On Associative Conformal Algebras of Linear Growth

arXiv:math/0005258v1[math.QA]25May2000ONASSOCIATIVECONFORMALALGEBRASOFLINEARGROWTHAlexanderRetakhAbstract.Weintroducethenotionsofconformalidentityandunitalassociativeconformalalgebrasandclassifyfinitelygeneratedsimpleunitalassociativeconformalalgebrasoflineargrowth.Thesearepreciselythecompletealgebrasofconformalendomorphismsoffinitemodules.0.Introduction.Thesubjectofconformalalgebrasisarelativelyrecentdevelopmentinthetheoryofvertexalgebras[K1].TherelationofLieconformalalgebrastovertexalgebrasissimilartothatofLiealgebrasandtheiruniversalenvelopingalgebras.SemisimpleLieconformalalgebrasoffinitetypewereclassifiedin[DK]andsemisimpleassociativealgebrasoffinitetypein[K2].Associativeconformalalge-brasappearinconformalrepresentationtheory.ThecompletealgebrasofconformalendomorphismsoffinitemodulesCEndN(calledinthispapertheconformalWeylalgebras)areaparticularexample(see[K1,2.9]).NoticethatthesearealgebrasarenotfiniteasmodulesoverC[∂].Thesepaperisconcernedwithassociativealgebrasoffinitegrowth(butnotnecessarilyoffinitetype).Werequiretheconformalalgebratobeunital,thatis,topossessanelementthatactsasaleftidentitywithrespecttothe0-thmultiplicationandwhoselocalitydegreewithitselfis1.Inparticular,thismeansthatits0-thcoefficientisthe(left)identityofthecoefficientalgebra.Onecanthenusetheclassificationofassociativealgebrasoflineargrowthobtainedin[SSW]toclassifyaclassofunitalconformalalgebras.MainTheorem.LetCbeafinitelygeneratedsimpleunitalassociativeconformalalgebra.IfChasgrowth1,thenitisaconformalWeylalgebraCEndN.Noticethatthereisamarkeddifferencefromthecaseofgrowth0wheresimpleassociativeconformalalgebrasarejustthecurrentalgebrasoverthematrixalgebras([K2,4.4],seealsoRemark4.3below).TypesetbyAMS-TEX12ALEXANDERRETAKHThepaperisorganizedasfollows.Thefirstchapterisdevotedtopreliminarymaterialonconformalalgebraswhere,forthemostpart,welooselyfollowthetreatmentof[K2].ThesecondchapterdiscussestheconceptofGelfand-Kirillovdimensionforconformalalgerbasandrelatesittothedimensionofthecoefficientalgebra.Thethirdchapterisdevotedtothediscussionofunitalconformalalgebras.Herethemainresult(Proposition3.5)relatesthepresenceofconformalidentitytothestructureofthecoefficientalgebra.ThefourthchaptercontainstheproofoftheMainTheoreminTheorem4.6.Wealsoclassifyunitalassociativealgebrasofgrowth0inTheorem4.2(theproofisdifferentfrom[K2]whichcontainsthegeneralcase).IamextremelygratefultoEfimZelmanovforintroducingmetothesubjectofconformalalgebrasandguidingmethroughallstagesofthisresearchandtoMichaelRoitmanforhelpfuldiscussionsandhiscircle-drawingTEXpackage.1.PreliminariesonConformalAlgebras.Westartwithamotivationfortheconceptoftheconformalalgebra.LetRbea(Lieorassociative)algebra.OnecanconsiderformaldistributionsoverR,thatiselementsofR[[z,z−1]].Theseappear,forinstance,inthetheoryofoperatorproductexpansionsinvertexalgebras.Obviously,itisimpossibletomultiplytwoformaldistributions;however,onecanintroduceaninfinitenumberofbilinearoperationsthatactasa“replacement”formultiplication.Leta(z),b(z)betwoformaldistributions.Foranintegern≥0defineanotherformaldistributiona(z)nb(z)=Res|z=0a(z)b(w)(z−w)n(1.1)calledthen-thproductofa(z)andb(z).(Thosefamiliarwithvertexalgebraswillobservethatwedonotconsiderthe−1st,i.e.theWick,producthereasitsdefini-tionrequiresarepresentationofR.)Ifonewritesa(z)explicitlyasPa(n)z−n−1,itfollowsthat(a(z)nb(z))(k)=Xj∈Z+(−1)jnja(n−j)b(k+j).(1.2)Inthetheoryofvertexalgebrasonewantstoconsideronlythemutuallylocalseries,i.e.a(z)andb(z)suchthata(z)nb(z)andb(z)na(z)arezeroforn0.Wesimplysaythata(z)andb(z)arelocalifa(z)nb(z)=0fornN(a,b)andcallminimalsuchN(a,b)thedegreeoflocalityofa(z)andb(z).Intermsofcoefficientsthisbecomes:nXj=0(−1)jnja(n−j)b(k+j)=0,n0,k∈Z.(1.3)Itispossibletorewritethisstatementasfollows:ONASSOCIATIVECONFORMALALGEBRASOFLINEARGROWTH3Lemma1.1.[K1,2.3]Theseriesa(z)andb(z)arelocaliffa(z)b(w)(z−w)n=0fornN(a,b).Intermsofcoefficients:nXj=0(−1)jnja(l−j)b(m+j)=0,nN(a,b),l,m∈Z.(1.4)Apartfromlocalityonealsowantstotakeintoaccounttheactionof∂=∂/∂z.Theseconsiderationsuggestedthefollowingdefinition,firststatedin[K1].Definition1.2.AconformalalgebraCisaC[∂]-moduleendowedwithbilinearproductsn,n∈Z≥0,thatsatisfythefollowingaxiomsforanya,b∈C:(C1)(locality)anb=0fornN(a,b);(C2)(Leibnitzrule)∂(anb)=(∂a)nb+an(∂b);(C3)(∂a)nb=−nan−1b.Clearlyalgebrasofformaldistributionsclosedwithrespecttotheactionof∂satisfytheseaxioms.Thebilinearproductnisusuallycalledthemultiplicationofordern.Ingeneral,N(a,b)6=N(b,a).Moreover,evenifa,bandcarepairwiselocal,anbandcneednotbesuch.Onecan,however,establishsomecorrelationsbetweendifferentdegreesoflocality.Forinstance,itfollowsfrom(C3)thatN(∂a,b)=N(a,b)+1andfrom(C2)and(C3)thatN(a,∂b)=N(a,b)+1.Itfollowsaswellthat(∂a)0b=0foralla,b.Yet,onecanhardlymakedeeperstatementsaboutconformalalgebraswithoutrestrictingattentiontolessgeneralcases.TwosuchcasesarealgebrasofformaldistributionsoverLieandassociativealgebrasclosedwithrespecttothederivation∂.Heretakingproductsdoesnotdestroylocality:Lemma1.3.(Dong’slemma)[Li,K1]Leta,bandcbepairwisemutuallylocalformaldistributionsovereit