Cohomology and Support Varieties for Lie Superalge

arXiv:0806.3740v1[math.RT]23Jun2008COHOMOLOGYANDSUPPORTVARIETIESFORLIESUPERALGEBRASOFTYPEW(n)IRFANBAGCI,JONATHANR.KUJAWA,ANDDANIELK.NAKANOAbstract.Boe,KujawaandNakano[BKN1,BKN2]recentlyinvestigatedrelativecoho-mologyforclassicalLiesuperalgebrasanddevelopedatheoryofsupportvarieties.ThedimensionsofthesesupportvarietiesgiveageometricinterpretationofthecombinatorialnotionsofdefectandatypicalityduetoKac,Wakimoto,andSerganova.InthispaperwecalculatethecohomologyringoftheCartantypeLiesuperalgebraW(n)relativetothedegreezerocomponentW(n)0andshowthatthisringisafinitelygeneratedpolynomialring.ThisallowsonetodefinesupportvarietiesforfinitedimensionalW(n)-supermoduleswhicharecompletelyreducibleoverW(n)0.Wecalculatethesupportvarietiesofallsim-plesupermodulesinthiscategory.RemarkablyourcomputationscoincidewiththepriornotionofatypicalityforCartantypesuperalgebrasduetoSerganova.WealsopresentnewresultsontherealizabilityofsupportvarietieswhichholdforbothclassicalandCartantypesuperalgebras.1.Introduction1.1.Letg=g¯0⊕g¯1beafinitedimensionalsimpleLiesuperalgebraoverthecomplexnumbersC.In1977KacprovidedacompleteclassificationoftheseLiesuperalgebras(cf.[Kac]).ThesimplefinitedimensionalLiesuperalgebrasaredividedintotwotypesbasedontheirdegree¯0part:theyareeitherclassical(wheng¯0isreductive)orofCartantype(otherwise).TheLiesuperalgebrasofCartantypeconsistoffourinfinitefamiliesofsuperalgebras:W(n),S(n),˜S(n)andH(n).LetusfirstsummarizewhatisknownfortheclassicalLiesuperalgebras[BKN1,BKN2].ThefirstfundamentalresultisthattherelativecohomologyringH•(g,g¯0;C)isafinitelygeneratedcommutativering.Notethatthisresultcruciallydependsonthereductivityofg¯0.Byapplyinginvarianttheoryresultsin[LR]and[DK],itwasshownundermildconditionsthatanatural“detecting”subalgebrae=e¯0⊕e¯1ofgarisessuchthattherestrictionmapincohomologyinducesanisomorphismR:=H•(g,g¯0;C)∼=H•(e,e¯0;C)W,whereWisafinitepseudoreflectiongroup.Thevectorspacedimensionofthedegree¯1partofthedetectingsubalgebraandtheKrulldimensionofRbothcoincidewiththecom-binatorialnotionofthedefectofgpreviouslyintroducedbyKacandWakimoto[KW].TheDate:June23,2008.2000MathematicsSubjectClassification.Primary17B56,17B10;Secondary13A50.ResearchofthefirstauthorwaspartiallysupportedbyNSFgrantDMS-0401431.ResearchofthesecondauthorwaspartiallysupportedbyNSFgrantDMS-0734226.ResearchofthethirdauthorwaspartiallysupportedbyNSFgrantDMS-0654169.12IRFANBAGCI,JONATHANR.KUJAWA,ANDDANIELK.NAKANOfactthatRisfinitelygeneratedcanbeemployedtodefinethecohomologicalsupportva-rietiesV(e,e¯0)(M)andV(g,g¯0)(M)foranyfinitedimensionalg-supermoduleM.ThevarietyV(e,e¯0)(M)canbeidentifiedasacertainsubvarietyofe¯1usingarankvarietydescription[BKN1,Theorem6.3.2].FortheLiesuperalgebrag=gl(m|n)thesupportvarietiesofallfinitedimensionalsimplesupermoduleswerecomputedin[BKN2].Aremarkableconse-quenceofthiscalculationisthatthedimensionsofthesupportvarietiesofagivensimplesupermodule(overgore)concideswiththecombinatoriallydefineddegreeofatypicalityofthehighestweightasdefinedbyKacandSerganova.1.2.InthispaperwedemonstratethatonecanalsouserelativecohomologyandsupportvarietiesinthesettingoftheCartantypesuperalgebraW(n).RecallthattheLiesuper-algebraW(n)istheLiesuperalgebraofsuperderivationsoftheexterioralgebraΛ(n)onngenerators.SinceΛ(n)=⊕kΛk(n)hasanaturalZ-gradinggivenbytotaldegree,oneobtainsaZ-gradingontheLiesuperalgebra,W(n)=Ln−1i=−1W(n)i,whereD∈W(n)isofdegreeiifD(Λk(n))⊆Λi+k(n)forallk∈Z.ThezerogradedcomponentW(n)0isisomorphictogl(n)andW(n)∼=Λ(V)⊗V∗asagl(n)-module,whereVisthenaturalndimensionalrepresentationofgl(n).ThecrucialdifferencebetweentheCartantypesuperalgebrasandtheclassicalsuperal-gebrasisthattheg¯0componentisnolongerreductive.However,aswasdescribedaboveforW(n),theCartantypeLiealgebrasoftypesW(n),S(n),andH(n)admitaZ-grading:g=⊕i∈Zgi.ThegradingiscompatiblewiththeZ2-gradinginthesensethat⊕ig2i=g¯0and⊕ig2i+1=g¯1.Furthermore,thebracketrespectsthegrading(i.e.[gi,gj]⊆gi+jforallinte-gersi,j).Inparticular,g0isareductiveLiealgebraunderthisbracket.Itisthennaturaltoconsiderthecategoryofg-supermoduleswhicharefinitelysemisimpleoverg0.Allfinitedimensionalsimpleg-supermodules,forexample,areobjectsinthiscategory.Furthermore,aswewillsee,thereductivenessofg0impliesthecohomologyringforthiscategoryisafinitelygeneratedalgebra.Thepaperisorganizedasfollows.Set(g,g0)=(W(n),W(n)0)andG0∼=GL(n)tobetheconnectedreductivegroupsuchthatLie(G0)=g0andsuchthattheactionofG0ongdifferentiatestotheadjointactionofg0ong.InSections2–3webrieflyreviewbasicfactsonrelativecohomologyprovedin[BKN1]andonW(n)from[Ser]thatwillbeneededforthispaper.Section4isdevotedtoapplyingtheseresultstothepair(g,g0).Inparticular,weusetherepresentationtheoryofgl(n)toshowinTheorem4.3.1thatR:=H•(g,g0;C)canbeidentifiedwitharingofinvariantsand,consequently,isfinitelygenerated.WealsoprovethatwhenMisafinitedimensionalg-supermodule,H•(g,g0;M)isafinitelygeneratedR-module.Thispropositionisthekeyfirststeptodevelopingatheoryofcoho-mologicalsupportvarieties.InSection5weinvokeinvarianttheoryresultsduetoLunaandRichardson[LR]toconstructadetectingsubsuperalgebraf=f¯0⊕f¯1f