; TeX output 1999.11.03:07557 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups"N cmbx12Intro`ductionIn2the rstcrhapterwehaveencounteredquantummonoidsandstudiedtheirroleasmonoidsopSeratingonquanrtumspaces.The\elements"ofquantummonoidsopSeratingonquanrtumspacesshouldbeunderstoodasendomorphismsofthequanrtumspaces.9Inetheconstructionofthemrultiplicationforuniversalquantummonoidsofquanrtum;spaceswehaveseenthatthismultiplicationisessentiallythe\compSosition"ofendomorphisms.WVe3are,}horwever,primarily3interestedinautomorphismsandweknowthatauto-morphismsshouldformagroupundercompSosition.EThiscrhapterisdevotedto ndinggroupstructuresonquanrtummonoids,i.e.8tode neandstudyquantumgroups.Thisiseasyinthecommrutativesituation,?Hi.e. iftherepresenrtingalgebraofaquanrtummonoidiscommutative. -Thenwecande neamorphismthatsendselemenrtsrofthequantumgrouptotheirinverses.Thiswillleadustothenotionofanealgebraicgroups.Inthenoncommrutativesituation,4however,itwillturnoutthatsucrhaninversionmorphismX(ofquanrtumspaces)doSesnotexist.OItwillhavetobSereplacedbyamorecomplicatedconstruction.ThrusquantumgroupswillnotbSegroupsinthesenseofcategoryEtheoryV.StillwrewillbSeabletoperformoneofthemostimportanrtandmostbasicconstructionsingrouptheoryV,theformationofthegroupofinrvertibleelementsofcamonoid. pInthecaseofaquanrtummonoidactinguniversallyonaquantumspace& thiswillleadtothegoSod& de nitionofaquanrtumautomorphismgroupofthequanrtumspace.InordertoharvetheappropriatetoSolsforinrtroducingquanrtumgroupswe rstinrtroSduce%HopfalgebraswhichwillbSetherepresentingalgebrasofquantumgroups.FVurthermorewreneedthenotionofamonoidandofagroupinacategory.Wewillsee,horwever,thatޗquantumgroupsareingeneralnotgroupsinthecategoryofquantumspaces.WVe rststudythesimplecasesofanealgebraicgroupsandofformalgroups.TheywillharveHopfalgebrasasrepresenrtingobjectsandwillindeedbSegroupsinreasonableBcategories.Thenwrecometoquantumgroups,W(andconstructquantumautomorphismgroupsofquanrtumspaces.Arttheendofthechapteryoushould#!", cmsy10 #knorwwhataHopfalgebrais,/^*o cmr931 *7 &e321:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSY# #knorwwhatagroupinacategoryis,# #knorwthede nitionandexamplesofanealgebraicgroupsandformalgroups,# #knorw>thede nitionandexamplesofquantumgroupsandbSeabletoconstruct #quanrtumautomorphismgroupsforsmallquantumspaces,# #understandEwhryaHopfalgebraisareasonablerepresentingalgebraforaquan- #tumgroup.! 7 &e1.pHOPF!ALGEBRASGL33Y1.׈HopfAlgebrasThe=di erencebSetrween=amonoidandagroupliesintheexistenceofanadditionalmapg cmmi12SX:;G3gt7!gn92K cmsy8|{Ycmr81o2GforagroupGthatallorwsforminginverses.Thismapsatis estheequationS׹(gn9)gË=UR1orinadiagrammaticform?'E^G:f1g:2fd&*PO line10-Í2cmmi8" ?G❔:2fd&*Pά-Y1@Ǡ@fesǠ??;ٴGGH4GGd32fdME ά-ׁ̃Sr}idǠ@fe4?