictheoryissimilartothatoffinite-dimensionalLiegroups.Howeverthisisinadequateformanyapplications,becausemanynaturalexamplesofinfinite-dimensionalLiegroupsarenotBanachmanifolds.InsteadoneneedstodefineLiegroupsmodeledonmoregenerallocallyconvextopologicalvectorspaces.InthiscasetherelationbetweentheLiealgebraandtheLiegroupbecomesrathersubtle,andseveralresultsaboutfinite-dimensionalLiegroupsnolongerhold. Theliteratureisnotentirelyuniforminitsterminologyastoexactlywhichpropertiesofinfinite-dimensionalgroupsqualifythegroupfortheprefixLieinLiegroup.OntheLiealgebrasideofaffairs,thingsaresimplersincethequalifyingcriteriafortheprefixLieinLiealgebraarepurelyalgebraic.Forexample,aninfinite-dimensionalLiealgebramayormaynothaveacorrespondingLiegroup.Thatis,theremaybeagroupcorrespondingtotheLiealgebra,butitmightnotbeniceenoughtobecalledaLiegroup,ortheconnectionbetweenthegroupandtheLiealgebramightnotbeniceenough(forexample,failureoftheexponentialmaptobeontoaneighborhoodoftheidentity).Itisthe"niceenough"thatisnotuniversallydefined. Someoftheexamplesthathavebeenstudiedinclude: Thegroupofdiffeomorphismsofamanifold.Quitealotisknownaboutthegroupofdiffeomorphismsofthecircle.ItsLiealgebrais(moreorless)theWittalgebra,whosecentralextensiontheVirasoroalgebra(seeVirasoroalgebrafromWittalgebraforaderivationofthisfact)isthesymmetryalgebraoftwo-dimensionalconformalfieldtheory.DiffeomorphismgroupsofcompactmanifoldsoflargerdimensionareregularFréchetLiegroups;verylittleabouttheirstructureisknown. Thediffeomorphismgroupofspacetimesometimesappearsinattemptstoquantizegravity. Thegroupofsmoothmapsfromamanifoldtoafinite-dimensionalLiegroupisanexampleofagaugegroup(withoperationofpointwisemultiplication),andisusedinquantumfieldtheoryandDonaldsontheory.Ifthemanifoldisacirclethesearecalledloopgroups,andhavecentralextensionswhoseLiealgebrasare(moreorless)Kac–Moodyalgebras. Thereareinfinite-dimensionalanaloguesofgenerallineargroups,orthogonalgroups,andsoon.[24]Oneimportantaspectisthatthesemayhavesimplertopologicalproperties:seeforexampleKuiper'stheorem.InM-theory,forexample,a10dimensionalSU(N)gaugetheorybecomesan11dimensionaltheorywhenNbecomesinfinite. Seealso[edit] AdjointrepresentationofaLiegroup Haarmeasure Homogeneousspace ListofLiegrouptopics RepresentationsofLiegroups Symmetryinquantummechanics Notes[edit] Explanatorynotes[edit] ^"Archivedcopy"(PDF).Archivedfromtheoriginal(PDF)on2011-09-28.Retrieved2014-10-11.{{citeweb}}:CS1maint:archivedcopyastitle(link) Citations[edit] ^ArthurTresse(1893)."Surlesinvariantsdifférentielsdesgroupescontinusdetransformations".ActaMathematica.18:1–88.doi:10.1007/bf02418270. ^Borel(2001). ^Rossmann2001,Chapter2. ^Hall2015Corollary3.45 ^abHall2015 ^Rossmann2001 ^Stillwell2008 ^T.Kobayashi–T.Oshima,Definition5.3.harvnberror:notarget:CITEREFT._Kobayashi–T._Oshima(help) ^ThisisthestatementthataLiegroupisaformalLiegroup.Forthelatterconcept,fornow,seeF.Bruhat,LecturesonLieGroupsandRepresentationsofLocallyCompactGroups. ^Helgason1978,Ch.II,§2,Proposition2.7. ^Hall2015Theorem3.20 ^ButseeHall2015,Proposition3.30andExercise8inChapter3 ^Hall2015Corollary3.50.Hallonlyclaimssmoothness,butthesameargumentshowsanalyticity. ^Hall2015Theorem5.20 ^Hall2015Example3.27 ^Hall2015Section1.3.4 ^Hall2015Corollary5.7 ^Hall2015Theorem5.6 ^Hall2015Section13.2 ^Hall2015Theorem3.42 ^Hall2015Theorem5.20 ^Hall2015PartIII ^Helgason,Sigurdur(1978).DifferentialGeometry,LieGroups,andSymmetricSpaces.NewYork:AcademicPress.p. 131.ISBN 978-0-12-338460-7. ^Bäuerle,deKerf&tenKroode1997 References[edit] Adams,JohnFrank(1969),LecturesonLieGroups,ChicagoLecturesinMathematics,Chicago:Univ.ofChicagoPress,ISBN 978-0-226-00527-0,MR 0252560. Bäuerle,G.G.A;deKerf,E.A.;tenKroode,A.P.E.