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1、DOI_HOPFMODULESFORHOPF丌-COALGEBRAANDBRAIDEDT．ALGEBRAABSTRACTThenotionsofHopf7r-coalgebras，where7／"isadiscretegroup，generalizethoseofHopfalgebras．Hopf'r-coalgebrawasusedbyV．G．TuraevtoconstructHennings—likeandKuperbergAikeinvariantsofprincipal丌一bundlesoverlinkcomplementsandover3-manifolds．Thispaperma
2、inlystudiesMaschketypetheoremandFrobeniuspropertiesofDoi-HopfmodulesforHopf_，r-aJgebras．AlsowegeneralizesuchresultstoentwinedⅡ一modules．V．G．互hraevintroducedthenotionofamodularcrossedv-category(orcalledT-categoryfollowingM．Ztmino[4])andshowedthatsuchacategorygivesrisetoathree-dimensionalhomotopyquant
3、umfieldtheorywithtargetspaceⅣ(霄，1)．Finally’westudythepropertiesof'I-algebraandgetallexampleofT-category：thecategoryofcorepresentation《T-algebra．Insectionl，wereviewthebasicdefinitions，notionsandexamplesusedinthisP印er·Insection2，firstly,weintroducetheconceptofanormalized7rwintegral．(／singit，weproveaM
4、aschketypetheoremofDoi-HopfmodulesforHopfr-coalgebras．Thatistheorem2．7：Let(EA，④beaDoi-Hopf7r-datum，ifthereexistsanormalized丌一integral{丸：Q固AI_《～loA1)口自of(日，A，G)，thenthefollowingassertionshold．(1)Amorphism“=ftk：心—十．^妃}口∈Fino．^／fjhasaretraction(resp．a(chǎn)section)ina^^j，iftheA1·linearmapul：Ⅳ口葉A如hasaretract
5、ion(resp．a(chǎn)section)inM山；(2)M∈。朋A，ifMissemisimpleasarightA1．module，thenMisseraisimpleas∞objectinc朋互．As蛐application，weobtainacharacterizationfor(H，A，C)Doi—Hopf_7r-modulestobeprojectiveasA—modules(CorollⅫy2．9)．Meanwhile，weapplyourresultstocharactertheseparabilityoftheforgetfulfunctorF’。朋A—÷朋^l(Theorem2
6、．12)．Attheendofthissectionwegeneralizethisresulttoentwinedw—modulesfTheorem2，is)．iijInsection3，wegiveSOmeequivalentconditionsfortheforgetfulfunctorF：。朋jH／Ⅵ^ItobeaFrobeniusfunctor．ThenOUl"mainresultisnowthefollowing(Theorem33)：Let(H，A，C)beaDoi-Hopfr-datum．AssumethatCisafinitetypeHopf丌-eoalgebra．Then
7、thefollowingstatementsareequivalent：(1)ThefunctorG={甌固·)口∈”：M^l—+cM互isaleftadjointoftheforgetfulfunctorF：c川j------4川^1；(2)Thereexistsz=￡c／oa‘∈a@A1whichsatisfiedforanya∈A1，。n=a2：(i．e．∑q-。(一1．1)oata(o，1)=∑ci@