`6JmÎulteWVewranttocarrythispropSertyovertoade nitionofquantumgroups. OWVeknowalreadythatquanrtummonoidsGarerepresentedbybialgebrasHV.Soan\inversemap"OshouldbSeamorphismS:G4!GOwithacertainpropertryV,hifGistobecomeaRquanrtumgroup,D|oranalgebrahomomorphismS:,|)"QH)"$' msbm10K̟:2fd)@ά-Íׄ" 1HU<:2fd.bά-(70Ǡ@feǠ??;P@2H HH H32fdJsά-ׁ'Sr} idSjǠ@fe?`6?;9reSymmetrically{wrede nearightHopfalgebrffa{HV.A{Hopfalgebra{isaleftandrighrtHopfalgebra.8ThemapSiscalleda(left,righrt,two-sided)antipffode.UsingtheSwreedlernotation(A.6.3)thecommutativediagramabSovecanalsobSeexpressedbrytheequation򍍍!u cmex10XS׹(a(1) \|)a(2)ι=URn9"(a)b%for7alla 2HV. Observre7thatwedonotrequirethatS߹: H^4!Hoisanalgebrahomomorphism.Problem2.1.1.nR1.LetH8bSeabialgebraandSp2HomZ"(HF:;HV).ThenSisananrtipSodefforH(andHisaHopfalgebra)i S[=isatrwofsidedinrversefforidinthealgebra(Homy(HF:;HV);;n9")(seeA.6.4).8InparticularSisuniquelydetermined.2.LetXHbSeaHopfalgebra.ThenSb/isananrtihomomorphismofalgebrasandcoalgebrasi.e.8S\inrvertstheorderofthemultiplicationandthecomultiplication".3.]LetHandK~,bSeHopfalgebrasandletfԢ:Hy4! )KbSeahomomorphismofbialgebras.8ThenfGSH n=URSK;f,i.e.8f2iscompatiblewiththeanrtipSode.De nition2.1.2.vaùBecauseofProblem2.1.13.evreryhomomorphismofbialge-brasbSetrweenHopfalgebrasiscompatiblewiththeantipSodes.jSowrede neahomo-morphism/ofHopfalgebrffastobSeahomomorphismofbialgebras.nThecategoryofHopfalgebraswillbSedenotedbryK-Hopf ."g7 &e341:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYProp`osition2.1.3.OLffetHlbeabialgebrawithanalgebrageneratingsetX.KnLetS:7!H$w!HV2op}bffe&analgebrahomomorphismsuchthatPWS׹(x(1) \|)x(2)=7!n9"(x)forallxUR2X.fiThen35S isaleftantipffode35ofHV.X+- cmcsc10Proof.@_Assume@>RHiU@(A): Ǡ*Ffe=LǠ? gThegroupalgebraKGis(ifitexists)uniqueuptoisomorphism.ItisgeneratedaslanalgebrabrytheimageofG.wThemap2:G4!iU@(KG)KGlڹisinjectiveandtheimageofGinKGisabasis.#7 &e1.pHOPF!ALGEBRASGL35YTheDAgroupalgebracanbSeconstructedasthefreevrectorspaceKGwithbasisGandxthealgebrastructureofKGisgivrenbyKG KG3g+# h7!gn9h2KGxandtheunitË:URK3 h7! e2KG.ThegroupalgebraKGisaHopfalgebra.ThecomrultiplicationisgivenbythediagramCj⍒0Gj⍒=RKGd{fd% ά-TDH`f tׁ @ t @ t @ t @Q>@Q>RH"њKG KGW"Ǡ*FfeTǠ?`<withfG(gn9):=g 6g&whicrhde nesagrouphomomorphismf:G4!U@(KG6 KG).Thecounitisgivrenbyj⍒x&Gj⍒ꄺKG4̟{fd% ά-қH`7fRܟׁ @Rܟ @Rܟ @Rܟ @\>@\>RH""'K󞊟Ǡ*FfeѼǠ?]C<"bwherefG(gn9)UR=1forallgË2G.4MOneshorwseasilybyusingtheuniversalpropSertyV,߯thatiscoassoSciativreandhascounit".De neanalgebrahomomorphismSZ:KG4!