; TeX output 1995.11.13:0830 ;\9Ǎ\97[R,N cmbx12RECONSTRUCTIONٚOFHIDDENSYMMETRIES FK`y cmr10BODOUUP*AREIGIS&XQ cmr12revisedVVers.8SeptemrbSer16,1995Q-- cmcsc10Contents1.Intro`ductionB?12.Braidedcategoriesand(!", cmsy10C5-categories42.1.,JC5-categories2.2.,JBraidedcategories2.3.,JC5-monoidalcategories2.4.,JRigidcategories2.5.,JCoadjoinrtcoactions2.6.,JC|{Ycmr80-generatedcoalgebras3.Reconstructionprop`erties-153.1.,JReconstructionofcoalgebras3.2.,JReconstructionofbialgebras3.3.,JReconstructionofmorphisms3.4.,JApplications4.Existencetheoremsinreconstructiontheoryq 224.1.,JRestrictedreconstruction4.2.,JFinitereconstruction5.Hiddensymmetries7325.1.,JFVunctorsofconrtrolcategories5.2.,JHiddensymmetriesofthebasecategory6.App`endixon+ msbm10K-additivecategoriesandC5-categoriesT39References^~42.1.}Introduction Groups'g cmmi12Gareoftenobtainedasgroupsofsymmetries(orautomorphisms)ofmathematicalRstructureslikreavectorspace(overa xed eldK)ortwovectorspacestogether"withalinearmapbSetrween"themorawholediagramofvrectorspaces,B=wherea?symmetryofsucrhadiagramisafamilyofautomorphismsoneforeachvectorspacewhicrharecompatiblewiththelinearmapsofthediagram(anaturalautomorphism).ThisproScessofconstructingthegroupofsymmetriesisaspecialcaseofthenotionof(TVannakXa-Krein)reconstruction.J1* ;\92FBODOUUP*AREIGISǍm\9 ConrverselygivenagroupGoneconsidersitsrepresentationsGx k!xGL(Vp)invrectorspacesVaoverthe eldK.һAllrepresentationsofGformthecategory!ppmsbm8K Vӟ2cmmi8GvkMofmoSdules,'^whicrhwemayconsiderasahugediagramofvectorspaces.ThecategoryK8G XzM׃hasanadditionalinrterestingstructure{thetensorproSductV[ KWyIoftworepresenrtationsisagainarepresentationinacanonicalwayV,^IK +GMisamonoidalcategoryV.VA spSecial consequenceofreconstructiontheoryisthefactthatGmarybSeMrecorveredasthefullgroupofthosesymmetriesofthishugediagramwhicharecompatiblewiththetensorproSduct.N Thisprocessseemstobetheinrverseofthe rst:one.aInamoregeneralsettingthereare,horwever,subtle:deviations.aOnemaryreconstructmruchlargergroupsofsymmetriesthanwhatonestartedoutwith.L MoreBgenerallywreknowthatalgebrasA,iLiealgebras#%n eufm10gandHopfalgebrasHcanbSereconstructedfromtheircategoriesofmodules.(FVorthereconstructionofanalgebraqAoneactuallyneedsnotonlythecategoryofA-moSdulesA-Mod_butalsotheunderlyingJfunctor!Ë:URA-MoSd&2!VVecQ.ThenJA(asanalgebra)canbSereconstructed(upGLtoisomorphism)asend\(!n9),^uthecategoricalendoftheunderlyingfunctor.NFVorthe2reconstructionofaHopfalgebraH oneneedsinadditionthemonoidalstructureofcHV-MoSdV. ThenthefullHopfalgebrastructurecanbSereconstructed[DM82}n,Pra81 ,Ul89m]. ThisstandsinpSoinrtedcontrasttoanothersimilarresult,FtheMoritatheorems[Ba68],c#whicrhshowthattheknowledgeofthecategoryofmoSdulesA-Mod ^ofanalgebraAdoSesnotdetermineAuptoisomorphism. AswreremarkedbSefore,½theforgetfulfunctor!:{A-Mod':o!VVecYisessenrtialintheproScessofreconstruction.Inparticularonehastoconsiderrepresenrtationsofthegivrenobjects(algebras,lgroups,Liealgebras,Hopfalgebras)invrectorspaces.RepresenrtationsO-incategoriesofobjectswitharicherstructurelikesupServectorspaces,?-spaces,gradedvrectorspaces,comoSdulesoverHopfalgebrashaveadi erentbSeharvior. InsteadofthebasecategoryVVec;wrewishtouseanarbitrarybraidedmonoidalcategoryˌA.,TherearemanryexamplesforsuchabasecategoryAsuchasL-MoSdi,thecategoryofmoSdulesorveraquasitriangularHopfalgebraL,YL-Comod(V,thecategoryoftcomoSdulesorvertacoquasitriangularHopfalgebraL,2LRAL ӨYDUV,thecategoryofYVetter-DrinfeldmoSdulesorveraHopfalgebrawithbijectivreantipSode,}ordysCA,thecategoryofdyslecticmoSdulesorveracommrutativealgebrainabraidedmonoidalcategoryCDm[Pra95J]. ,WVe8studythefollorwingproblem:givenanalgebra,abialgebra,oraHopf~|algebraHkinA,rcanitbSereconstructedfromH A,theunderlyingfunctor!N:ڟH A 53!A< andthemonoidalstructure?- A;spSecialcaseisthereconstructionofasupSeralgebrafromitssuperrepresenrtations. The4surprisingsolutionofthisproblemshorwsthatoneusuallyreconstructsanobjectendY(!n9)fromtheunderlyingfunctor!:HH AH !HAinAthatismruchbiggerthanӟHV.Inthegroupcase(forH1=kgG)thisamounrtstoadditionalsymmetrieswhicrh;wecallhiddensymmetries.Aconcretecaseofsuchahiddensymmetryisgivrenin5.1.8.KInthe(Hopf8)algebracasethesituationismorecomplexbutwealsotalkabSouthiddensymmetries.,IncertaincaseswredescribepreciselytheadditionalhiddensymmetriesbryasmashproSductdecompositionofthereconstructedobject. WVe-OconrtroltheproScessofreconstructionbyacontrolcategoryCwhichopSeratesone!:5HO(A5 =!5A. KWithdi erenrtchoicesofthecontrolcategoryCNweobtaindi erenrtureconstructedobjectsendK cmsy8C*(!n9)fromoneandthesameunderlyingfunctor!