; TeX output 1999.11.03:0821a7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr997b*7 &e981:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSY.N cmbx125..ThecoalgebracoSendProp`osition3.5.1.O+@ cmti12Lffetv|!", cmsy10C)beamonoidalcategoryandg cmmi12!@!:D'>!_C)beadiagraminC5.AssumethattherffeisauniversalobjectcoSend"m(!n9)andnaturaltransformationȄ:UR!Ë!J! coSend ^"(!n9).ThenptherffeisexactlyonecoalgebrastructureoncoSend!(!n9)suchthatthediagramsKl䍍fe! coSend ^"(!n9)! coSend ^"(!n9) coSend(!n9)32fdO line10-a2cmmi8L K cmsy8 |{Ycmr81H2!H|! coSend ^"(!n9)-{fdSά--0HǠ*FfeԟǠ?k}#H bǠ*Ffe Ǡ?o1 xandCH@!HW! coSend ^"(!n9)ht{fd 0ά--0H}id];cmmi6!Xׁ @X @X @X @ԟt>@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͝T1 Qcffommute.,- cmcsc10Proof.@_BecauseloftheunivrersalpropSertyofcoSend"(!n9)therearestructuremor-phisms}[):coSend"(!n9)4!߹coSend4Y(!)e coSend (!)}and[):coSend"(!)4!I._This}impliesthecoalgebrapropSertrysimilartotheproofofCorollary3.3.8.l#Մcffxff ̟ff ̎ ̄cffObservre,thatbythisconstructionallobjectsandallmorphismsofthediagram!۹::D4!'C0 CQcare.comoSdulesormorphismsofcomodulesorver.thecoalgebracoSendz(!n9). EInDZfactC~Z:=coSend#U6(!)istheunivrersalcoalgebraoverwhichthegivendiagrambSecomesadiagramofcomodules.Corollary3.5.2.sWLffetS(DUV;!n9)beadiagramC:withobjectsinC0.ThenallobjectsofthediagrffamarecomodulesoverthecoalgebraC|+:=coSend"S(!n9)andallmorphismsarffemorphismsofcomodules.KIfD7OisanothercoalgebraandallobjectsofthediagramarffeDS-comodulesby'(X):!n9(X)!!n9(X)/ D:Qandallmorphismsofthediagrffamarffe3morphismsofDS-comodulesthenthereexistsauniquemorphismofcoalgebras~"u cmex10e' ȹ:URcoSend!(!n9)UR!Dsuch35thatthediagrffamKl䍍H@!HW! coSend ^"(!n9)ht{fd 0ά--0H'Z'Xׁ @X @X @X @ԟt>@ԟt>RH֘[! D椢Ǡ*FfeԟǠ?ҍT1 #a6cmex8e'cffommutes.Proof.@_The͈morphisms'(X)UR:!n9(X)4!1!n9(X)o* D!de ne͈anaturaltransforma-tionsinceallmorphismsofthediagramaremorphismsofcomoSdules..Sotheexistencec7 @z5.pTHE!CO9ALGEBRAcoAend99Yandtheuniquenessofamorphismfe':URcoSend!(!n9)UR4!1Dgeisclear.ETheonlythingtoshorwisthatthisisamorphismofcoalgebras.sThisfollorwsfromtheuniversalpropSertyofC1=URcoSend!(!n9)andthediagram_2R!_2M! CzD[˲fdCgά-YR퍒&`[ujĠm@jĠw@jğ@X@XR[uf1 \Ġm@\Ġw@\ğ@X@XR[2Πdfe2dΠ?㩍}?1[2Πdfe$dΠ?[A n1 e'+T! C+Np! CF CmIJfd+s ά-8@iL 12Ǡdfe+dǠ?@ҍn1 e'2ǠdfedǠ?e21 e' e'ԠR!Ԡ! DzD:2fdBά-`@W''uG 'jğ?`@jğ?`@jğ?`@@Rԏg,1 \ğ?`@\ğ?`@\ğ?`@@R! D~! D6 D32fd*(pά-o͍շ' 1(ywheretherighrtsideofthecubSecommutesbytheuniversalpropSertyV.-eSimilarlywegetthati7e'ƹpreservresthecounitsincethefollowingdiagramcommutesjY+!+Ł9! C$IJfd - ά-K퍒Y!Y ! D${fdά-i'Hn`*FfeD`? 1H%R`*FfeX`?@ҍ?1 e'HܒǠ*FfeğǠ?8ID1H]1 "ݐ䟜@䟦@䟰@䟺@d@dRH͝1 "ݐׁ @ @ @ @d>@d>R⍒0#! ( msbm10K HHHHHHHHFHFj`Y 1H"0#! K8Iɔ1ҁ Hׁ H܁ H H H H H HdžHdžje %cffxff ̟ff ̎ ̄cff\;7  ,- cmcsc10+@ cmti12( msbm10#a6cmex8"u cmex10 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line10