(KG)2op ŹbryDZ@j⍒Gj⍒ZKGl{fd% ά-%LH`f|ׁ @| @| @| @">@">RH (KG)2op(*Ǡ*Ffe[\Ǡ?` Swith.fG(gn9)P:=g21uwhicrh.isagrouphomomorphismf:PG4!U@((KG)2op). zqThenPropSosition1.3shorwsthatKGisaHopfalgebra.$The precedingexampleofaHopfalgebragivresrisetothede nitionofparticularelemenrts~inarbitraryHopfalgebras,thatsharecertainpropSertieswithelementsofagroup.8WVewilluseandstudytheseelemenrtslateroninchapter5.De nition2.1.6.vaùLetHebSeaHopfalgebra.)XAnelemenrtgË2URHF:;g6=0iscalledagrffoup-like35element꨹ifd?(gn9)UR=g g:Observrexthat"(gn9)UR=1xforeachgroup-likeelementgSinaHopfalgebraHV.7%InfactwreohavegË=URr(" 1)(gn9)="(g)gË6=0ohence"(gn9)=1.4xIfthebaseringisnota eldthenoneaddsthispropSertrytothede nitionofagroup-likeelement.Problem2.1.2.nR1.oWLetKbSea eld.Shorwthatanelementx'2KGйsatis es(x)UR=x x꨹and"(x)UR=1ifandonlyifxUR=gË2G.2. QShorwxthatthegroup-likeelementsofaHopfalgebraformagroupundermrultiplicationoftheHopfalgebra.$07 &e361:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYExample2.1.7.oQ(Universal$EnvelopingAlgebras)0hA0VLiesRalgebrffaconsistsofavrectorspace-%n eufm10gtogetherwitha(linear)multiplicationg go3x y7!o[x;yn9]2gsucrhthatthefollowinglawshold:'8RʍKY[x;x]UR=0;KY[Ox;[yn9;z]]xѫ+[yn9;[z;x]]+[z;[x;yn9]]UR=0(Jacobiidenrtity).'AZhomomorphism=ofLiealgebrffasZf\:g4!AhisalinearmapfsucrhthatfG([x;yn9])=[fG(x);f(yn9)].8ThrusLiealgebrasformacategoryK-Lie.An+impSortanrtexampleistheLiealgebraassociatedwithanassociativrealgebra(withunit).8IfAisanalgebrathenthevrectorspaceAwiththeLiemultiplication ʒ[x;yn9]UR:=xyyx(1)"hyisy/aLiealgebradenotedbryA2LGع. ThisconstructionofaLiealgebrade nesacovXariantfunctor-P2Lrz:URK-Alg4!-`K-Lie.8Thisfunctorleadstothefollorwinguniversalproblem.LetugbSeaLiealgebra.ٰAnalgebraU@(g)togetherwithaLiealgebrahomomor-phism\E:g4!FU@(g)2L ܹiscalleda(the)universalgenvelopingalgebrffaofg,x[ifforevreryalgebra{AAandforevreryLiealgebrahomomorphismfQ:URg4!1A2L there{AexistsauniquehomomorphismȽofalgebrasgË:URU@(g)4!1AȽsucrhthatthefollowingdiagramcommutesXȱ㍒Cgȱ㍒pU@(g)2L{fd!wЍά- X7H`Jfԟׁ @ǁԟ @сԟ @ہԟ @T>@T>RH#ӎA2LG:͂Ǡ*FfeǠ? 4g!TheGunivrersalenvelopingalgebraU@(g)is(ifitexists)uniqueuptoisomorphism.Itisgeneratedasanalgebrabrytheimageofg.The7univrersalenvelopingalgebracanbSeconstructedasU@(g)UR=Tƹ(g)=(x݁ yKy xi[x;yn9])ywhereTƹ(g)UR=Kigg g:::isythetensoralgebra.{ThemapUR:g4!1U@(g)2Lisinjectivre.TheunivrersalenvelopingalgebraU@(g)isaHopfalgebra.8ThecomultiplicationisgivrenbythediagramR]H egHےU@(g){fd$ά-WH`-fItׁ @It @It @It @Տ>@Տ>RHRU@(g) U(g)"Ǡ*FfeTǠ?`z%=7 &e1.pHOPF!ALGEBRASGL37Ywith*fG(x)v}:=x 1+1 xwhicrhde nesaLiealgebrahomomorphismf|:v}g4!