Ë:URH nAUR !URA.8WVestudythepropSertiesofend1Cw(!n9).P ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIES[W3Ǎm\9 ThepsecondsectionofthispapSerisdevrotedtosomebasicnotionsfromthetheoryofbraidedmonoidalcategoriesCPandthenotionofC5-categories.Themostinrterestingexamples.forCncarethecategoriesofmoSdulesresp.rcomodulesorver.Hopfalgebraswithanadditionalstructureknorwnasaquasitriangularstructureresp.braiding,oneguexampleisthecategoryofsupServrectorspaces,ƩfurthermorethecategoryofYVetter-DrinfeldmoSdules.( Inthethirdsectionwrestudythegeneralalgebraicstructureofreconstructedobjects5inabraidedmonoidalcategoryV.We5separatethediscussionofthepropSertiesofcreconstructedobjectsstrictlyfromtheexistencetheoremsthataretreatedinthefourthsection.okWVeharvedecidedtobaseourinrvestigationsoncoalgebrasand(righrt)1comoSdulesinsteadofalgebrasand(left)modules,Bbecausethefundamenrtalstructure%ctheoremforcomoSdulesmakrescertainconstructionsinvectorspacesinthiscasemrucheasier.SowestudythecoSendofafunctor! H:B| M!AastheunivrersalnaturalYtransformation!J !H!C G U=andshorwthatsuchauniversalcoSend"(!n9)H=Ur2A9carriesthestructureofacoalgebra,abialgebra,orevrenaHopfalgebradepSendingonthepropertiesof!n9.Thesestructuresonthecoendofafunctorharvealready!bSeenstudiedinvXariouspapers[Mj93b g,Mj94a$,Mj94b$+,Pra93l,Scrh92a':].4OurtecrhniquesLallowustorestricttheclassofnaturaltransformations(bythenotionofC5-morphisms.wandbrychoSosingdi erentcontrolcategoriesC5),Twhichgivesusafamilyofosdi erenrtuniversaltransformations! !7Y!sJ UC 2parametrizedbythechoiceofconrtrol9categoryCrnandthusdi erentcoalgebras,]bialgebras,or9HopfalgebrasUCm.Thistecrhniquewas rstusedin[Pa78J,Pa81].Nowweexpandthistechniquetothecaseofbraidedmonoidalcategoriestostudythestructureofhiddensymmetries.Itturnsoutthatsomeofthestructureisconnectedwithcoadjoinrtcoactions,cosmashproSductsandtransmrutation. InHthefourthsectionwreshowunderwhichconditionsacoalgebraCinAcanbSeĔreconstructedfromthecategoryofCܞ-comodulesA2C ,inAandthefunctor!6y:A2C tC!URA.EIfyCVtisacoalgebraintheC5-monoidalcategoryA=C- and!Ë:A2C tC!AishtheunderlyingfunctorthenC=,coSend!߀C'M?(!n9)([Pra81J]andCor.K4.3withC0 =,C5).OneRwrouldliketohaveamoregeneraltheoremoftheform:ifC.isacoalgebrainan0@ cmti12arbitrffary/9C5-monoidalcategoryAand!8@:A2C >!A/9istheunderlyingfunctorthenC=)coSend"C(b(!n9). vThis,horwever,isnottruebSecauseofhiddensymmetries,whicrhlivreinpartinA(ormorepreciselyincoSend!8C&X(id :URA !A),nseesection5).Ouruseofconrtrolcategoriesallowsustoreconstructacoalgebrafromitsrepresentations(comoSdules)%inanarbitrarymonoidalcategoryA,namelyCLM=ocoend!#)A(n(!:oA2C G!A)AF(nobraidingofAisneeded).jThisfactallorwsusnowtowonderabSoutandstudytheadditionalhiddensymmetriesappSearingincoend"PoC'.(!n9).OVVariousre nemenrtsofthisreconstructioncanbSefoundinTheorems4.1and4.2. Thesecondexistencetheoremdealswiththe(re-)constructionofCpJinthecaseofanarbitraryC5-functor!Ë:URB !A.BConditionsforsucrhareconstructionhavebSeendevrelopSedinmanypapSers,*suchas[Ul89,*Ul90]inthecaseAUR=VVecQ.Inoursituationwrecanprovethefollowing:mif!Z:BJ _!C$isaC0-functorwhichfactorsthroughthe>categoryC0ofrigidobjectsinCandifCiscoScompletethenUCqAacmr60 t=coend!Cq0*(!n9)exists.WAwreakvXariantofthistheoremwasprovedin[Mj93b g](Theorem2.2)whereB0ismessenrtiallyacategorywithonlya nitenumbSerofobjectsandmorphisms,theconrtrolfcategoryistrivial, andonlyrigidobjectsarereconstructed(formoredetailsseetheremarksin4.1.2). InAsection vreweshowasoneofthemainresultsofthispapSerthattheuniversal ;\94FBODOUUP*AREIGISǍm\9objectcoSend MC&; (!n9)UR=UCforafunctor!Ë:URB !AtendstodecompSoseinrtoacosmashproSduct[ofaHopfalgebrawithacoalgebra.1Inparticularwreshowthefollowing.1IfH9isabraidedHopfalgebrainabraidedmonoidalcategoryDUV,1CЁisacoalgebrainDUV2Hn,ʇand!c:*(DUV2H)2C !DUV2H [istheunderlyingfunctor.R+ThenthereisacanonicalisomorphismɍZ{fQ:URcoSend!̟D(X/(!n9)UR !URcoSend!̟D(id ʤ>DD cmmi10 z,,and.PlTheminimalrequirementsforcoherenceareobviousinmostcases.PlW*edonotfurtherinvestigateUUthem.2 ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIES[W5Ǎm\92.1.2. LetoA(withmA 36:URA4 A !AoanduA 36:IF .!A)bSeanalgebrffa(amonoid)in3A,i.e.themrultiplicationmA isassoSciativeandunital(withunitmorphismuA).InthesituationAUR=M2BN>,sucrhanalgebraAiscalledaB-cffomodulealgebra.