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1、ZMP-HH/11-9HamburgerBeitr¨agezurMathematikNr.408June2011MODULARINVARIANTFROBENIUSALGEBRASFROMRIBBONHOPFALGEBRAAUTOMORPHISMSJ¨urgenFuchsa,ChristophSchweigertb,CarlStigneraaTeoretiskfysik,KarlstadsUniversitetUniversitetsgatan21,S–65188KarlstadbOrganisationseinheitMathematik,Universit¨atHa
2、mburgBereichAlgebraundZahlentheorieBundesstra?e55,D–20146HamburgAbstractForany?nite-dimensionalfactorizableribbonHopfalgebraHandanyribbonautomorphismofH,weestablishtheexistenceofthefollowingstructure:anH-bimoduleFωandabimodulemorphismZωfromLyubashenko’sHopfalgebraobjectKforthebimoduleca
3、tegorytoFω.Thismorphismisinvariantunderthenaturalactionofthemappingclassgroupoftheone-puncturedtorusonthespaceofbimodulemorphismsfromKtoFω.WefurthershowthatthebimoduleFωcanbeendowedwithanaturalstructureofacommutativesymmetricFrobeniusalgebraarXiv:1106.0210v3[math.QA]16Jul2012inthemonoid
4、alcategoryofH-bimodules,andthatitisaspecialFrobeniusalgebrai?Hissemisimple.ThebimodulesKandFωcanbothbecharacterizedascoendsofsuitablebifunctors.ThemorphismZωisobtainedbyapplyingamonodromyoperationtothecoproductofFω;asimilarconstructionfortheproductofFωexistsaswell.Ourresultsaremotivated
5、bythequesttounderstandthebulkstatespaceandthebulkpartitionfunctionintwo-dimensionalconformal?eldtheorieswithchiralalgebrasthatarenotnecessarilysemisimple.11IntroductionOneremarkablefeatureofcomplexHopfalgebrasistheirintimateconnectionwithlow-dimen-sionaltopology,includinginvariantsofkno
6、ts,linksandthree-manifolds.TheseconnectionsareparticularlywellunderstoodforsemisimpleHopfalgebras.Therepresentationcategoryofasemisimplefactorizable?nite-dimensional(weak)Hopfalgebraisamodulartensorcategory[NTV]andthusallowsonetoconstructathree-dimensionaltopological?eldtheory.Asaconseq
7、uence,itprovides?nite-dimensionalprojectiverepresentationsofmappingclassgroupsofpuncturedsurfaces.IthasbeenshownbyLyubashenko[Ly1,Ly3]thatsuchrepresentationsofmappingclassgroupscanbeconstructedfornon-semisimplefactorizableHopfalgebrasHaswell.Thisconstructionisinfactpurelycatego