(1997).A.vanGroesen;E.M.deJager(eds.).FiniteandinfinitedimensionalLiealgebrasandtheirapplicationinphysics.Studiesinmathematicalphysics.Vol. 7.North-Holland.ISBN 978-0-444-82836-1–viaScienceDirect. Borel,Armand(2001),EssaysinthehistoryofLiegroupsandalgebraicgroups,HistoryofMathematics,vol. 21,Providence,R.I.:AmericanMathematicalSociety,ISBN 978-0-8218-0288-5,MR 1847105 Bourbaki,Nicolas,Elementsofmathematics:LiegroupsandLiealgebras.Chapters1–3ISBN 3-540-64242-0,Chapters4–6ISBN 3-540-42650-7,Chapters7–9ISBN 3-540-43405-4 Chevalley,Claude(1946),TheoryofLiegroups,Princeton:PrincetonUniversityPress,ISBN 978-0-691-04990-8. P.M.Cohn(1957)LieGroups,CambridgeTractsinMathematicalPhysics. J.L.Coolidge(1940)AHistoryofGeometricalMethods,pp304–17,OxfordUniversityPress(DoverPublications2003). Fulton,William;Harris,Joe(1991).Representationtheory.Afirstcourse.GraduateTextsinMathematics,ReadingsinMathematics.Vol. 129.NewYork:Springer-Verlag.doi:10.1007/978-1-4612-0979-9.ISBN 978-0-387-97495-8.MR 1153249.OCLC 246650103. RobertGilmore(2008)Liegroups,physics,andgeometry:anintroductionforphysicists,engineersandchemists,CambridgeUniversityPressISBN 9780521884006doi:10.1017/CBO9780511791390. Hall,BrianC.(2015),LieGroups,LieAlgebras,andRepresentations:AnElementaryIntroduction,GraduateTextsinMathematics,vol. 222(2nd ed.),Springer,doi:10.1007/978-3-319-13467-3,ISBN 978-3319134666. F.ReeseHarvey(1990)Spinorsandcalibrations,AcademicPress,ISBN 0-12-329650-1. Hawkins,Thomas(2000),EmergenceofthetheoryofLiegroups,SourcesandStudiesintheHistoryofMathematicsandPhysicalSciences,Berlin,NewYork:Springer-Verlag,doi:10.1007/978-1-4612-1202-7,ISBN 978-0-387-98963-1,MR 1771134Borel'sreview Helgason,Sigurdur(2001),Differentialgeometry,Liegroups,andsymmetricspaces,GraduateStudiesinMathematics,vol. 34,Providence,R.I.:AmericanMathematicalSociety,doi:10.1090/gsm/034,ISBN 978-0-8218-2848-9,MR 1834454 Knapp,AnthonyW.(2002),LieGroupsBeyondanIntroduction,ProgressinMathematics,vol. 140(2nd ed.),Boston:Birkhäuser,ISBN 978-0-8176-4259-4. T.KobayashiandT.Oshima,LiegroupsandLiealgebrasI,Iwanami,1999(inJapanese) Nijenhuis,Albert(1959)."Review:Liegroups,byP.M.Cohn".BulletinoftheAmericanMathematicalSociety.65(6):338–341.doi:10.1090/s0002-9904-1959-10358-x. Rossmann,Wulf(2001),LieGroups:AnIntroductionThroughLinearGroups,OxfordGraduateTextsinMathematics,OxfordUniversityPress,ISBN 978-0-19-859683-7.The2003reprintcorrectsseveraltypographicalmistakes. Sattinger,DavidH.;Weaver,O.L.(1986).Liegroupsandalgebraswithapplicationstophysics,geometry,andmechanics.Springer-Verlag.doi:10.1007/978-1-4757-1910-9.ISBN 978-3-540-96240-3.MR 0835009. Serre,Jean-Pierre(1965),LieAlgebrasandLieGroups:1964LecturesgivenatHarvardUniversity,Lecturenotesinmathematics,vol. 1500,Springer,ISBN 978-3-540-55008-2. Stillwell,John(2008).NaiveLieTheory.UndergraduateTextsinMathematics.Springer.doi:10.1007/978-0-387-78214-0.ISBN 978-0387782140. HeldermannVerlagJournalofLieTheory Warner,FrankW.