(InthecaseAUR=MBN>,sucrhanalgebraAiscalledaB-moffdule35algebraX[Sw69|].H2.1.3. Theh4categoryB)A bvifandonlyifPisarighrtB-comoSduleandaprighrtA-moSdulesuchthats2(pa)4=Pmp(0) \|a(0) \6p(1)a(1),0apB-A-HopfmoSdule.G9AvrectorspaceP,isin(MBN>)A hi PisarighrtB-moSduleandarightA-moSdulesuchthat(pa)bUR=P (pb(1) \|)(ab(2)),i.e.aB#A-moSdule.2.1.5. FVurthermorewletC(withC t:URC1 I!C kCandw"C t:C1 I!I)bSeacoalgebrainA.6InthesituationAUR=M2BN>,sucrhacoalgebraCIiscalledaB-cffomoduleD,sucrhacoalgebraCFiscalledaB-moffdule35coalgebra.2.1.6. ThecategoryBJF=A2C of(righrt)Cܞ-cffomodules(PS;` :P !Pq MCܞ)inAiso"a(left)A-categoryV,AsinceX^ PcarriesthestructureofarighrtCܞ-comoSdulebyX+ P Lq!URX (PLn Cܞ)PUR԰n:=(X P) Cܞ.2.1.7. A0vrector1spacePisin(M2BN>)2C ifandonlyifPisarighrtB-comoSduleandarighrtCܞ-comoSdulesuchthat>$ ۟9P۟9}PLn BEb8҄fd>bO line10-ԻX.BPLn Cc(PLn Cܞ) B*32fd"Gά-a3X.ByXfeڟ?ՍJX.CmXfe2?ՍX.C7 id\X.B]Acommrutes,Ri.e.P\isyaB#2c.yCܞ-comoSdule,acomoSduleorverythecosmashproduct.Avrector͸spacePo~isin(MBN>)2C Pi PisarighrtB-moSduleandarightCܞ-comoSdulesuchthatC(pb)UR=Pp(0) \|b(1)$ p(C;1)ߤb(2).捍De nition2.2.YLetB\OandB]m20*bSeC5-categories.IAfunctor!Ë:URB !B]m20togetherwithacoherenrtnaturalisomorphism:UR!n9(X+ P) !X !n9(P)iscalledaC5-functor.! Observre/thatourassumptiononcoherenceimpliesinparticularthatn920.LetAbSeanalgebrainVVec .-QItcanbSeconsideredasaB-modulealgebrabrythetrivialactionab':=a"(b).ELetBܔ:=(MBN>)A.ConsidertheC5-functor!`:'(MBN>)A ] d!MB with!n9(P)=P,YtheCforgetfulfunctor.CThenforanrya2ACthemorphism'a:!Z D!!n9,'a(P):!n9(P) #_!!n9(P),L'a(p)=paisanaturaltransformationandinfactaC5-morphism.9FVorbanrybw2center#y(B)bthemorphism'b:w!$ ;}!!n9,'b"(P):!(P) D!!n9(P),+'b"(p)UR=pbKisanaturaltransformation,butingeneralitisnotaC5-morphism.If~'b1ZisaC5-morphismthenforthespSecialcrhoiceX=QB,sP4=B^ A,x=1BN>,andp=1B 9 A1A 'lwreIhavePb(1)G Ab(2) 1A Դ=ПP b(1) b(2) 1Ab(3)SL=ПP xb(1) pb(2)SL='b"(xl. p)9!=xl. 'b(p)9!=xl. pb9!=1B l l.b 1A,Mandhence(b)9!=1l. bwhicrhimpliesFb'= d1B ( 2K).MKConrverselyforb'= d1B itiseasytoseethat'biisaC5-morphism.瀍2.2.Braidedacategories.FVorLthede nitionandstudyofmorecomplicatedob-jects,likrebialgebrasandHopfalgebrasinC5,wreassumethatthemonoidalcategoryC isbrffaided(oraquasitensorcategory)withanaturalisomorphismofbifunctorsX&;Y|:XM YP԰=Y: X,dtheKbrffaiding,sucrhthat(1Y = X&;ZTQ)(X&;Y 1Z8)=X&;Y Zand(X&;Z 1YP)(1X 1 Yx;Z)UR=X Yx;ZϾ.2.2.1. A &quasitriangulargstructurffe ^oruniversalRJ-matrix~p[Dr86J]forabialgebraBX=(B;m;u;;")inVVecOisaninrvertibleelementRn=URPR1j R2V2URBE BsucrhthataÛ(1)#8bUR2BX:W(b)=RJ(b)R21 u;Û(2)#( 1)(RJ)UR=R13 R23;Û(3)#(1 )(RJ)UR=R13 R12whereR12 UZ=URR 1BN>,R13=URPR1j 1B R2,andR23=UR1B RJ.瀍2.2.2. Acffoquasitriangular``structureH([Scrh92b$:]De nition2.4.4and[LVT91-])orbrffaid-ingisaconrvolution-invertiblehomomorphismr:URBE BX E!KsucrhthatÛ(1)#mo=URr6mrS21 ;Û(2)#rS(m 1)UR=r213 Sr223;Û(3)#rS(1 m)UR=r213 Sr212:瀍2.2.3. IfpB7visquasitriangularthenMB isabraidedmonoidalcategorywithX&;Y(x yn9)UR=P (yR2j xR1)[Dr86J].2.2.4. IfB\iscoSquasitriangularthenM2B 9isabraidedmonoidalcategorywithX&;Y(x} yn9)ez=P(y(0)g }x(0) \|)rS(x1= y1) ([Scrh92b$:]Remark2.4.6;seealsothelastparagraphin[Pra81J]). ?ff< s6 ^2|sThere AarewellknownstandardmethoGdstohandlethesettheoreticdicultiesofthis construction.Z ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIES[W7Ǎm\92.2.5. 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WVesarythatacoactionad3l:BC "9!C+ OBwforagivenzy%:BC "9!B,qBwabialgebra,isaright35cffoadjointcoactioniftheequationinLemma2.13holds. Moregenerallyifzt:C/ C!Bis?-inrvertiblethenacoadjointcoactioncanbSeconstructedinthesamewrayasabSorve.93WVedon'tknorwiftherearemoregeneralconditionsBforzt2:OC F!BsucrhBthatarightcoadjointcoactionexistsnorwhetheritisuniquethen.e΍2.5.3. In3"thedualsituationletf:ЯH ^!A3"bSeanalgebrahomomorphismwithaHopfI algebraHV.TherighrtadjointactionahUR=PfG(S(h1))`af(h2)I iscrharacterizedbrytheequationPxfG(h1)(ah2)UR=afG(h).2.6.C0-generated$coalgebras.WVe7stillneedanothersomewhatmoregeneralsetup. LetcCbSeamonoidalcategoryV,~C0#beafullmonoidalsubcategoryofC5. InthissituationwreconsideraspSecialtypSeofcoalgebrainC5.`ݍDe nition2.14._LetC12URCbSeacoalgebrasatisfyingthefollorwingconditions:Û(1)#CFisacolimitinCofadiagramofobjectsCiOinC0.Û(2)#AllMmorphismsX\# jiz M6:URX Ciz M6 !URX CG> MareMmonomorphisms#in}dC0whereX@2OC0,M2Cand}dthei:OCi H!CZare}dtheinjectionsofthe#colimitdiagram.Û(3)#EvreryCiOisasubScoalgebraofCFviai,:URCi!Cܞ.Û(4)#If:(PS;P g:URP Lq!PCa Cܞ)isacomoSduleorverCandP2URC0,thenthereexists#aCiOinthediagramforCFandamorphismP]ߍ8ӍPP8Ӎ=PLn CiҲ&8҄fd ά-ӌ(sX.P1;iPLn CՍX.P@@@R̟Xfe?n1 8:i ⍑#commrutes.ThenCFiscalledaC0-generffated35coalgebra.vx ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIESVW15Ǎm\92.6.1. IfOC0V=URC thentheconditionsinthepreviousde nitionaretriviallysatis ed.If+'C* =vVVecandC0 6=vvrecȔ,{Gthecategoryof nite-dimensionalvrectorspaces,thenevreryXcoalgebrainCisaC0-generatedcoalgebrabythefundamentaltheoremforcoalgebras8([Sw69|]Thm.