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1、JOURNALOFPUREANDAPPLIEDALGEBRAEISBVIERJournalofPureandAppliedAlgebra98(1995)245-278TanglesandHopfalgebrasinbraidedcategoriesVolodimirLyubashenkoDepartmentofMathematics,UniversityofYork,HeslingtonYorkYOI5DD,UKCommunicatedbyJ.D.Stashe$received13June1991;revised15March1992AbstractProper
2、tiesofthecategoryofribbonorframedtanglesareusedtostudyHopfalgebrasinbraidedmonoidalcategories.Aninnertraceforribbonbraidedcategoriesisinvestigated.IntegralsforHopfalgebrasarediscussed.1.Introduction1.0.RecentworkofMooreandSeiberg[21]revealedthepresenceoftensorcatego-riesinconformalfi
3、eldtheories.Thesecategoriesaresemisimplewithsomeadditionalstructures-modulartransformationsSandT.Thegoalofthepresentandsub-sequentpapersistoshowthatanyabelianribbonbraidedcategory(notnecessarilysemisimple)withsomenon-degeneracyassumptionbehavessimilarly.Inparticular,thereisaHopfalgeb
4、raHinsuchacategorywithoperationsS,T:H+H,satisfyingmodularidentitites.Thispaperisanexpandedversionofthefirstpartof[14].InSection1wecollectthebasicdefinitionsoftensorcategories,braidedcategories,autonomouscategories,andtangles.InSection2weinvestigatethesecondcohomol-ogygroupofbraidedca
5、tegoryandprovefundamentalresultsconcerningthetraceofanendomorphism.Alsoweconstructthefreeribboncategoryandshowhowtobuildabeliantensorcategories.InSection3weexplainhowthefundamentalstructuresofcoalgebrasandHopfalgebrascarryovertothenon-symmetrictensorcategories.1.1.Iapologizeforusings
6、ometermsandnotationsof[14].Ihavetogivesomesynonymsforthenamesofcategoricalnotionsandtorecallthesenotions.Atensorcategory=cuTegory(%T,0)isacategoryV?withafunctor0:Sf?x59+%T.Amonoidalcategory[151=associativetensorcategory=cATegory(W,0,a,I,m)isOnleavefromDepartmentofMathematicsMethodsof
7、SystemsAnalysis,KievPolytechnicInstitute,37Peremogyprosp.,252056Kiev,Ukraine.0022-4049/95/$09.5001995-ElsevierScienceB.V.Allrightsreserved.SSDI0022-4049(95)00044-J246V.LyubashenkolJournalofPureandAppliedAlgebra98(1995)245-278atensorcategorywithanassociativityconstrainta:X0(Y0Z)+(X@Y)
8、@Zsatisfying