(1983),FoundationsofdifferentiablemanifoldsandLiegroups,GraduateTextsinMathematics,vol. 94,NewYorkBerlinHeidelberg:Springer-Verlag,doi:10.1007/978-1-4757-1799-0,ISBN 978-0-387-90894-6,MR 0722297 Steeb,Willi-Hans(2007),ContinuousSymmetries,Liealgebras,DifferentialEquationsandComputerAlgebra:secondedition,WorldScientificPublishing,doi:10.1142/6515,ISBN 978-981-270-809-0,MR 2382250. LieGroups.RepresentationTheoryandSymmetricSpacesWolfgangZiller,Vorlesung2010 Externallinks[edit] MediarelatedtoLiegroupsatWikimediaCommons vteManifolds(Glossary)Basicconcepts Topologicalmanifold Atlas Differentiable/Smoothmanifold Differentialstructure Smoothatlas Submanifold Riemannianmanifold Smoothmap Submersion Pushforward Tangentspace Differentialform Vectorfield Mainresults(list) Atiyah–Singerindex Darboux's DeRham's Frobenius GeneralizedStokes Hopf–Rinow Noether's Sard's Whitneyembedding Maps Curve Diffeomorphism Local Geodesic Exponentialmap inLietheory Foliation Immersion Integralcurve Liederivative Section Submersion Typesofmanifolds Closed (Almost) Complex (Almost) Contact Fibered Finsler G-structure Hermitian Hyperbolic Kählermanifold Kenmotsu Hermitianmanifold Liegroup Liealgebra Manifoldwithboundary Oriented Poisson Prime Quaternionicmanifold (Pseudo−, Sub−) Riemannian Rizza (Almost) Symplectic Tame TensorsVectors Distribution Liebracket Pushforward Tangentspace bundle Torsion Vectorfield Vectorflow Covectors Closed/Exact Covariantderivative Cotangentspace bundle DeRhamcohomology Differentialform Vector-valued Exteriorderivative Interiorproduct Pullback Riccicurvature flow Riemanncurvaturetensor Tensorfield density Volumeform Wedgeproduct Bundles Cotangentbundle Fiberbundle (Co) Fibration Jetbundle Principalbundle Subbundle Tangentbundle Tensorbundle Vectorbundle Connections Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Paralleltransport Related Classificationofmanifolds Gaugetheory History Morsetheory Movingframe Singularitytheory Generalizations Banachmanifold Diffeology Diffiety Fréchetmanifold K-theory Orbifold Secondarycalculus overcommutativealgebras Sheaf Stratifold Supermanifold Topologicallystratifiedspace AuthoritycontrolNationallibraries Spain France(data) Germany Israel UnitedStates Japan Other FacetedApplicationofSubjectTerminology SUDOC(France) 1 Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Lie_group&oldid=1082143188" Categories:LiegroupsManifoldsSymmetryHiddencategories:CS1maint:archivedcopyastitleHarvandSfnno-targeterrorsArticleswithshortdescriptionShortdescriptionmatchesWikidataCommonscategorylinkisonWikidataArticleswithBNEidentifiersArticleswithBNFidentifiersArticleswithGNDidentifiersArticleswithJ9UidentifiersArticleswithLCCNidentifiersArticleswithNDLidentifiersArticleswithFASTidentifiersArticleswithSUDOCidentifiersPagesthatuseadeprecatedformatofthemathtags Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages العربيةБеларускаяCatalàČeštinaDanskDeutschEspañolEsperantoفارسیFrançais한국어BahasaIndonesiaInterlinguaItalianoעבריתNederlands日本語NorskbokmålਪੰਜਾਬੀPolskiPortuguêsRomânăРусскийSlovenčinaSlovenščinaСрпски/srpskiSuomiSvenskaTürkçeУкраїнськаTiếngViệt粵語中文 Editlinks