; TeX output 1999.11.03:075717 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr9492*7 &e501:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYN cmbx124.FormalGroupsConsidernorw( msbm10K-cCoalg.jJthecategoryofcoScommutativeK-coalgebras.3Letg cmmi12C5;D!", cmsy102K-cCoalg+z.Then74Cd D¹isagainacoScommrutative74K-coalgebrabryProblemA.11.4.Infactthisholdsalsofornon-commrutativeK-algebras,butinK-cCoalg/ePwrehaveNSProp`osition2.4.1.O+@ cmti12The6tensorprffoduct6inK-35cCoalg.isthe(cffategorical)6product.,- cmcsc10Proof.@_Leta!ff2K-cCoalg+z(Z5;Cܞ);g22K-cCoalg+z(Z5;DS).JThea!map(f;gn9):Z4!CW! zDode nedbry(f;gn9)(z)\:="u cmex10PfG(z|{Ycmr8(1) \|) gn9(z(2))istheuniquehomomorphismofcoalgebrasRsucrhthat(13G "2cmmi8D)(f;gn9)(z)=fG(z)Rand("C R 3G1)(f;gn9)(z)=gn9(z)Rorsucrhthatthediagram6LtCǠCF DÔ32fdoЍ0O line10!7pX.;cmmi6CD$32fdά-!7pX.Dꃀfi?`i?`i?`᤟᤟ uGfgڤ?`@ڤ?`@ڤ?`@b@bR .ZoǠ@feآDǠ?`R0T(fh;gI{)commrutes,where;pC(c d)UR=(1 ")(c d)UR=c"(d)andpD(c d)UR=(" 1)(c d)UR="(c)d꨹arehomomorphismsofcoalgebras.ڣcffxff ̟ff ̎ ̄cffSothecategoryK-cCoalg/ePhas niteproSductsandalsoa nalobjectK.NSDe nition2.4.2.vaùAqformalTgrffoup۹isagroupinthecategoryofK-cCoalg0ofcoScommrutativecoalgebras.Aformal.groupGde nesaconrtravXariant.representablefunctorfromK-cCoalgtoGr ,thecategoryofgroups.Prop`osition2.4.3.OLffetH6Z2IK-35cCoalg+5.5H$arepresentsaformalgroupifandonly35iftherffearegivenmorphismsinK-cCoalgeahrUR:H HB!gHF:;u:K!H;S):HB!gHsuch35thatthefollowingdiagrffamscommute?9Q*H H?IḦ́32fd?ά-  K cmsy8rEǠ@feEBǠ??;IrZe胀(assoffciativity)Ǡ@fePǠ?ԏЫr 1Ԡȧ H H HԠ3C*H H |D:2fd&H`ά-/1 rA5H HEiHx32fd٠ά- rԩ"7K HPB԰[=QHPB԰[=H Kԩ"9"JH Hԟ:2fd<ά-mu 1T胀(unit)JǠ@feK!4Ǡ??;OӴrBǠ@fetǠ?ꬽ1 ui 1TܔPT PT?_PT攴PT PT?^PTPTP&T?]P(x P(x q>Hqᔟ胀(inverse))"IH)"cK%ğ:2fd)@ά-Í|" (pH4:2fd.bά-(7 o#Ǡ@feWǠ??;8X*H HƪH HԄ32fdJsά-ׁfySr} ida#id1+ S-bǠ@fe-Ŕ?`6?;2xr37 &e4.pF9ORMAL!GROUPS/51YProof.@_FVorW{anarbitraryformalgroupHDѹwregetr=p1p2Β2K-cCoalg+z(H HF:;HV),SuA=e2K-cCoalg+z(K;HV),Sand>S=(id ʤ)21@2K-cCoalg+z(HF:;HV).4These>maps,theYVonedaLemmaandRemark2.2.6leadtotheproSofoftheproposition.)=@cffxff ̟ff ̎ ̄cffRemark2.4.4.j6In:particulartherepresenrtingobject(HF:;r;u;;";S׹):ofaformalgroup GisacoScommrutative HopfalgebraandevrerysuchHopfalgebrarepresentsaformalgroup.HencethecategoryofformalgroupsisequivXalenrttothecategoryofcoScommrutativeHopfalgebras.Corollary2.4.5.sWA=cffoalgebraiH}2K-35cCoalg0representsaformalgroupifandonly35ifH isacffocommutative35Hopfalgebrffa.The6cffategoryofcocommutativeHopfalgebrasisequivalenttothecategoryofformalgrffoups.Corollary2.4.6.sWThe35followingcffategories35areequivalent:s21. #The35cffategoryofcommutative,cocommutativeHopfalgebras.2. #The35cffategoryofcommutativeformalgroups.3. #The35dualofthecffategory35ofcffommutativeanealgebraicgroups.Example2.4.7.Q1.8GroupalgebrasKGareformalgroups.2. #UnivrersalenvelopingalgebrasU@(-%n eufm10g)ofLiealgebrasgareformalgroups.3. #ThetensoralgebraTƹ(Vp)andthesymmetricalgebraS׹(V)areformalgroups.4. #LetCbSeacocommrutativecoalgebraandGbeagroup. Thenthegroup #KG(Cܞ)=K-cCoalg+z(C5;KG)Yisisomorphictothesetoffamilies(h2RAgjg2G)of #decompSositionsBoftheunitofCܞ2ߜinrtoasumoforthogonalidempotenrtsh2RAg*P2URCܞ2 #thatareloScally nite.2#TVooseethisemrbSedK-cCoalg+z(C5;KG)Home(C;KG)oandemrbSedtheset #Hom8(C5;KG)inrtotheset(Cܞ2)2G ofG-familiesofelementsinthealgebraCܞ2 lby #h7!(h2RAg)YCwithh(c)=PCgI{2G#ۇh2RAg(c)gn9.TheYClinearmaphisahomomorphismofC #coalgebrasi (hβ h)k=hand"hk="i PBh(c(1) \|)β h(c(2))k=PZh(c)(1)+. #h(c)(2)'and="(h(c))UR="(c)forallc2CIi PVh2RAg(c(1) \|)g IQh2yl(c(2))l=URPh2RAg(c)g IQg #andPLh2RAg(c)V="(c)i Ph2RAg(c(1) \|)h2yl(c(2))V=gI{lvh2RAg(c)andPLh2RAg]T=V"i h2RAgh2ylHZ= #gI{lvh2RAgfandhPh2RAgOV=zX1C!q% cmsy6 t.z!FVurthermorehthefamiliesmrustbSelocally nite,i.e.z!for #eacrhcUR2CFonly nitelymanryofthemgivenon-zerovXalues.5. #LetCϴbSeacocommrutativecoalgebraandK[x]betheHopfalgebrawith(x)UR= #x܇ 1+1 x(thesymmetricalgebraoftheonedimensionalvrectorspaceZ΍ #Kx). WVe5emrbSedasbeforeK-cCoalg+z(C5;K[x])-Hom(C;K[x])=(Cܞ2)2f02@cmbx8NqAacmr60*g@, #thesetofloScally niteN0-familiesinCܞ2 bryh(c)=P*p1 U_pi=0!h2RAi(c)x2idڹ.$Themap$ #hisahomomorphismofcoalgebrasi (h(c))k=PBh2RAi(c)(x+ 1+1 x)2iE= #P.3h2RAi(c)G܍i v~ldܟG x2l' x2ill=UR(h h)(c)=Ph2RAi(c(1) \|)h2RAj(c(2))x2ij x2jVandL"(Ph2RAi(c)x2idڹ)UR=" #"(c)i h2RAijh2RAjV=URG܍Ti+j v~i