Lie algebra theory without algebra

arXiv:math/0702016v2[math.DG]15Mar2007LiealgebratheorywithoutalgebraS.K.DonaldsonFebruary2,2008DedicatedtoProfessorYuI.Manin,onhis70th.birthday.1IntroductionThisisanentirelyexpositorypiece:themainresultsdiscussedareverywell-knownandtheapproachwetakeisnotreallynew,althoughthepresentationmaybesomewhatdifferenttowhatisintheliterature.Theauthor’smainmotivationforwritingthispiececomesfromafeelingthattheideasdeservetobemorewidelyknown.LetgbeaLiealgebraoverRorC.AvectorsubspaceI⊂gisanidealif[I,g]⊂I.TheLiealgebraiscalledsimpleifitisnotabelianandcontainsnoproperideals.AfamousresultofCartanassertsthatanysimplecomplexLiealgebrahasacompactrealform(thatistosay,thecomplexLiealgebraisthecomplexificationoftheLiealgebraofacompactgroup).ThisresultunderpinsthetheoryofrealLiealgebras,theirmaximalcompactsubgroupsandtheclassificationofsymmetricspaces.Inthestandardapproach,Cartan’sresultemergesafteragooddealoftheory:theTheoremsofEngelandLie,Cartan’scriterioninvolvingthenondegeneracyoftheKillingform,rootsystemsetc.Ontheotherhandifoneassumesthisresultknown–bysomemeans–thenonecanimmediatelyreadoffmuchofthestandardstructuretheoryofcomplexLiegroupsandtheirrepresentations.Everythingisreducedtothecompactcase(Weyl’s“unitariantrick”),andonecanproceeddirectlytodevelopthedetailedtheoryofrootsystemsetc.In[3],CartanwroteJ’aitrouv´eeffectivementunetelleformepourchacundestypesdegroupessimples.M.H.Weylad´emontr´eensuitel’existencedecetteformeparuneraisonnementg´en´erals’appliquant`atouslescas`afois.Onpeutsedeman-dersilescalculsquil’ontconduit`acer´esultatnepourraientpasencoresesimplifier,ouplutˆotsil’onnepourraitpas,paruneraissonnementapriori,1d´emontrerceth´eor`eme;unetelled´emonstrationpermettraitdesimplifierno-tablementl’expositiondelatheoriedesgroupessimples.Jenesuisacet´egardarriv´e`aaucunr´esultat;j’indiquesimplementl’id´eequim’aguid´edansmesrecherchesinfructueuses.ThedirectapproachthatCartanoutlined(inwhichheassumedknownthenondegeneracyoftheKillingform)wasdevelopedbyHelgason(seepage196in[4]),andacompleteproofwasaccomplishedbyRichardsonin[14].Inthisarticlewerevisittheseideasandpresentanalmostentirelygeometricproofoftheresult.ThisisessentiallyalongthesamelinesasRichardson’s,soitmightbeaskedwhatwecanaddtothestory.Onepointisthat,guidedbymoderndevelopmentsinGeometricInvariantTheoryanditsrelationswithdifferentialgeometry,wecannowadaysfitthisintoamuchmoregeneralcontextandhencepresenttheproofsina(perhaps)simplerway.Anotheristhatweareabletoremovemoreofthealgebraictheory;inparticular,thenondegeneracyoftheKillingform.WeshowthattheresultscanbededucedfromageneralprincipleinRiemanniangeometry(Theorem4).TheargumentsapplydirectlytorealLiegroups,andinourexpositionwewillworkmainlyinthatsetting.Intherealcasethecrucialconceptisthefollowing.SupposeVisaEuclideanvectorspace.ThenthereisatranspositionmapA7→ATontheLiealgebraEndV.Wesayasubalgebrag⊂EndVissymmetricwithrespecttotheEuclideanstructureifitispreservedbythetranspositionmap.Theorem1LetgbeasimplerealLiealgebra.ThenthereisaEuclideanvectorspaceV,aLiealgebraembeddingg⊂End(V),andaLiegroupG⊂SL(V)withLiealgebrag,suchthatgissymmetricwithrespecttotheEuclideanstructure.Moreover,anycompactsubgroupofGisconjugateinGtoasubgroupofG∩SO(V).Weexplainin(5.1)belowhowtodeducetheexistenceofthecompactrealform,inthecomplexcase.Theorem1alsoleadsimmediatelytothestandardresultsaboutrealLiealgebrasandsymmetricspaces,aswewilldiscussfurtherin(5.1).TheauthorthanksProfessorsMartinBridson,FrancesKirwan,ZhouZhangandXuhuaHeforcommentsontheearlierversionofthisarticle.2MoregeneralsettingConsideranyrepresentationρ:SL(V)→SL(W),whereV,Warefinite-dimensionalrealvectorspaces.LetwbeanonzerovectorinWandletGwbetheidentitycomponentofthestabiliserofwinSL(V).ThenwehaveTheorem2IfVisanireduciblerepresentationofGwthenthereisaEuclideanmetriconVsuchthattheLiealgebraofGwissymmetricwithrespecttothe2Euclideanstructure,andanycompactsubgroupofGwisconjugateinGwtoasubgroupofGw∩SO(V).NowwewillshowthatTheorem2impliesTheorem1.GivenasimplerealLiealgebrag,considertheactionofSL(g)onthevectorspaceWofskewsymmetricbilinearmapsfromg×gtog.TheLiebracketofgisapointwinW.ThegroupGwistheidentitycomponentofthegroupofLiealgebraautomorphismsofg,andtheLiealgebraofGwisthealgebraDer(g)ofderivationsofg,thatis,linearmapsδ:g→gwithδ[x,y]=[δx,y]+[x,δy].TheadjointactiongivesaLiealgebrahomomorphismad:g→Der(g).Thekernelofadisanidealing.Thisisnotthewholeofg(sincegisnotabelian)soitmustbethezeroideal(sincegissimple).Henceadisinjective.IfUisavectorsubspaceofgpreservedbyGwthenanyderivationδmustmapUtoU.InparticularadξmapsUtoUforanyξing,so[g,U]⊂UandUisanideal.SincegissimpleweseethattherecanbenopropersubspacepreservedbyGwandtherestrictionoftherepresentationisirreducible.ByTheorem2thereisaEuclideanmetricongsuchthatDer(g)ispreservedbytransposition.NowwewanttoseethatinfactDer(g)=g.Forα∈Der(g)andξ∈gwehave[adξ,α]=adα(ξ),sogisanidealinDer(g).ConsiderthebilinearformB(α1,α2)=Tr(α1α2)onDer(g).Thisisnondegenerate,sinceDer(g)ispreservedbytranspositionandB(α,αT)=|α|2.WehaveB([α,β],γ)+B(β,[α,γ])=0forallα,β,γ∈Der(g).Thusthesubs