ON TWISTED TENSOR PRODUCTS OF ALGEBRAS

ESITheErwinSchrodingerInternationalPasteurgasse6/7InstituteforMathematicalPhysicsA-1090Wien,AustriaOnTwistedTensorProductsofAlgebrasAndreasCapHermannSchichlJirVanzuraVienna,PreprintESI163(1994)November23,1994SupportedbyFederalMinistryofScienceandResearch,AustriaAvailableviaanonymousftporgopherfromFTP.ESI.AC.ATONTWISTEDTENSORPRODUCTSOFALGEBRASAndreasCapHermannSchichlJirVanzuraAbstract.Theproblemsconsideredinthispaperaremotivatedbynon{commuta-tivegeometry.StartingfromtwounitalalgebrasAandBoveracommutativeringKwedescribealltriples(C;iA;iB),whereCisaunitalalgebraandiAandiBareinclusionsofAandBintoCsuchthatthecanonicallinearmap(iA;iB):AB!Cisalinearisomorphism.WediscusspossibilitiestoconstructdierentialformsandmodulesoverCfromdierentialformsandmodulesoverAandB,andgiveadescriptionofdeformationsofsuchstructuresusingcohomologicalmethods.1.IntroductionAlthoughtheproblemsweconsiderarefrompurealgebra(andtopologicalal-gebra),themotivationcomesfromnon{commutativedierentialgeometry:Thesimplequestionwestartedfromis,giventwoalgebraswhicharesupposedtode-scribesome\spaces,whatisanappropriaterepresentativeoftheproductofthetwo\spaces?Thinkingofthecommutativecaseonewouldbeledtoconsideringthe(topological)tensorproductofthetwoalgebras.Butinthenon-commutativecasethismeansthatoneassumesthatfunctionsonthetwofactorscommutewitheachother,althoughthefunctionsontheindividualfactorsdonotcommuteamongthemselves,andweseenoreasontoassumethis.Inthispaperwestudyalgebras,whichareinacertainsenseveryclosetothetensorproductofthegivenones,andinparticulardeformationsofthetensorproduct.Itshouldalsoberemarkedthatspecialexamplesofsuchalgebras,notablythenon{commutativetwotoriandmoregenerallycrossedproductsofC{algebrasbygroups,alreadyplayanimportantroleinnon-commutativegeometry.Theproblemmayaswellbeviewedasaquestionofdecompositionsofgivenal-gebras:Supposethataunitalalgebrais,asalinearspace,thetensorproductoftwosubalgebras.Whatdoesthissayaboutthealgebrastructure?Fromthispointofviewtheanalogousproblemsfordiscretegroups,Liegroups,LiealgebrasandHopfalgebrashavebeenstudied(seee.g.[Majid,1990],[Michor,1990]and[Takeuchi,1981]),oftenunderthenameofmatchedpairsorfactorizationofstructures.InthestudyoftheHopfalgebracasethebasicconditions2.4(1)foralgebrastructureshavebeenobtained(c.f.[Majid,1994,7.2.3]).Itturnsoutthatthecaseofalgebras1991MathematicsSubjectClassication.16E4016S1016S3516S8058B30.Keywordsandphrases.Non-commutativegeometry,twistedtensorproducts,twistingmaps.PartsofthisworkweredoneduringastayofthethirdauthorattheErwinSchrodingerinter-nationalInstituteforMathematicalPhysicsandavisitoftherstauthorinBrnoandOlomouc.TypesetbyAMS-TEX12ANDREASCAPHERMANNSCHICHLJIRIVANZURAisthemostcomplicatedone,sinceinallothercasestheproblemreducestothestudyofmutualactionsofthetwofactorsoneachotherwhicharecompatibleinacertainsense,whileinthealgebracasesuchareductionisnotpossible.Anyhow,forourworkthepointofviewofdecompositionsislessimportant,sinceourmainaimisthestudyofdeformations.Wewillstudytheproblemwithoutassumingthatthealgebrasareendowedwithtopologies.Infact,allconstructionscanbecarriedoutpreciselyinthesamewayincategoriesofvectorspacesandlinearmaps,suchthattheHom{functorL(;)liftstothecategory,andwhichadmitatensorproduct^suchthatthereisanaturalisomorphismL(E^F;G)=L(E;L(F;G))(i.e.inmonoidallyclosedcategories).ThisisthecaseforexampleinthecategoryofBanachspacesandcontinuouslinearmapswiththeprojectivetensorproductor,moregenerally,inthecategoryofconvenientvectorspacesandboundedlinearmapswiththebornologicaltensorproduct(c.f.[Frolicher{Kriegl,1988]).2.TwistedtensorproductsThroughoutthispaperwexsomecommutativeringKwithunit.LateronwhenwewillstudydeformationswewillspecializetoK=RorC.Weassumeallalgebrastobeunitalandallhomomorphismstopreserveunits.2.1.Denition.LetAandBbealgebrasoverK.AtwistedtensorproductofAandBisanalgebraCtogetherwithtwoinjectivealgebrahomomorphismsiA:A!CandiB:B!Csuchthatthecanonicallinearmap(iA;iB):AKB!Cdenedby(iA;iB)(ab):=iA(a)iB(b)isalinearisomorphism.AnisomorphismoftwistedtensorproductsisanisomorphismofalgebraswhichrespectstheinclusionsofAandB.2.2.Thereisasimplewaytoconstructcandidatesfortwistedtensorproductsasfollows:Let:BA!ABbeaK{linearmapping,suchthat(b1)=1band(1a)=a1.ThenonABdeneamultiplicationby:=(AB)(AB).WewriteABforidAidB.ThisisalsojustiedbythefactthatthisisthefunctorABappliedtothemap.NextdeneiA:A!ABbyiA(a):=a1andlikewiseiB:B!AB.Thesearealgebrahomomorphismsbytheconditionson.Obviously,ifthemultiplicationisassociative,then(AB;)isatwistedtensorproductofAandB.Nowtheassociativityofthemultiplicationcanbecharacterizedintermsofasfollows:2.3.Proposition/Denition.Supposethat(b1)=1band(1a)=a1.Thenthemultiplicationisassociativeifandonlyifwehave:(BA)=()(BA)AmappingwhichsatisestheseconditionsiscalledatwistingmapforAandB,andwedenotethealgebra(AB;)byAB.Proof.Letusrstassumethatisassociative.Wealsowriteforthemultipli-ONTWISTEDTENSORPRODUCTSOFALGEBRAS3cation.Sinceisas