#r2.2.1)anditsgeneralizationtothefundamenrtaltheoremforcomoSdules.0Ѝ2.6.2. WVe|denotebryC25CRA0 OwthecategoryofCܞ-comoSdulesinC0. ThenC25CRA0isaC0-categoryandtheforgetfulfunctor!Ë:URC25CRA0 (}x!C0isaC0-functor.2.6.3. ItisaneasyexercisetoshorwforaC0-generatedcoalgebra,thatthe(PS;PandofaC-functor!Ë:URB !A.}At`morphismt[;]UR:(B]m;!n9) !(B]m20+;!n920(!n9;! {`)andinducesaunivrersalfactorizationof!XthroughthecategoryofcomoSdulesA2C. & ThestudyofC5-functorsasconductedherehasmanrypropSertiesincommonwithsimilar;resultsforgeneralfunctors. +Infactgeneralcategories,functorsandnat-ural#transformationsmaryalsobSeconsideredasC5-categories,,C-functors,resp.=PC-morphismsforthemonoidalcategoryCwithoneobjectI+andonemorphismid LI.J ManryT\ofthefollowingpropSositionsarewellknownforthecaseofamonoidalcategoryzC=IfIgandcanbSeprorvedzbystandarduniversalabstractnonsense.ESowreQonlysketchtheideaoftheproSofs. Thereare,however,subtleQdicultiesandrestrictions4withrespSecttobraidingsthatdonotoccurinthecaseofasymmetricconrtrolcategoryC5.퍍Prop`osition3.3.bIf b6NatߟC"(!n9;!v" { )isrffepresentable,ДthentherffepresentingobjectC=coSend"xC(:7(!n9)'isacffoalgebra'inA.B#Thiscffoalgebra'isuniquelydetermineffduptoisomorphisms35ofcffoalgebras. Furthermorffeơeveryobject!n9(P)UR2AơwithP2URB$isaCܞ-comoduleviaȄ:UR!n9(P) !!n9(P) Cand35everymorphism!(fG)isamorphismofCܞ-cffomodules. Everyzobjeffct!n9(X OP)UR2AzwithXF2URC-andP2BisisomorphicasaCܞ-cffomoduleto35X+ !n9(P)withthestructurffeinducedby!n9(P).퍍Prffoof.#RThecomrultiplicationandcounit"areuniquelyde nedby(1! Kw 8),D=( 1C)]and(1!* ")Ȅ=UR21RA! \|.8Thelastclaimfollorwssince]isaC5-morphism. yff٘ ̍ ff ̄ ffffff٘ & WVe`willencounrtersituationsofcomoSdules(P;#?:P ^!Pf Cܞ)`inAwherewrewrantI^toknorwifthiscomoSdulecomesaboutasinthepreviousProposition.USowrede ne  ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIESVW17Ǎm\9De nition3.4.YLetNatSC S(!n9;!2 x{X)UbSerepresenrtablebyC12URA.0ThenacomoSdule(PS;#:P !Py Cܞ)GAinAcanbelifteffdRalong!zifthereisanobjectQ2BandGAacomoSdule}isomorphism(!n9(Q);s2)PUR԰n:=(P;#).In}thiscasethecomodule(P;#)iscalledliftable35along!XandQUR2Balifting.3.2.Reconstructionofbialgebras.Assumednorw,thatthebasecategoryAisa C5-braidedC-monoidalcategoryV. (ItwillbSeclearfromtheconrtextwhichtensorproSductisbeingused,sowresimplyuse forthetensorproductinA.) ConsidernaC5-functor!Ë:URB !A.Thennthebifunctors! i!=UR!n922:B iB !Aand!n922 aM`:/B:B| !AareC5-bifunctorsascanbSeeasilycrhecked._ThesetsNat@Ch(!n922.=;!n922= YM@)!IofC5-bimorphismsdepSendfunctoriallyonM,Ii.e.wrehaveafunctor!NatnC8-(!n9 2.=;!n9 2 {)UR:A !Set:LetNatCaj(!n9;!7 8{)bSerepresenrtablewithuniversalC5-morphismȄ:UR!Ë !!7 8Cܞ.&SIngeneralthemorphism( 2V:=UR(1!* co)(P t Q/)UR= s20ڍP.: Q]:&΍ Themorphism2s20RAP.: Q R$':UR!n9(P)>a !(Q) !(RJ)UR !UR!(P' Q RJ) BPrespSonsibleforassoSciativitryisaC5-trimorphism. The4coSquasitriangularstructurer:BW B r!Iis4de nedbrytheC5-bimorphismn91cA ʵ(!n9(Q);!(P))!(BE(PS;Q))D:!n9(P)p !(Q)D O!D!(P) !(Q) I\and thebraidequationE!}fdfe fdfe/(1fdfeWWfdfeMfdfeҟfdfe)фfdfe !fdfe7ńfdfe҅bÄfdfefdfe뫶؄fdfekfdfe}0Lfdfeb0 fdfeF۟W7fdfe(ϟ}fdfeŸ}fdfe՟ȧfdfe+fdfefdfeuY4VfdfeKVfdfe xքfdfeg5fdfeׄfdfeyӄfdfe])fdfe(fdfeo7fdfe!Tefdfe-&_fdfe%3fdfe ƭfdfeʄfdfeұifdfej;fdfe fdfe⻄fdfe=fdfefdfe$+afdfe88fdfeNџfdfeg 搄fdfeEτfdfea열fdfemq/fdfezK_fdfeg&5fdfe!DfdfeG%ݿfdfen뺆fdfe☑fdfe/vfdfe՟Tfdfe!T4fdfeRß fdfe=fdfe㻍fdfe͟܄fdfe,wfdfe-}fdfeʟfdfeɣ){fdfeѸYfdfe yfdfe떟fdfe_wfdfe]fdfe*LufdfeF|fdfedџsfdfeşfdfeqfdfeT>fdfeoofdfe.ޟfdfe`mfdfeXfdfe1kfdfe afdfeIifdfefdfeэgfdfeO#fdfefMTefdfenfdfecfdfe]fdfe嶁afdfeFfdferw_fdfe!&_fdferfdfeafdfefdfe3dcfdfe3fdfeݟefdfeߟfdfe빟gfdfe#qfdfekAifdfe]wfdfe8GkfdfefdfeAmfdfe^Nfdfe?ofdfeMfdfeMqfdfeٍfdfe\sfdfeX5+fdfeufdfefdfe}wfdfe.ΟifdfeY9yfdfe熨fdfe-{fdfeђfdfer-w}fdfewnfdfefdfelfdfefdfe9jfdfeifdfeoUhfdfeZWfdfeBffdfe&+fdfe\dfdfefdfebfdfe`fdfel`fdfe=֟N߄fdfe ^fdfe\݄fdfeş\fdfebۄfdfe"AZfdfeୟqلfdfe'XfdfeReׄfdfegVfdfeF3ՄfdfedџdTfdfe ӄfdfe3RfdfeZ фfdfe&PfdfeKPfdfeMBфfdfeRtRfdfeZ0CӄfdfeeTfdfetՄfdfeןVfdfe՟ׄfdfeQXfdfeΑ لfdfeIZfdfe=ۄfdfe4m\fdfe\̟^݄fdfes.^fdfeV߄fdfeu`fdfeПfdfeWglbfdfe';fdfe5 dfdfefdfeZffdfeǟyfdfeşIhfdfe=fdfe[jfdfe fdfe>lfdfe"Vfdfe&nfdfeKwnfdfeL駚fdfeO"fdfeT]fdfe[)4>fdfecaӄfdfen„fdfe{+fdfe涄fdfe/fdfe< fdfeeɄfdfeIքfdfe=fdfe fdfe%ٟ(fdfeC,fdfecRnfdfeߟwfdfefdfeϝfdfe[Gfdfe! 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EW{%S)ismrultirepresentable,thenBX=URcoSend!(!)isabialgebrainA.8Thisbialgebraisuniquelydetermineduptoisomorphismsofbialgebras. IfinadditionBHisbraided,thencoSend!"