(U@(g) U(g))2LGع.8ThecounitisgivrenbyLj{HٝgHa4U@(g)T{fd$ά-%LH`fׁ @ @ @ @^,>@^,>RH""KcZǠ*Ffe񖌟Ǡ?]CI ";withMwfG(x)=0forallx2g.aMOneshorwseasilybyusingtheuniversalpropSertyV,f+that_TiscoassoSciativreandhascounit".De neanalgebrahomomorphismS:U@(g)4!(U@(g))2op ŹbryGHT5gHU@(g)T{fd$ά-ɟH`vfDׁ @ĒD @ΒD @ؒD @ğ>@ğ>RHz(U@(g))2opǠ*Ffe$Ǡ?`äSMwithbfG(x)ɞ:=xwhicrhisaLiealgebrahomomorphismf:g4!ɹ(U@(g)2op)2LGع. ThenPropSosition1.3shorwsthatU@(g)isaHopfalgebra.(Observre,[that79themeaningofUxinthisexampleandthepreviousexample(groupofN units,funivrersalenvelopingalgebra)istotallydi erent,finthe rstcaseUcanbSeappliedtoanalgebraandgivresagroup,inthesecondcaseU׹canbSeappliedtoaLiealgebraandgivresanalgebra.)9The precedingexampleofaHopfalgebragivresrisetothede nitionofparticularelemenrts~inarbitraryHopfalgebras,thatsharecertainpropSertieswithelementsofaLiealgebra.8WVewillusetheseelemenrtslateroninchapter5.De nition2.1.8.vaùLetqHǹbSeaHopfalgebra. v2=H 5bSeagroup-likreelement.Anelementx=2H 5is2calledaskewuprimitiveorgn9-primitive35element꨹if(x)UR=x 1+g x:9Problem2.1.3.nRShorwUthatthesetofprimitiveelementsPƹ(HV)UR=fx2Hj(x)=x 1+1 xg꨹ofaHopfalgebraHisaLiesubalgebraofHV2L5..Prop`osition2.1.9.OLffet]HbeaHopfalgebrawithantipodeS.\Thefollowingareeffquivalent:1.fiSן22-=id.2.fiPS׹(a(2) \|)a(1)ι=URn9"(a)35foralla2HV.3.fiPa(2) \|S׹(a(1))UR=n9"(a)35forallaUR2HV.&I7 &e381:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYProof.@_LetSן22-=id.8TheneʍF/PTS׹(a(2) \|)a(1)㷹=URSן22r۹(PS׹(a(2) \|)a(1))=S׹(PS(a(1) \|)S22r۹(a(2)))=URS׹(PS(a(1) \|)a(2))=S׹(n9"(a))="(a);bryusingProblem2.1.1.Conrverselyassumethat2.8holds.Thenʍ_oKS]Sן22r۹(a)a=URPS׹(a(1) \|S22r۹(a(2))UR=S׹(PS(a(2))a(1)a=URS׹(n9"(a))="(a):iThrusSן22){andid7areinversesofSiwintheconvolutionalgebraHom0)(HF:;HV),;henceSן22-=URid .Analogouslyoneshorwsthat1.8and3.areequivXalenrt.l@cffxff ̟ff ̎ ̄cff8ICorollary2.1.10.zIfHisacffommutativeHopfalgebraoracocommutativeHopfalgebrffa35withantipodeS,thenS22-=id.Remark2.1.11.qN6Kaplansky:TenconjecturesonHopfalgebrasIn2asetoflecturenotesonbialgebrasbasedonacoursegivrenatChicagouniversityin,[1973,|madepublicin1975,KaplanskyformrulatedasetofconjecturesonHopfalgebrasthatharvebSeentheaimofinrtensiveresearch.s21. #IflpCIisaHopfsubalgebraoftheHopfalgebraBvthenBisafreeleftCܞ-moSdule.2#(YVes,Sif=Hœis nitedimensional[Nicrhols-ZoSeller];Noforin nitedimen- #sionalrHopfalgebras[ObSerst-Scrhneider];BX:URCgisnotnecessarilyfaithfully at #[Scrhauenburg])2. #CallacoalgebraCadmissibleifitadmitsanalgebrastructuremakingitaHopf #algebra.FThecuconjecturestatesthatC@isadmissibleifandonlyifevrery nite #subsetofCFliesina nite-dimensionaladmissiblesubScoalgebra.2#(Remarks.)C(a)=GBothimplicationsseemhard.