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1、Infinite-DimensionalRepresentationsof2-GroupsJohnC.Baez1,AristideBaratin2,LaurentFreidel3,4,DerekK.Wise51DepartmentofMathematics,UniversityofCaliforniaRiverside,CA92521,USA2MaxPlanckInstituteforGravitationalPhysics,AlbertEinsteinInstitute,AmM¨uhlenberg1,14467Golm,Germany
2、3LaboratoiredePhysique,EcoleNormaleSup´erieuredeLyon´46All´eed’Italie,69364LyonCedex07,France4PerimeterInstituteforTheoreticalPhysicsWaterlooON,N2L2Y5,Canada5DepartmentofMathematics,UniversityofCaliforniaDavis,CA95616,USAAbstractA‘2-group’isacategoryequippedwithamultipl
3、icationsatisfyinglawslikethoseofagroup.Justasgroupshaverepresentationsonvectorspaces,2-groupshaverepresentationson‘2-vectorspaces’,whicharecategoriesanalogoustovectorspaces.Unfortunately,Lie2-groupstypicallyhavefewrepresentationsonthefinite-dimensional2-vectorspacesintro
4、ducedbyKapranovandVoevodsky.Forthisreason,Crane,SheppeardandYetterintroducedcertaininfinite-dimensional2-vectorspacescalled‘measurablecategories’(sincetheyarecloselyrelatedtomeasurablefieldsofHilbertspaces),andusedthesetostudyinfinite-dimensionalrepresen-tationsofcertainLi
5、e2-groups.Herewecontinuethiswork.Webeginwithadetailedstudyofmeasurablecategories.Thenwegiveageometricaldescriptionofthemeasurablerepresen-tations,intertwinersand2-intertwinersforanyskeletalmeasurable2-group.Westudytensorproductsanddirectsumsforrepresentations,andvarious
6、conceptsofsubrepresentation.Wedescribedirectsumsofintertwiners,andsub-intertwiners—featuresnotseeninordinarygrouprepresentationtheory.Weclassifyirreducibleandindecomposablerepresentationsandinter-twiners.Wealsoclassify‘irretractable’representations—anotherfeaturenotseen
7、inordinarygrouprepresentationtheory.Finally,wearguethatmeasurablecategoriesequippedwithsomeextrastructuredeservetobeconsidered‘separable2-Hilbertspaces’,andcomparethisideatoatentativedefinitionof2-HilbertspacesasrepresentationcategoriesofcommutativevonNeumannalgebras.arX
8、iv:0812.4969v1[math.QA]29Dec20081Contents1Introduction32Representationsof2-groups72.1Fromgroupsto2-groups.....