(!n9)iscoSquasitriangularinA. IfBHisrigidthencoSend!"(!n9)isaHopfalgebrainA. FVurthermoreforanryobjectsPS;QUR2BetheB-comodulestructureon!n9(P)= !(Q)isde nedbrythemultiplicationoncoSend!"(!n9).E3.4.2. WVe/spSecializeC|=GA0 3withAabraidedmonoidalcategoryandA0afull(braided)LmonoidalsubScategoryV.;WeLwillcallthiscaserffestricted reconstruction.Let!4:ƧB$ ym!A0AbSe-=anA0-functor.Let9:Ƨ! 9!!F6 C bSe-=aunivrersalA0-morphism.ThenthepropSositionsofthissectionspecializeto: IfKNatAq0"I(!n9;! Q{mM)Kisrepresenrtable,*thentherepresentingobjectC1=URcoSend!̟Aq0,!(!n9)iseacoalgebrainA.Thiscoalgebraisuniquelydetermineduptoisomorphismsofcoalgebras. FVurthermore|evreryobject!n9(P)UR2A0hwith|P2BisaCܞ-comoSduleviaȄ:!n9(P) !!n9(P) CFandevrerymorphism!(fG)isaCܞ-comoSdulemorphism. LetB#SbSeA0-monoidaland!Ë:URB !A0beanA0-monoidalfunctor.,IfNatAq0"|(!n9;!Ѕ bL{BH)̐ismrultirepresentable,then̐BX=URcoSend!̟Aq0,!(!n9)isabialgebrainA.كThisbialgebraisuniquelydetermineduptoisomorphismsofbialgebras. IfinadditionBHisA0-braided,thencoSend!"Aq0-w(!n9)iscoSquasitriangularinA. IfBHisrigidthencoSend!"Aq0-w(!n9)isaHopfalgebrainA. 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Norwweconstructthesecondmap.ELet';:! [!~t !! [! M@.EDe nelimg Q!Q'(Bid;Bjf )='(B;B):B] B!B B Mastheuniquelydeterminedmorphismsothat9Ϫq Bi BjqBi Bj MO8҄fd^ά-ӻo'(B8:i,r;B8:j)yBE B^ BE B M32fd$ ά-؍'(Bd;B)ßXfe?< 48:i,r 8:jXfeM?eG߽8:i,r 8:j 1X.MN2commrutes.^HObserveLuthatBE ?B{isthecolimitoftheBiR BjsincebryassumptionthetensorproSductpreservrescolimits.8LetN24Ӛ(')UR:=(  1M )'(B;B)UR:BE BX E!B B M6 !URM:AsintheproSofof4.1(')isanHV-comodulemorphism. WVeRdharve(fG)=(H  1M )(1Bd B< f)(1B ? Bd;Bj8 1BN>)(B BN>)=fG(H  1Bd B)Bd B=%fG.ObservredthatP 6:P $!P BisaB-comoSdulemorphismwiththeB-structureonPLn BcomingfromtheoneofB.8Sowrehavek̞&~(')((PS;P);(Q;Q/))7e=UR(1P.: Qr   1M )(1P.: Q '(B;B))(1P t Bd;Q 1BN>)(P Q/)7e=UR(1P.: Qr   1M )(1P.: Q limg b!b'(Bk#;Bl!)),OU(1P t Bd;Q 1BN>)(P Q/)7e=UR(1P.: Qr   1M )limg Q!Q[(1P.: Q '(Bk#;Bl!))(1P t Bi?k;Q7 1Bi?l ()]OU(1P t i 1Q . jf )(P6J􏍒PLn Q􏍒WPLn Q BE B8҄fdڍά-v~q2PLn QpPLn Q B32fdά-OMX.P Q*Xfe,\?UM2c1X.P QXfe!?]Ս 1X.P Q EemX.Bucommrutative,~sincet!g .!Ë=UR!n9(- -) ,g UR !!n9(- -) BzistaC0-bimorphismofbifunctors.TheunitisgivrenbyUR:IF .!I# 2]BX=B.%>ThenewmultiplicationcanbSedescribedbrythebraiddiagramCЍI\fd ]⎍vTfe336fe33©Qfe]㎎}fdfefdfe(1fdfeϟWfdfeMfdfeJfdfeфfdfez !fdfekw7ńfdfeZbÄfdfeHfdfe4.؄fdfefdfeLfdfeo0 fdfeSW7fdfeG}fdfe:}fdfeoMȧfdfeKp+fdfe%fdfeџ4Vfdfe#Vfdfexքfdfezߟ5fdfeK`ׄfdfeӄfdfew)fdfe'fdfey7fdfe@TefdfeK&_fdfeL3fdfeOƭfdfeT]ʄfdfe[)ifdfec;fdfen fdfe{+⻄fdfefdfe/fdfeafdfe8fdfeIfdfe搄fdfe τfdfe%ٟ열fdfeCq/fdfecK_fdfeߟ&5fdfefdfeϝݿfdfe[뺆fdfe! fdfeLvfdfezMTfdfe̟4fdfe; fdfefdfeDfdfe{E܄fdfewfdfeK}fdfeMBfdfeR){fdfeZ0Yfdfeeyfdfetfdfeןwfdfe՟fdfeLufdfeΑ|fdfeIsfdfe=fdfe4mqfdfe\̟>fdfesoofdfeVfdfeumfdfeПfdfeWg1kfdfe'afdfe5ifdfefdfeZgfdfeǟ#fdfeşTefdfe=fdfe[cfdfe fdfe>afdfe"Ffdfew_fdfe&_fdfefdfeafdfefdfedcfdfe3fdfeoUefdfeZWfdfeBgfdfe&qfdfeAifdfefdfekfdfe`fdfelmfdfe=֟Nfdfe ofdfe\fdfeşqfdfebfdfe"\sfdfeୟ+fdfe'ufdfeRefdfegwfdfeFifdfedџ9yfdfe fdfe3{fdfeZ fdfew}fdfe6feٝ ]⍄ ]fe{ ӄ ]fefefeٝk6feٝ ]⍄ ]fe{ ӄ ]fefefeٝ݅P6feٝ ]⍄ ]fe{ ӄ ]fefefeٝA6feٝ ]⍄ ]fe{ ӄ ]fefefeٝAfeٝ ]⍄ ]feI#ifsbm ӄ ]fefefeٝ~}fdfe}fdfe{(1fdfev9WfdfeomMfdfeffdfe\ фfdfeOk !fdfe@7ńfdfe0gbÄfdfefdfe ؄fdfeMfdfeLfdfeٟ0 fdfeW7fdfe}fdfef}fdfeDȧfdfe ڟ+fdfefdfe;4VfdfeVfdfe}xքfdfePI5fdfe ʟׄfdfe[ӄfdfe)fdfe􆑟fdfeOQ7fdfeTefdfe!&_fdfe"3fdfe$ƭfdfe)ǟʄfdfe0ifdfe9L;fdfeC fdfeP⻄fdfe_fdfeofdfe afdfeh8fdfe쬳fdfe搄fdfe'τfdfeC열fdfeOq/fdfe9\K_fdfe[I&5fdfe&fdfeݿfdfeş뺆fdfesfdfe"vfdfeOTfdfe64fdfe fdfefdfeofdfeP܄fdfewfdfe!}fdfe"fdfe'){fdfe/Yfdfe:yfdfeIxfdfe[Awfdfep?fdfeLufdfe|fdfe³sfdfe䧟fdfe ןqfdfe26>fdfe]ݟoofdfefdfeߟmfdfe:fdfe,џ1kfdfehafdfeifdfefdfe/ogfdfex1#fdfe/TefdfePfdfeeşcfdfevfdfecafdfepFfdfew_fdfe~&_fdfe}Tfdfex{afdfepffdfeedcfdfeV3fdfeDefdfe/fdfegfdfeqfdfeMAifdfeYfdfe)kfdfemʟfdfeB#mfdfe@Nfdfe!ofdfeƟfdfes/qfdfe7ofdfea\sfdfe+fdfepufdfe'ϟfdfeџwfdfe󌰟ifdfe:;9yfdfe䊟fdfe򋝟{fdfe/tfdfew}fdfeD7ife݄ fefe7ife- ]fee{fPe{LQe{Pe{Q0Ӈ= ]⍒SuP ]⍒yQ ]⍒dB ]⍒(P ]⍒yQ ]⍒iBx ;\926FBODOUUP*AREIGISǍm\9where@thebraidmorphismontheleftistheoneinC+andthebraidmorphismonetherighrtistheoneinC52Hy. X ThezproSofshorwsthattheuniversalmorphismis(1P "Bd;QZ 1Q/)(P Q/):! !Ë !UR! ! B, B.FAnanalogousresultholdsformrultifunctors! :::\ !Ë=UR!n92nandC0-morphisms'm:!n92n O!!n92n G 1M@.DThisprorvesthatwithasuitableelemento2URBn intheArtinbraidgroupthemorphismDAWn:=URWs2 n p:!n9 n k4!!n9 n1 B nistheunivrersalC0-morphismforallnUR2N,inparticularcoSend!"Cq0*(!n92n)UR=B2nCV. yff٘ ̍ ff ̄ ffffff٘Oj4.1.2. TheproSofof[4.1resp. 04.2prorvidesaproofforthe\represenrtabilityas-sumptionformoSdules"(in[Mj90O]3.2)inavrerygeneralsettingforthefunctorNat@Cq0k(!n92n;!n92n |{x)%insteadofthefunctorNat(!n92n;!n92n |{x)._A spSecialcaseofTheorem4.2is[Mj93aK]PropSositionA.4ifoneusesC=URVVecQ. 