; TeX output 1999.11.03:0819[7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr991\*7 &e921:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYN cmbx124.[FinitereconstructionTheEendomorphismringofavrectorspaceenjoysthefollowinguniversalpropSertyV.Itisavrectorspaceitselfandallowsahomomorphismg cmmi12q:End~(Vp)!", cmsy10 V^4!V..Itisunivrersaly5withrespSecttothispropertryV,i.e.ifZUӹisavectorspaceandfQ:URZ V4!`Visahomomorphism,+thenthereisauniquehomomorphismg3:ŬZJ4!End-B(Vp)sucrhthatD xEnd-7(Vp) V2V\Ğ32fd&O line10-gۍ H`tf@ԟ>RYZF VŸǠ*FfeǠ?q-2cmmi8gI{ K cmsy8 |{Ycmr81+commrutes.ThealgebrastructureofEndg(Vp)comesforfreefromthisunivrersalpropSertyV.IfHwrereplacethevectorspaceVbyadiagramofvectorspaces! :D54!Vec.wegetasimilarunivrersalobjectEnd(!n9).>AgaintheuniversalpropSertyinducesauniquealgebrastructureonEndg(!n9).Problem3.4.1.nR1.@녤>RYZF VӒǠ*FfeğǠ?q-"gI{ 1commrutes).8WVecallEandUR:E^ V4!`Va,@ cmti12veffctor35spaceactinguniversallyonVp.2.*LetfEعand(:E= &V4!V1bSefavrectorspaceactinguniversallyonVp.*Showthat"EuniquelyhasthestructureofanalgebrasucrhthatVnbSecomesaleftE-module.3. Let!Ƴ:XzD4!Vec6 7bSeadiagramofvrectorspaces.ShorwthatthereisaunivrersalkvectorspaceE andnaturaltransformation1b:E !4!!+(sucrhthatforeacrhU^vectorspaceZ1andeachnaturaltransformationfQ:URZVW y!Ë4!2j!×thereisauniquehomomorphismgË:URZ14!EsucrhthatLE^ !V!T32fd ά-gۍH`"f˷dׁ @շd @߷d @d @>@>RYVZF !KҟǠ*FfeǠ?q- gI{ 1׍commrutes).8WVecallEandUR:E^ !Ë4!2j!Xaveffctor35spaceactinguniversallyon!n9.]7 &e[:4.pFINITE!RECONSTR9UCTION93Y4.J Let EO#and:E "!4!! EbSe avrectorspaceactinguniversallyon!n9.J Showthat_EAvuniquelyhasthestructureofanalgebrasucrhthat!bSecomesadiagramofleftE-moSdules.{Similar^considerationscanbSecarriedoutforcoactionsV4!`V( C:or!Ë4!2j! Cand#HacoalgebrastructureonCܞ.Thereisonerestriction,1phorwever.WVe#Hcanonlyuse nitedimensionalvrectorspacesVordiagramsof nitedimensionalvectorspaces.ThiswillbSedonefurtherdorwn.Aswrehaveseen,???WVeiwrantto ndauniversalnaturaltransformation:,|!4!!n coSend 3(!n9).FVorthispurpSosewreconsidertheisomorphisms>RMorh'CmD(!n9(X);!(X) M@)PUR԰n:=Mor%5C*(!n9(X) j !n9(X);M@)thatmaregivrenbyf7!Y(ev / 1)(1ߛ fG)mandasinverseg7!Y(1ߛ gn9)(db. 1). 0WVe rstGdevreloptechniquestodescribSethepropertiesofanaturaltransformation :!4!! MθaspropSertiesoftheassociatedfamilygn9(X)k:!(X)2 !(X)k4!M@."eWVewillseethatgË:UR!n92{K M!4!2jMwillbSeacffone.)ThenwrewillshowthatisauniversalnaturalytransformationifandonlyifitsassoSciatedconeisunivrersal.OIntheliteraturethisiscalledacoSend!".Throughout thissectionassumethefollorwing.FLetD_ubSeanarbitrarydiagramscrheme. eLetCW=bSeacocompletemonoidalcategorysucrhthatthetensorproductpreservrescolimitsinbSotharguments.(~LetC0ybSethefullsubcategoryofthoseobjectsinaC_thatharveaaleftdual.$Let!Ë:URD4!CbSeadiagraminCsucrhthat!n9(X)UR2C0leforallXF2URDUV,Li.e./!>.isgivrenbyafunctor!0V:URD4!C0./WVecallsuchadiagrama nitediagrffam;yinC5.+RFinallyforanobjectM2Clet!O M:D464!,CbSethefunctorwith(! M@)(X)UR=!n9(X) M@..