(p(b)=GThereisacorrespSondingconjecturewhere\Hopfalgebra"isreplacedbry=G\bialgebra".)(c)=GThereisadualconjectureforloScally nitealgebras.)2#(Noresultsknorwn.)3. #AHopfalgebraofcrharacteristic0hasnonon-zerocentralnilpSotentelements.2#(FirstUcounrterexamplegivenby[Schmidt-Samoa].@IfHګisunimoSdularand #not3semisimple,Fe.g.aDrinfel'ddoubleofanotsemisimple nitedimensional #Hopfealgebra,thentheinrtegralsatis es=6=0,22FA="()=0esinceDS(HV) #isnotsemisimple,candaj="(a)="(a)=asinceDS(HV)isunimodular #[Sommerh auser].)4. #(Nicrhols).8 Let(xbSeanelementinaHopfalgebraH~withantipSode(S׹.8 Assume #thatforanryainHwehaveW5uXbidxS׹(ci)UR="(a)x #whereaUR=Pbi cidڹ.8Conjecture:xisinthecenrterofHV.2#(axUR=Pa(1) \|x"(a(2))UR=Pa(1) \|xS׹(a(2))a(3))UR=P"(a(1) \|)xa(2)ι=xa:)'U7 &e1.pHOPF!ALGEBRASGL39Y2#In?theremainingsixconjecturesH-isa nite-dimensionalHopfalgebraorver #analgebraicallyclosed eld.5. #IfHCissemisimpleoneitherside(i.e."eitherHorthedualHV2 VGissemisimpleas #analgebra)thesquareoftheanrtipSodeistheidentityV.2#(YVesfifcrhar(K)=0[Larson-Radford],+Uyesifchar(K)islarge[Sommerh au- #ser])6. #The(tsizeofthematricesoSccurringinanryfullmatrixconstituentofHʹdivides #thedimensionofHV.2#(YVesisHopfalgebraisde nedorverZ[Larson];9ingeneralnotknorwn;work #bry[Montgomery-WitherspSoon],[Zhru],[Gelaki])7. #IfHissemisimpleonbSothsidesthecrharacteristicdoesnotdividethedimen- #sion.2#(Larson-Radford)8. #IfthedimensionofHisprimethenHiscommrutativeandcoScommrutative.2#(YVesincrharacteristic0[Zhu:81994])2#Remark.8Kac,Larson,andSwreedlerhavepartialresultson5{8.2#(WVasalsoprorvedby[Kac])2#Inkthetrwok nalconjecturesassumethatthecrharacteristicdoSesnotdivide #thedimensionofHV.9. #ThedimensionoftheradicalisthesameonbSothsides.2#(CounrterexampleXby[Nichols];ccounterexampleinFVrobSenius-Lusztigkernel #ofUq(slC(2))[Scrhneider]) 10. #TheredXareonlya nitenrumbSerdX(uptoisomorphism)ofHopfalgebrasofagivren #dimension.2#(YVesforsemisimple,cosemisimpleHopfalgebras:8Stefan1997)2#(Counrterexamples:8[Andruskiewitsch,Schneider],[Beattie,others]1997)(d7 &e401:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYeF2.{MonoidsandGroupsinaCategoryBeforewreuseHopfalgebrastodescribSequantumgroupsandsomeofthebSetterknorwn#groups,Ȃsuchasanealgebraicgroupsandformalgroups,ȂweintroSducetheconcept/gofageneralgroup(andofamonoid)inanarbitrarycategoryV.UsuallythisconceptJisde nedwithrespSecttoacategoricalproductinthegivrencategoryV.ButinʊsomecategoriesthereareingeneralnoproSducts..+Still,onecande netheconceptofagroupinavrerysimplefashion.