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Bused`intheconstructionofB(uptoaprecedingisomorphism).VSincewrearenowconsideringzatensorproSductoftrwozdiagramsandthecolimitthereofandsincetensorproSductspreservrecolimitswehaveprovedtheclaimedresult. yff٘ ̍ ff ̄ ffffff٘nq4.2.1. UsingKtheresultsofsection3wreobtainuniquelydeterminedcoalgebra,kbial-gebra,andHopfalgebrastructures(depSendingonthegivrenfunctor)oncoend"EJ(!n9)andcoSend!"Cq0*(!n9).:5.HiddensymmetriesǍ Insection4.1wrestudiedunderwhichcircumstancescoalgebrasandbialgebras(pSossiblywithatransmrutedmultiplication)canbSereconstructedfromtheircate-gories@ofcomoSdulesandthefunctor!U:C52C N!C5.:WVe@sarwthattheyareobtainedas/5therepresenrtingobjectcoSend!⯟C'Pn(!n9)ofNatoޟCݝ(!;!G T{P):C}6 ҏ!Set.In/5thissectionwre;willseethatthisreconstructiondepSendsstronglyonthechoiceofthecontrolcategoryC5.7IfC)isdecreasedtoacategoryDcoSend!D((!n9)>bSecomeslarger.3WVewillseethatundercertainconditionsthere-constructedobjectdecompSosesinrtoacosmashproductwhereonefactorrepresenrtsthe\hiddensymmetries".ō5.1.Functorsofcontrolcategories.WVeeconsiderofabraidedmonoidalfunctorFc:URD !Cofconrtrolcategories.5.1.1. IfBHisaC5-categoryvia UR:C]B !B]m,thenBbSecomesaDUV-categorybry~lI UR:DBF(1g  !]lC]B 1 g  !B^with=~assoSciativitrymorphism O(QG 1)Q:(Xԑ Yp) P p!Xԑ (Y~ P)=~andunaryactionn9(& 1)UR:ID P Lq!PnforXJg;Y2D?andP2B]m.5.1.2. If;IM:B E!B]m20fisaC5-functor,qthenbSecomesalsoaDUV-functor. *If:BTB]m20@i!jB]m2020 isZaC5-bifunctor,vthenbSecomesalsoaDUV-bifunctor.Inbothcasesthestructuremorphismsҩresp.8Aremainuncrhanged.Ǎ If;20#:UB K!B]m20lareC5-functorsand=: !20isaC5-morphism,thenisalsoaDUV-morphism.5.1.3. IfAisaC5-monoidalcategoryV,thenitbSecomesalsoaDUV-monoidalcategory.Ǎ TheabSorveobservXationsgiveimmediatelyProp`osition5.1.bIfFc:URD !Cisabrffaidedmonoidalfunctor, thenitinducesan\underlying"35functorA ϯ(F1)UR:A (C5) !A(DUV).!   ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIESVW33Ǎm\95.1.4. Lete(B]m;!n9)UR2A (C5)andconsiderA (F1)(B]m;!n9)=(B;!n9)ewiththeinducedstruc-ture%morphisms.2AssumethatNatΟC(!n9;! {d)andNatΟDh1(!;! {d)arerepresenrtablebry coSend#]C),(!n9) resp. coSend)M D0n(!). ThenA(coSendzC#!9(!);M@)P԰ =rfd6^ά-捒ނLAP! CAPL[! Cܞ22Y32fd'`ά-uR.L 1 @= H H j @ 1 fP H H jЎhHHjЎLW1 fЎhPԬHHjЎLݹ1 f fЎaV̄Afe̬?ό@ @[Afe?̼ŽXЎV̄AfeH̬?Yꈸ1  @PvAfe?V1 !Íand\ٍ[!ٍ! D?>rfd]ȍά-ҍƇ@ @۟9[! Cό퟽Q_Q BQq"Qq"s @! IK⍍=mߟ DSߟ DSmߟ DSߟ DSmߟ DSߟ DSfSfw @p:1 f_ Bٱ}"ٱ}"+ @.ݍG}1 "x D Dx D Dx D D/şXfe2?, 2g1 "cshorwthatfQ:URD 9!CFisacoalgebramorphism.f Let7gF:L!n9(eC D) -!C9bSeaCܞ-comoduleisomorphism.Thenthefollorwingdiagramcommrutes^'/ziR!n9(eC D)/ !n9(eC D) DzΟ>rfdhά-ҍ@//I+ DP԰=SD>rfd0\ά-`"gI{ 1 @i9!n9(eC D) CόDQ_Q BQ"Q"s @AlXfet?pt1 f @8Afel.?#1 f @'3\Afe'f?+ef @9ڸCIܟXfe}?7(gAlXfet?ϝCgI{ 1NQYQQQsڟCF CqI+ CP1԰J=ܙC5: v32fd{ά-C" 1ThelorwermorphismofthediagramistheidentityhencefisaretractioninA. yff٘ ̍ ff ̄ ffffff٘^ɍ5.1.5. ObservrethatthemorphismcoSend"F*9(!n9)/:coSend"rD*2(!) Q!coSend"rC(Q1(!)istheuniquelyde nedmorphismsucrhthat(1!* coSend ^"F'^(!n9))@n=URs2." h ;\934FBODOUUP*AREIGISǍm\95.1.6. The9precedingtheoremshorwsthatthereconstructedcoalgebraDgw.r.t.&sDisolargerthanCܞ.&"WVeconsidertheadditionalpartinDashiddensymmetriesinthesenseȽdescribSedintheinrtroduction. ItisresponsibleforDUV-morphisms'U:!= !!Qn 5MwhicrharenotC5-morphismsorforcertainelementsinNatɫD(!n9;!Qn 5M@)whicharenotconrtainedinNatşC}(!n9;! rcM@)./AswehaveseenthispartofD"tendstosplito ..5.1.7. An-exampleofahiddensymmetrycanbSeobtainedforsuperalgebrarepre-senrtations.OThis|isdualtotheabSoveconsiderations.OGivenanalgebraAconsideredas;asupSeralgebra(A;0). )ConsiderA"=C5,the;categoryofsupervrectorspaces(KZ2-ComoSd),WtheAcategoryBF=CA ofsuperA-modules,Wandtheforgetfulfunctor!:oCA Mq!C5.gLetF:D=VVec% zr!CFbSethefunctorwhicrhsendseachvectorspaceVtowCthesupServrectorspace(V;0).޲AnyDUV-morphism'D:! 8!!|iswCdescribSedbyitsimageunder'UR2NatD^(! I;!n9)P԰n:=A(I;endGDq(!n9)). Thenaturaltransformation'\:PI"{!Pgivrenby(p0;p1)\7!(p0;p1)isasymmetryforallrepresenrtationsofA(anaturalautomorphismof!n9),6&whichisnotRinducedbrythemultiplicationwithanyelementofA.8AQmultiplicationwithanelemenrt.aUR=(a;0)2AonA-moSdules(P0;P1)inCA wrouldhavetosatisfy(p0;p1)aUR=(p0a;p1a)r=(p0;p1);forallcrhoicesof(p0;p1);whicrhisnotpSossible./Thenaturaltransformation'is,9horwever,aDUV-morphismandthruscomesfrommultiplicationwith(anelemenrtbUR2endbD(!n9),Ain(factfromtheelemente1_oket2UR(KZ2)2VendbD(!n9)wheree1;et|risthedualbasisto1;tUR2K[t]=(t22j1)=KZ2.5.1.8. WVe}applytheprecedingexampleinrepresenrtationtheoryofgroups.IfweconsiderیrepresenrtationsofagroupGinvectorspacesovera eldK,Wi.e. thecategory'MKG Xz,6theneacrhelementg+62GinducesaC5-monoidalautomorphism'g:! "#!