čRemark3.4.1.j6ConsidercthefollorwingcategoryxS"u cmex10eDtq.FVoreachmorphismf:z:{X*4!YutherePisanobjectVe*fׂ2wGeD.TheobjectcorrespSondingtotheidenrtityP1X :X4!X{isdenotedbrywDeXI2xuebDp.'FVoreachmorphismfGa:bX4! YhinD:NtherearetwomorphismsQf1Ï:V,e*f4!w$5!e!6X09ǹandf2:V,e*f4!w"Te!6Y.inx6eD .kFVurthermoretherearetheidenrtities1fq:V~e*URf K4!V!U|e*,*f*)yinx`eD .9Since therearenomorphismswithweXasdomainotherthan(1X)i~:weX74!w&le#nhX2*andU1f:Ve*f4!V!e*f*swre onlyhavetode nethefollowingcompSositions(1X)i fj揹:=fjf .Then8xv3eD bSecomesacategoryandwrehave1'#a6cmex8eX r۹=UR(1X)1V=(1X)2.]ߍWVede neadiagram!n92 !Ë:x˅eURDA4!!Cݹasfollorws.8IffQ:URXF4!YisgiventheniO}(!n9  !n9)(V)Re*f)UR:=!(Yp) j !(X)>andXAʍ!n9(f1)UR:=!(fG)2j !(1X);4-!n9(f2)UR:=!(1YP)2j !(fG):^7 &e941:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYThemwcolimitof!n921 !۰consistsofanobjectcoSend" (!n9)32C togethermwwithafamilyofmorphisms(XJg;X)UR:!n9(X)2j !(X)UR4!1coSend2w(!)sucrhthatthediagrams\Gxf2MorXD;?!n9(Zi q(fG)) j !(Qu(f))%a'؍X2Ob D^!n9(X) j !(X)@cc+ܟfd[ά- pFcFc+ܟfd[ά-׍-q&&pcoSend|r(!n9)@Kƌ2fd Pά-A{+- cmcsc10Proof.@_This{isjustareformrulationofRemarkA.10.11,&sincethecolimitmayalsobSebuiltfromthecommrutativesquaresgivrenaborve.ObservrethatfortheconstructionofthecolimitnotallobjectsofthediagramharvetobSeusedbutonlythoseoftheform!n9(X)2j !(X).{cffxff ̟ff ̎ ̄cff*cTheorem3.4.3.p(Tannak@a-Krein)Lffet!Ë:URD!fgC0VCFbea nitediagram.1BThenthereexistsanobjectcoSend!G9(!n9)UR2Cand#{anaturffaltransformationȄ:UR!Ë!J! coSend ;0(!n9)suchthatforeachobjectM62URCand:feffachnaturaltransformation'<:!!N ! mM{Jthere:fexistsauniquemorphism~e' ȹ:URcoSend!(!n9)UR!Mtsuch35thatthediagrffamRH@!HW! coSend ^"(!n9)ht{fd 0ά--0H'Z'Xׁ @X @X @X @ԟt>@ԟt>RHSj! M椢Ǡ*FfeԟǠ?ҍT1 e'0cffommutes.*cProof.@_LetjcoSend#n(!n9)p2C)togetherjwithmorphisms(V)Re*f)p:!(Yp)2pR N!(X)4!3coSendz(!n9)YbSethecolimitofthediagram!2o' @!;0:xC*eD04!&C5.Sowregetcommutative_#7 &e[:4.pFINITE!RECONSTR9UCTION95Ydiagramsr䍍!n9(X)2j !(X)3!I{(f)-:UT 1lΠlΠlΠlΠlΠÇlΠė쟯Fė쟯F*(X&;X)ylF`HylF`HylF`H ylF`HylF`H!ylF`H"H"j!n9(Yp)2j !(Y)ꃀ1 !I{(f)l?`Hl?`Hl?`Hl?`Hl?`HÇl?`HėHėjꃀ(Yx;Y)ylǠylǠylǠ ylǠylǠ!ylǠ"?"?*\n!n9(Yp)2j !(X)&coSendDc(!n9)@ :2fdx`ά-̰獒o@(㎍erf)FforeacrhfQ:URXF4!YinC5.FVor@X2xCwrede neamorphisms2(X):!n9(X)4!P}!n9(X) coSend!F(!)@bry(1 (XJg;X))(db. 1)UR:!n9(X)4!1!n9(X)_ !n9(X)2c !n9(X)UR4!1!(X)_ coSendu(!n9).ThenxwregetasinCorollary3.3.5(XJg;X)UR=(1 ev@)(1 s2(X)).WVeshorwthat]ڹisanaturaltransformation.8ForeacrhfQ:URXF4!YthesquareW9s!n9(Yp) !(Yp)2h!n9(Yp) !(X)2:\32fd/逍ά-W`J(1 !I{(f)-:HFIH !n9(X) !(X)2ܟ{fdQِά-HDdbǺX.;cmmi6XH*Ǡ*Ffe \Ǡ?e~dbX.YHǠ*Ffe9Ǡ?`!I{(f) 1$commrutesbyCorollary3.3.9.8Thusthefollowingdiagramcommutes|1N!n9(X)!n9(X) !(X)2j !(X)PܟA2fd=ά-偍ddboyL 1,!n9(X) coSend ^"(!) !A2fdά-1 (X&;X)L!n9(Yp) !(Yp)2j !(X)!n9(Yp) !(X)2j !(X)al:2fd ά-έ''ЀJJ1 !I{(f)-:UT 12+q!n9(Yp)R!n9(Yp) !(Yp)2j !(Yp)P32fd@<0ά-ׁe3 dbp 1-`!n9(Yp) coSend ^"(!): L32fdά-1 (Yx;Y)? zǠM@fe?SǠ?){!I{(f)XzǠM@feY)Ǡ?],!I{(f) 1񍍑i\dbt 1K< QU