&HWVewillstartwiththatde nitionandthenshowthatitcoincideswiththeusualnotionofagroupinacategoryincasethatcategoryhas niteproSducts.De nition2.2.1.vaùLet5CbSeanarbitrarycategoryV. LetG2CbSe5anobject. WVeuse)thenotationG(X)t:=Mor~C =(XJg;G))forallXf2tC5,yG(fG):=Mor~C(f;G))forallmorphismsfQ:URXF4!Y8finC5,andfG(X):=MorOC(XJg;f)forallmorphismsfQ:G4!1G209.G!͹togetherwithanaturaltransformationmf:G(-)~G(-)f4!gG(-)!iscalledagrffoup(monoid)inthecategoryC5,\ifEfthesetsG(X)togetherwiththemrultiplicationm(X)UR:G(X)G(X)UR4!1G(X)aregroups(monoids)forallXF2URC5.Let"(G;m)and(G209;m20)"bSegroupsinC5.XOA morphismf':g(G4!G20[in"CWiscalledahomomorphism35ofgrffoups꨹inC5,ifthediagramsMHyHQG(X)G(X)H&*G(X)ޮ|{fd$ 0ά-`ƪm(X)G209(X)G20(X)G209(X)32fdά-"&m-: q% cmsy60(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)8鍹commruteforallXF2URC5.Let?(G;m)and(G209;m20)?bSemonoidsinC5.A9morphismf:G4!OG20xin?Ctiscalledahomomorphism35ofmonoidsinC5,ifthediagramsHQG(X)G(X)H&*G(X)ޮ|{fd$ 0ά-`ƪm(X)G209(X)G20(X)G209(X)32fdά-"&m-:0(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)8鍹andApHfg]CW~uׁ̟ ̟ ̟ ̟ ی>ی> Hnu-:07,ׁ A7, A7, A7, AZl>AZl>U2JG(X)nG209(X)̶32fd_`ά- f(X)ŶcommruteforallXF2URC5.Problem2.2.4.nR1)IfasetZf6togetherwithamrultiplicationmUR:ZfZ14!Zisamonoid,]then9theunitelemenrteUR2ZIis9uniquelydetermined.Ifitisagroupthenalso)l7 &eph2. %MONOIDS!ANDGR9OUPSINACA:TEGORYc41YtheVinrversei :Z?4!mZ3isuniquelydetermined.}Unitelemenrtandinversesofgroupsarepreservredbymapsthatarecompatiblewiththemultiplication.2):FindanexampleofmonoidsYandZHandamapf%:݁Yy4!pZwithfG(y1y2)݁=fG(y1)f(y2)forally1;y2V2URYp,butfG(eYP)6=eZ8.3)bIf(G;m)isagroupinCbandiX :-G(X)4!aG(X)bistheinrverse,thenbiisanatural6itransformation.#TheYVonedaLemmaprorvidesamorphismS:CG4!G6isuchthatiX r۹=URMorOC(XJg;S׹)UR=S(X)forallXF2URC5.Prop`osition2.2.2.OLffetC6beacategorywith nite(categorical)products. WA2nobjeffctoGinCcarriesthestructuremq:G(-35)(G(-)q!HG(-)oofagrffoupinCifandonly=iftherffearemorphismsmgw:GGgw!5[G,?uugw:E!rG,and=SN:G!5[Gsuchthatthe35diagrffamsU)YCGGGYjGGl{fdQά-)-m1O\GGbGt 32fd*Fά-ÍmH`\*Ǡ*Ffe`\Ǡ?͝I{1mHѪǠ*FfeܟǠ?]C\mY|E^GPUR԰n:=GPUR԰n:=GEYLoGG>cl{fd ά-)-;1u"GGXgfG̞32fd_ά-Í!mHǠ*FfeǠ?͝ABu1H\֊Ǡ*Ffe] Ǡ?]Ca,Ǡ?1mi?-H:Ǡ*FfelǠ?(mi?-ecommrutesifandonlyifMorѨC?g(-;m(m R1))UR=MorOC(-;m)(Mor5C(-;m) R1)UR=m̹-j(m̹-s1)_R=m̹-j(1m̹-)_R=MorOC(-;m)(1MorK Cʹ(-;m))_R=MorOC(-;m(1m))ifandonlyif*|"7 &e421:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYm(m1)UR=m(1m)ifandonlyifthediagramMY#GGGYmLGG۠ܟ{fdQά-)-ۙm1GGdGϦ|32fd*Fά-ÍZmH^Ǡ*Ffe̟Ǡ?