^!with'g(p):=pgn9,iswhere!:MKG  d!M=VVec!=Cistheforgetfulfunctor.ObservrethatanynaturaltransformationoffunctorsintoCHisaC5-morphismbry{?Theorem6.4. ConverselygivenanyC5-monoidalautomorphism'+:!md ½!!thereispreciselyoneg C2 Gwith'='g.vThrusGcanbSereconstructedfromitsrepresenrtations,i.e.8from!Ë:URMKG%!M. NorwC4considerrepresentationsofGinsupServectorspacesoverK,di.e.thecategoryAB$oftrwo-gradedB$vectorspaces.?ULetCp=;M2KZq2#=A,XFL:D?=VVecu!o!C=M2KZq28,and!X:AKGk!A.JWVemaryconsiderKGasaHopfalgebra(KG;0)inAandharveu(p0;p1)gb=(p0gn9;p1gn9)uwithasuitableG-structureonP0 5andP1separatelyV.Thenveacrhelementg2BGinducesaC5-monoidalautomorphism'g :! 5!!n9.%FVoranryyC5-monoidalautomorphismthereispreciselyoneg2GGwith'='g.FVortheDUV-monoidalautomorphism'`:!K #!!with'(P0;P1)(p0;p1)`:=(p0;p1)thereis,thorwever,nogR?2Gwith'='g.Sointhiscasethegroupofsymmetries(ofmonoidal/automorphismsof!n9)isabiggergroupthantheonewrestartedoutwith.Thegivren'isanexampleofahiddensymmetryV.5.1.9. WVesconsidernorwthesituationofamorphism[;]>:(B]m;!n9) !(B]m20+;!n9209w-D9ٻCɀ8҄fdbά-Ԇf6}ffKDS20fKrCܞ20Ş32fdά-ݚfǟ-:0NXfe?4yHߟXfe|?{z>΍or>΍NhcoSendkCq;([;])coSendzF%(!n9)UR=coSend!̟F(g(! 0΍(where!n9( )(PS;Q)UR:=!(PLn Q))andtheunarypropSertryisprovedsimilarlyV.5" IfthebialgebrasCܞ20`andDS20װareHopfalgebrasthenbryCorollary߰3.8CisaCܞ20-comoSdule8coalgebrabrythecoadjointcoactionw.r.t.λtheinducedcoalgebramorphismz5:URC1 I!Cܞ20M^andDisaDS20!-comoSdulecoalgebrabrythecoadjointcoactionw.r.t. the$  ;\936FBODOUUP*AREIGISǍm\9induced{coalgebramorphismynC: DS !DS20!. +FVurthermore{fG20 B:DS20 !w*!Cܞ20 &isanepimorphismandabialgebra(Hopfalgebra)morphism.$x)5.2.Hidden"symmetriesofthebasecategory.ConsiderGabraidedmonoidalfunctor(F:ZD H!C7]andamorphism[;]:(B]m;!n9) !(B]m20+;!n920(e.g.s220^qisaunivrersalDUV-morphism).Let=EPTbSeacoalgebrainB]m20andletUR: ! { EbSe=aC5-morphismcompatiblewiththestructureofE(e.g.8aunivrersalC5-morphism).Theorem5.4.S8InthesetupgivenabffoveDS20hv F!n920rfd爍ά-ҍxOE!n920DS20`q !n920rfd:ά-5!I{-:0B;>!n920PL 1Ix!n920rfdčά-5 CRL̟-:0 1AAM%!n920rfdčά-5 CRL̟-:0 1AAM%!n920?ЍѮf f& < ;\938FBODOUUP*AREIGISǍp^commrutes. ItO(isnoweasytoseethatDS20 m !n920fdfesoofdfeVfdfeumfdfeПfdfeWg1kfdfe'afdfe5ifdfefdfeZgfdfeǟ#fdfeşTefdfe=fdfe[cfdfe fdfe>afdfe"Ffdfew_fdfe&_fdfefdfeafdfefdfedcfdfe3fdfeoUefdfeZWfdfeBgfdfe&qfdfeAifdfefdfekfdfe`fdfelmfdfe=֟Nfdfe ofdfe\fdfeşqfdfebfdfe"\sfdfeୟ+fdfe'ufdfeRefdfegwfdfeFifdfedџ9yfdfe fdfe3{fdfeZ fdfew}fdfe6feٝ ]⍄ ]fe{ ӄ ]fefefeٝk6feٝ ]⍄ ]fe{ ӄ ]fefefeٝ݅P6feٝ ]⍄ ]fe{ ӄ ]fefefeٝA6feٝ ]⍄ ]fe{ ӄ ]fefefeٝAfeٝ ]⍄ ]feI#ifsbm ӄ ]fefefeٝ~}fdfe}fdfe{(1fdfev9WfdfeomMfdfeffdfe\ фfdfeOk !fdfe@7ńfdfe0gbÄfdfefdfe ؄fdfeMfdfeLfdfeٟ0 fdfeW7fdfe}fdfef}fdfeDȧfdfe ڟ+fdfefdfe;4VfdfeVfdfe}xքfdfePI5fdfe ʟׄfdfe[ӄfdfe)fdfe􆑟fdfeOQ7fdfeTefdfe!&_fdfe"3fdfe$ƭfdfe)ǟʄfdfe0ifdfe9L;fdfeC fdfeP⻄fdfe_fdfeofdfe afdfeh8fdfe쬳fdfe搄fdfe'τfdfeC열fdfeOq/fdfe9\K_fdfe[I&5fdfe&fdfeݿfdfeş뺆fdfesfdfe"vfdfeOTfdfe64fdfe fdfefdfeofdfeP܄fdfewfdfe!}fdfe"fdfe'){fdfe/Yfdfe:yfdfeIxfdfe[Awfdfep?fdfeLufdfe|fdfe³sfdfe䧟fdfe ןqfdfe26>fdfe]ݟoofdfefdfeߟmfdfe:fdfe,џ1kfdfehafdfeifdfefdfe/ogfdfex1#fdfe/TefdfePfdfeeşcfdfevfdfecafdfepFfdfew_fdfe~&_fdfe}Tfdfex{afdfepffdfeedcfdfeV3fdfeDefdfe/fdfegfdfeqfdfeMAifdfeYfdfe)kfdfemʟfdfeB#mfdfe@Nfdfe!ofdfeƟfdfes/qfdfe7ofdfea\sfdfe+fdfepufdfe'ϟfdfeџwfdfe󌰟ifdfe:;9yfdfe䊟fdfe򋝟{fdfe/tfdfew}fdfeD7ife݄ fefe7ife- ]fee{fPe{LQe{Pe{Q0Ӈ= ]⍒SuP ]⍒yQ ]⍒H ]⍒(P ]⍒yQ ]⍒H' Ġ ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIESVW39Ǎm\9In~particular#fsmH.:P#HR H=y !HkYis~amorphismofHV-comoSdulesunderthecoad-joinrt(coaction.`(ThismorphismhasbSeenstudiedin[Mj93b g]underthenotionoftransmrutation.)qK FVurthermorewrehavecoSend!"C' (!n9)UR=CFascoalgebrasinAagainbyTheorem4.1. BypTheorem5.4wrethusgetacanonicalcoalgebramorphismf5:HV#2c.yC"{!H䍑 eV#2csCԣwherethe rstcosmashproSductwrasdescribedaborveandthesecondcosmashproSductcomesfromthetransmrutationmultiplicationonHvoandthebraidinginAUR=DUV2Hn.8Asobservredin4.1.4thesetwocosmashproSductsarethesame.qK With`thesecoalgebrasthecanonicalmorphismfisde nedasintheproSofofTheorem5.4bry5\Ѝ2!Q! HV#2c.yC8҄fd*ά-2F:! Cڅ! H C^32fdά-CL̟-:0!I{ 1Xfe6?ğXfe?Ѝf1 fwithOtm=;(s220Ak!] k1)asabSorve,hsoOthattheuniquelydeterminedmorphismfunderthegivrenidenti cationsistheidentityV. yff٘ ̍ ff ̄ ffffff٘ Example5.7.QULetHBbSeacoquasitriangularHopfalgebraorverthe eldKandletCbSe"'anHV-comodulecoalgebra.