ON FUSION ALGEBRAS ASSOCIATED TO FINITE GROUP ACTI

pacificjournalofmathematicsVol.177,No.2,1997ONFUSIONALGEBRASASSOCIATEDTOFINITEGROUPACTIONSHidekiKosaki,AkihiroMunemasaandShigeruYamagamiInourpreviouspaper(seeKosakiandYamagami),fourkindsofbimodulesnaturallyattachedtocrossedproductsPoGPoHdeterminedbyagroup-subgrouppairGHwereidentiedwithcertainvectorbundlesequippedwithgroupactions.Inthepresentpaperwewilldescribethestruc-tureofthefusionalgebraofvectorbundlesandclarifyarela-tionshiptofusionalgebrasappearinginothercontexts.Someapplicationstoautomorphismanalysisforsubfactorswillbealsogiven.1.Introduction.Thenotionofbimodules(forvonNeumannalgebras)wasintroducedbyA.Connesandhasprovedveryusefulinmanyplacesinthetheoryofoperatoralgebras.TheJonesindextheory(andsubsequentsubfactoranalysis)isnoexception.Agivenfactor-subfactorpairMN(ofniteindex)naturallyprovidesuswithfourkindsofbimodules(i.e.,M-M,M-N,N-M,N-Nbimodules)viathebasicconstruction.ThesebimodulesareknowntocontainenormousamountofinformationonthepairMN,andhencestudyonbimodulesisindispensableforsubfactoranalysis.Forexample,knowinghowtensorproductssplitintoirreduciblebimodules(i.e.,fusionrule)isveryuseful.Animportantexampleoffactor-subfactorpairsisthecrossedproductpairM=PoGN=PoHarisingfromagroup-subgrouppairGH.Inthiscase,thefourkindsofbimodulesandtheinduction-restrictionprocedureforthem(i.e.,theprincipalandthedualprincipalgraphs([8]))canbeexplicitlydescribedintermsoftherepresentationtheoryforthepairGH(i.e.,theMackeymachine).Actually,inourpreviouspaper[22],viaacertaincrossedproductconstructionatensorcategoryofvectorbundleshavinggroupsasbasespaceswithbivariantactionsofsubgroupswasintroduced.Wethenshowedthat(basedontheouternessofanactioninquestion)thiscanbeidentifedwiththecategoryoftheabovementionedbimodules,andhencestudyofbimoduleswascompletelyreducedtothatinrepresentationtheory(seex2).269270H.KOSAKI,A.MUNEMASAANDS.YAMAGAMIThefusionruleforM-Mbimodulesisquiteeasyinthegroup-subgroupcase,andthepurposeofthepresentpaperistodeterminemuchmoresubtlefusionruleforN-Nbimodulestogetherwithsomeapplicationstothestudyonautomorphismsforsubfactors.ItturnsoutthatthefusionalgebraforN-NbimodulesisessentiallythesameasthealgebraofHeckeoperatorsconsideredbyT.Yoshida([40,41]).Wegiveaconcretedescriptionofthefusionalgebrabasedon[28],revealingtheabstractstructureasasemi-simplealgebra.MotivatedmainlybyVerlinde’swork([34]),fusionalgebrasinseveralcontextshavebeeninvestigatedsofar.Amongthese,thefusionalgebraoftheorbifoldmodelappearingin[5](seealso[1,2,27,35])withthetrivial3-cocycleisaspecialcaseofourfusionalgebraofvectorbundles,whereG=HHwiththediagonallyimbeddedH.Inx3weobtainamultiplicityformulaforanirreduciblebundleinthetensorproductofbundlesinquiteageneralsetting.Inx4westudythefusionalgebraofvectorbundlesandobtainaconcreterealizationofthisalgebra.Wealsopointoutthatthefusionalgebraoftheasymptoticinclusion(see[30])ofR0oGR0(whereR0isthehyperniteII1factor)isthecenterofthequantumdouble([6])ofthegroupHopfalgebra‘1(G).Thisfactforthefusionalgebraarisingfromanorbifoldmodel([1,2,5])isprobablyknowntoseveralpeople,especiallytoM.Wakui([35],seealso[4]).Then,inx5weidentifyelementsinthefusionalgebrawithoperatorsactingonacertaincharacterring(consideredin[40,41]).Thisenablesustoexpressthetensorproductveryexplicitlyintermsofrelevantrepresentations.Insubfactoranalysis,certaingroupsofautomorphismswithspecialpropertiesnaturallyappear(i.e.,forexamplethegroupofnon-stronglyouterautomorphisms([3,19,32]))andtheyareknowntoberelatedtoanalysisonbimodules.Inthenalx6,weexplicitlywritedownthesegroupsforaninclusionofxed-pointalgebrafactors.Throughoutthepaperwefollowthenotationsanddenitionsinourpre-viouspaper[22](whicharesummarizedinx2)whilebasicfactsontheindextheorycanbefoundintheoriginalarticles[8,14,18].TheauthorsaregratefultoY.Kawahigashiforinformingthemof[7,16]inpreparation.2.Preliminaries.Inthissectionwebrieyrecallbasicdenitionsandresultsin[22]partlytoxournotations.2.1.VectorBundles.Letbeadiscretegroup,anditsnitesubgroupswillbedenotedbyG;H;K;.Theunitinthegroupwillbedenotedby1throughoutthepaper.ByanH-KvectorbundleV(=HVK)wealwaysmeanavectorbundleofHilbertspacesoverthebasespacewithleftH-FUSIONALGEBRAS271andrightK-actionswhichextendthenaturalleftH-andrightK-actionsonthebasespaceandpreserveinnerproducts(inbreHilbertspaces).WeallowsomeofbresVgtobetrivial,ands(V)=fg2;Vg6=f0ggiscalledthesupportofV.Thedirectsumandtheadjointbundlearedenedinthenaturalway(V=K(V)Hwith(V)g=(Vg1),thedualspace)whilethetensorproductisdenedasfollows:LetV;WbeG-H,H-Kbundlesrespectively.TheoutertensorproductVWoveradmitstheH-actionh(vw)=(vh1)(hv):Hence,wegetthequotientbundleVHWoverthequotientspaceHof.TheproductbundleVHW=G(VHW)KisdenedasthepushforwardofVHWviathemapp:gHg02H!gg02.Hence,weget(VHW)g=MaHb2p1(g)(VaHWb);andwehavethenaturalG-Kactionfromtheoutside.ForH-KbundlesV;W,abundlehomomorphismT:V!WsatisfyingT(Vg)Wgforeachg2iscalledan(H-K)gaugetransformation,andthesetofallgaugetransformationsisdenotedbyHom(V;W)(andHom(V;V)=Hom(V)asusual).AbundleVisirre