Realizing Enveloping Algebras via Varieties of Mod

arXiv:math/0604560v2[math.QA]10Jan2008REALIZINGENVELOPINGALGEBRASVIAVARIETIESOFMODULESMINGDING,JIEXIAOANDFANXUDedicatedtoProfessorJ.A.Greenontheoccasionofhis80thbirthdayAbstract.ByusingtheRingel-Hallalgebraapproach,weinvestigatethestructureoftheLiealgebraL(Λ)generatedbyindecomposablecon-structiblesetsinthevarietiesofmodulesforanyfinitedimensionalC-algebraΛ.WeobtainageometricrealizationoftheuniversalenvelopingalgebraR(Λ)ofL(Λ).ThisgeneralizesthemainresultofRiedtmannin[17].WealsoobtainGreen’sformulain[4]inageometricformforanyfinitedimensionalC-algebraΛanduseittogivethecomultiplicationformulainR(Λ).1.IntroductionLetusrecallaresultofRiedtmannin[17].LetΛbeafinitedimensionalassociativeC-algebrawithunitelement.IfΛisoffiniterepresentationtype,letJbeasetofrepresentativesfortheisomorphismclassesofinde-composableΛ-modules.ThenP={LA∈JAμ(A):μ(A)∈N}isasetofrepresentativesfortheisomorphismclassesofallΛ-modules.LetR(Λ)bethefreeZ-moduleonthebasis{uA,A∈P}.Definethemultiplicationoftwobasisvectorsby(seeSection2in[17])uA•uB=XX∈Pχ(V(A,B;X))uX,whereχ(V(A,B;X))istheEuler-Poincar´echaracteristicofthevariety:V(A,B;X)={0⊆Z⊆X:Z∈modΛ,Z∼=A,X/Z∼=B}ThisisageometricversionofRingel-Hallalgebras(see[18])byconsideringtheEuler-Poincar´echaracteristicofthefiltrationvarietiestoreplacethefiltrationnumbersoverafinitefield(seealso[10]and[22]).ThenR(Λ)isanassociativeZ-algebrawithunitelement.TheZ-submoduleL(Λ)=LA∈JZuAofR(Λ)isaLiesubalgebrawithbracket[x,y]=x•y−y•x(wewillsimplywritexyforx•yinthebelow).Sheprovedthefollowingresultbyagradationmethod.ForanynumberingJ={A1,···,An},theuniversalenvelopingalgebraU(Λ)ofL(Λ),isfreelygeneratedasaZ-modulebytheclassesofthewordsTheresearchwassupportedinpartbyNSFofChinaandbythe973ProjectoftheMinistryofScienceandTechnologyofChina.12MINGDING,JIEXIAOANDFANXUAλ11···Aλnn,(λ1,···,λn∈N).ItisisomorphictothesubalgebraR(Λ)′ofR(Λ)generatedby{uA:A∈J},whichisthefreeZ-modulegeneratedbytheelements(Qnj=1(λj!))uAλ11⊕···⊕Aλnn.WenotethatasimilarpropertyforRingel-HallalgebrasH(Λ)hasbeenobtainedbyGuoandPengin[6]foranyfinitedimensionalalgebraΛoverafinitefield.InthispaperweextendRiedtmann’sresult.LetR(Λ)betheZ-modulegeneratedbythecharacteristicfunctionsofconstructiblesetsofstrati-fiedKrull-Schmidt(seeSection3.2forthedefinition)andL(Λ)betheZ-submoduleofR(Λ)generatedbythecharacteristicfunctionsofindecompos-ableconstructiblesets(seeSection3.2forthedefinition).Ourmainresult(Theorem4.2)provides,R(Λ)canberealizedastheenvelopingalgebraofL(Λ),fromwhichRiedtmann’sresultcanbededucedbyreplacingorbitsofmodulesbyconstructiblesets.Moreover,weproposeageometricversion(Theorem5.7)ofthe“degen-eratedform”ofthewell-knownGreen’sformulain[4]andextendsGreen’sformulatoanyfinitedimensionalassociativealgebra(notonlyhereditaryalgebra).Wenotethatanalternativeapproachistoapplytherestrictionfunctorgivenin[11].Thepaperisorganizedasfollows.InSection2werecallthebasicknowl-edgeaboutvarietiesofmodulesandconstructiblefunctions.InSection3wegivethedefinitionsofR(Λ)andL(Λ)foranyfinitedimensionalgebraΛoverCwhichcanbeviewedasanextensionofSection2.3in[17].Followingthis,westudytherelationbetweenR(Λ)andL(Λ)inSection4.Inparticular,Theorem4.2isRiedtmann’sresultwhenconsideringrepresentation-finitecase.InSection5,wegiveaformulainTheorem5.7whichcanbeviewedasavariantofGreen’sformula.InSection6,weprovethatthisformulainducesthecomultiplicationofR(Λ)asaHopfalgebra.2.BasicConceptsInthissection,wewillfixnotationsandrecallsomeelementaryconceptsinalgebraicgeometry.Thesetofalln-tupleswitheachcoordinateelementinthefieldCiscalledann-dimensionalaffinespaceoverC.Eachclosedset(withinducedtopology)ofanaffinespaceiscalledanaffinealgebraicvariety.ApointxofanaffinevarietyXissmoothifthedimensionofthetangentspaceofXatxisequaltothedimensionofXatx,denotedbydimTX,x=dimxX.Letϕ:X−→Ybeadominating(i.eϕ(X)isZariski-denseinY)regularmap,ifx∈Xandy=ϕ(x)∈Yisasmoothpoint,thenwesaythatϕissmoothatxifxisasmoothpointofXanddϕmapsTx,XontoTy,Y.LetXbeaZariskitopolgyspace,theintersectionofanopensubsetandaclosesubsetiscalledalocallyclosedsubset.AsubsetinXiscalledconstructibleifitisadisjointunionoffinitelymanylocallyclosedsubsets.Obviously,opensetsandclosesetsarebothconstructiblesets.REALIZINGENVELOPINGALGEBRASVIAVARIETIESOFMODULES3Definition2.1.AfunctionfonXiscalledconstructibleifXcanbedividedintofinitelymanyconstructiblesetssatisfyingthatfisconstantoneachsuchconstructibleset.LetObeanaboveconstructibleset,1Oiscalledacharacteristicfunctionif1O(x)=1foranyx∈O;1O(x)=0foranyx/∈O.Itisclearthat1Oisthesimplestconstructiblefunctionandanyconstructiblefunctionisalinearcombinationofcharacteristicfunctions.LetΛbeafinitedimensionalC-algebra.ByaresultofP.Gabrielin[5],thealgebraΛisgivenbyaquiverQwithrelationsR(uptoMoritaequivalence).LetQ=(Q0,Q1,s,t)beaquiver,whereQ0andQ1arethesetsofverticesandarrowsrespectively,ands,t:Q1→Q0aremapssuchthatanyarrowαstartsats(α)andterminatesatt(α).ThereareonlyfinitelymanysimpleΛ-modulesS1,···,Sn,uptoisomorphism.TheindexsetforsimplemodulesisdenotedbyI(wemayidentifyI=Q0).ForanyΛ-moduleM,wedenotebydimMthevectorinNIwhosei-thcomponentisthemultiplicityofSi