\Let!":(M2HD)2C _(!M2H ;kbe"'theforgetfulfunctor.ThenCcoSend(!n9)PUR԰n:=HV# c.yC5:Prffoof.#RUseg[PropSosition6.4toshorwcoend"(!n9))=coend! M,z(!n9).Thisg[specialcasecanalsobSederivredfrom[Mj94b g]withouttheuseofcontrolcategories. yff٘ ̍ ff ̄ ffffff٘ Theblastexampleshorwsthatthehiddensymmetriesasgivenin5.1.8arerepre-senrtedhbytheHopfalgebraHѾ(withthetransmutationmultiplication),"i.e.&!H=coSendzM(Q7(id :URM2H n!M2HD).ʍ+6.;AppendixonK-additiveca32tegoriesandC5-categoriesqK LetuKbSeacommrutativeuringandletC:=URK-mod cbethecategoryof nitelygen-eratedprojectivreK-moSdules.8ThenCisasymmetricmonoidalK-abeliancategoryV.qK LetAbSeacategorywithsplittingidempotenrts,Zi.e.foramorphismfQ:URXF .!XwithifG22 5X=-Uftherearemorphismsq:X t1!P Vandj(:P $t!X[withjq=fandqn9j%=UR1P.) Then0qË:XF .!P\isacoSequalizerof(1X;fG)andq)iandjhareuniqueuptoisomorphismsofP.lꍍLemma6.1.IULffetAf*:J+X; !X3/andgd:X; !X3/bffeidempotentswithfGgd=J+gn9fandusplittings(PS;qn9;j)uofftand(PƟ20o;q20X !K2n ڙandq=w:K2n w!X#forsomen2Nwithqn9j|=1X.FVorPq2AthemorphismjqF E P:URK2n] P Lq!URK2n] PZisanidempSotenrt(jqF E P)22V=jqF E PZsothereDisasplittingq EP:URK2n P Lq!URX7J P[ andDj P:URX7J P Lq!URK2n P[ (thrusde ningq P,jW{ PnandX+ P)withJD(jW{ P)(q P)UR=jq PnandeB(q P)(jW{ P)UR=1X PK:Inparticularq %ZP:URK2nͪ P Lq!URX PPisacokrernelof(1%Zjqn9) P:URK2nͪ P Lq!URK2nͪ Pandwrehaveacommutativediagram>܍ۧsK2nR Pۧ xK2nR P=8҄fd7+pά-x1(1jvqI{) PۧŽۧ*X+ P-8҄fdcά-{qI{ P"sK2nR P" xK2nR P=32fd7+pά-؍1(1jvqI{) P""*X+ P532fd0썑ά-ue qI{ PV˟Xfe?omjvqI{ PHXfe{?oS1jvqI{ P&@KXfe&s}?-(*r=oßjv PNYss+LetwjV>:kX G!K2nP,kq:K2n Q!Xandjӟ20$w:X G!K2m ;andqn920:K2m /!XbSetrwocrhoices ofarepresentationofXasadirectsummandofafreeK-moSdules.qThenthefollorwingdiagramcommuteserfd7+pά- (1jvqI{) Pbb%X+ PW>rfdcά-ӰqI{ PۧYK2m l PۧK2m l P 8҄fd4`ά-i(1jv-:0'qI{-:0B) P"[X+ P]8҄fd 䖍ά-ԆfqI{-:0B P"K2nR P"K2nR Pg32fd7+pά-؍(1jvqI{) P""%X+ PW32fdcά-ueӰqI{ P:Xfen'? @T:P fen'P?}imjvqI{ P,Xfe`? @T,P fe`P?}i_jvqI{ P @6$uAfe6W?قȍ:W1X.Xv P @Uadcjv-:0'qI{ Po# D{o# Dur˟%ur˟% @UaUjv-:0'qI{ Pa Da Dd%d% @N⍍((=%X DX D\K%\K% OdcjvqI{-:0B Pur˟@r˟@o#@o#ROUjvqI{-:0B Pd@d@a@aRH⍍((=\K@\K@%X@%XRand theisomorphismsbSetrween X #nP=and[X #nP](thecorrespSondingobjectfortheHsecondrepresenrtationofX)arisefromthisandthesymmetricdiagramwith(j;qn9;K2nP)and(jӟ20{ ;q20rfd ά- x'(K P.:)!n9(P)yL!n920(1X 8'(P))(x qn9).5`This)holdsforallxUR2XandKallq2!n9(P)sothat'(X (P)s21=(s20p^1| 1)(1X '(P))Kand'isaC5-morphism. yff٘ ̍ ff ̄ ffffff٘hReferences[Ba68]4#HigmanUUBass:q1': cmti10Algebr}'aicK-theory.W.A.Benjamin,NewY*ork{Amsterdam1968. [DM82]4#P*.Deligne,J.S.Milne: fT;annakian׈Cate}'gories.In:HoGdgeCycles,MotivesandShimura4#V*arieties.UUSpringerLNMath900(1982)101-228.[Dr86]4#V.UUG.Drinfeld:qQuantumgr}'oups.InICMProGceedingsBerkeley*,1986.[Dr89]4#V.R{G.Drinfeld:ZQuasi-HopfAlgebr}'asandKnizhnik-ZamolodchikovEquations.R{In:ZA.A.4#Belavin:qProblemsUUofmoGdernQFT,Springer1989.+ uF ;\9`WRECONSTRUCTIONUUOFHIDDENSYMMETRIESVW43Ǎm\9[KT92]4#Christian*Kassel,NVladimirT*uraev:qDoublemConstructionforMonoidalCate}'gories. 4#Publ.UUdel'Inst.Rech.Math.Av.507/P-294(1992).[L*T91]4#Richard?G.Larson,CJacobT*owbGer:fTwoDualClassesofBialgebr}'asRelatedto`Quan-4#tumGr}'oups'and`QuantumLieAlgebras'.CommunicationsinAlgebra19,(1991),4#3295-3345.[ML71]4#Saunders5MacLane::Cate}'gories.fortheWorkingMathematician.5Springer-V*erlagNew4#Y*ork,UUHeidelbGerg,Berlin,1971.[Mj90]4#ShahnMa8jid: R}'ankجofQuantumGroupsandBraidedGroupsinDualF;orm.4#DAMTP/90-44,UUEulerInst.ProgrammeonQuantumGroups,Leningrad,1990.[Mj93a]4#Shahn rMa8jid:T;r}'ansmutation(theoryandrankforquantumbraidedgroups. rMath.4#ProGc.UUCamb.Phil.Soc.113(1993)45-70.[Mj93b]4#ShahnUUMa8jid:qBr}'aidedgroups.J.ofPureandAppl.Alg.86(1993)187-221.[Mj94a]4#ShahnGMa8jid:jAlgebr}'asLandHopfalgebrasinbraidedcategories.GIn:jAdvqancesinHopf4#Algebras.UULNpureandappl.math.158(1994)55-105.[Mj94b]4#ShahnFAMa8jid:j=Cr}'ossProductsbyBraidedGroupsandBosonisation.FAJ.ofAlgebra1634#(1994)UU165-190.[Pa77]4#BoGdoPareigis: Non-additiveRingandMo}'duleTheoryII. !", cmsy10CW-Categories,zC-F;unctors4#andCW-Morphisms.UUPubl.Math.(Debrecen)24(1977)351-361.[Pa78]4#BoGdoh^Pareigis:Non-additivekRingandMo}'duleTheoryIII.Moritaequivalences.h^Publ.4#Math.UU(Debrecen)25(1978)177-186.[Pa81]4#BoGdoلPareigis:z%A cNon-Commutative Non-Co}'commutativeHopfAlgebrain\Nature".4#J.UUAlg.70(1981)356-374.[Pa93]4#BoGdo;Pareigis:":EndomorphismBialgebr}'asofDiagramsandofNon-CommutativeAlge-4#br}'as|andSpaces. cmmi10ٓRcmr7K`y cmr10< lcircle10O line10u cmex10