͝~ 1mHǠ*FfeLǠ?]Cm^ߍcommrutes.8InasimilarwayoneshowstheequivXalenceoftheotherdiagram(s).cffxff ̟ff ̎ ̄cffProblem2.2.5.nRLetCbSeacategorywith niteproducts.HShorwthatamorphismfQ:URG4!1G20inCݹisahomomorphismofgroupsifandonlyifJzYGGYIGI|{fd@ά-:mK̔G20xG20KG2032fdNά-.&m-:0HǠ*Ffe4̟Ǡ?`ffHڟǠ*Ffe Ǡ?`focommrutes.De nition2.2.3.vaùA.Ocffogroup.`(comonoid)GinCᕹisagroup(monoid)inC52op R,?Ni.e.anaGobjectGUR2ObC=ObC52op togetheraGwithanaturaltransformationm(X)UR:G(X)G(X)UR4!1G(X)whereG(X)UR=MorOCmr;cmmi6op&(XJg;G)=MorOC(G;X),sucrhthat(G(X);m(X))isagroup(monoid)foreacrhXF2URC5.Remark2.2.4.j6Let C7@bSeacategorywith nite(categorical)coproducts.  Anobject1GinCfcarriesthestructuremӹ:G(-)ɨG(-)4!_3G(-)1ofacogroupinCfifandonly%*iftherearemorphisms:G4!]G~qG,3":G4!I,3and%*Sk:G4!G%*sucrhthatthediagramsL#7GqGq9GqGqG[d32fdά- [q1bYBGbY}8GqGO\{fd*ά-ˍa\HZǠ*FfeǠ?` 1qHGXڟǠ*FfeG Ǡ?`;0ٙۄ8GqGٙI+qGPUR԰n:=GPUR԰n:=GqI32fdqά- "q1bYQ6GbYVGqG ܟ{fd_Zά-ˍ 0Hg Ǡ*FfegA,Ǡ?`k1q"HZǠ*FfeǠ?`+H8Ib1H+ʬQHtQ HQH+Q HrQ*HQ-ܟQ-ܟsNa bY4GbYտFI{fd K0ά-bYbY&Gbܟ{fd K0ά-"gGqGYGqG32fd6@ά- бw1qSkбwSr}q1H*Ǡ*Ffe\Ǡ?`H*Ǡ*Ffe\ׁ 6` zrlcommruteo^whererisdualtothemorphismde nedinA.2.ThemultiplicationsarerelatedbryX r۹=URMorOC(;X)UR=(X).Let"CoWbSeacategorywith nitecoproductsandletGandG20[becogroupsinC5.ThenElahomomorphismofgroupsf7й:G20 4!hGisamorphismf:G4!/G20inElCsucrh+-7 &eph2. %MONOIDS!ANDGR9OUPSINACA:TEGORYc43YthatthediagramE̴YVGYGG{fd@ά-ˍɆnK:G20KG20xG20532fdNά-.y*-:0H6Ǡ*FfeiǠ?`lffHzǠ*FfeǠ?` fcommrutes.8Ananalogousresultholdsforcomonoids.Remark2.2.5.j6Obrviously/similarobservXationsandstatementscanbSemadeforotheralgebraicstructuresinacategoryC5.CSoonecaninrtroSducevectorspacesandcorvectorO spaces,&monoidsandcomonoids,ringsandcoringsinacategoryC5. fThestructures(b)UR=1 b꨹arealgebrahomomorphisms.Scffxff ̟ff ̎ ̄cff(ÍSothecategoryK-cAlg"has nitecoproSductsandalsoaninitialobjectK.A'more( generalpropSertryofthetensorproductofarbitraryalgebraswrasalreadyconsideredin1.2.13.Observrethatthefollowingdiagramcommutes>%ԠnAԠȫ A A:2fd~`ά-͝;8qAacmr61ԠԠ`Ak̟:2fd~`쁠ά͝;xq2kS1X.A?`@?`@?`@Ɩ@ƖRkS~d1X.A?`?`?` gA؜*Ǡ@fe\Ǡ?bp݁rwhereristhemrultiplicationofthealgebraandbythediagramthecoSdiagonalofthecoproSduct.De nition2.3.2.vaùAn/anealgebrffaicgroup/isagroupinthecategoryofcom-mrutativegeometricspaces.ByrthedualitrybSetweenthecategoriesofcommutativegeometricspacesandcom-mrutativealgebras, ananealgebraicgroupisrepresenrtedbyacogroupinthecate-goryofK-cAlg"ofcommrutativealgebras.FVoranarbitraryanealgebraicgroupHwregetbyCorollary2.2.7MUR=1j2V2K-cAlgo(HF:;H HV);E3:"UR=e2K-cAlgo(HF:;K);jand4CjS)=(id ʤ) 12K-cAlg(HF:;HV):|ThesemapsandCorollary2.2.7leadtoProp`osition2.3.3.OLffetxH>2QK-35cAlg.3HisarepresentingobjectforafunctorK-35cAlg":N!3 GrHIBif35andonlyifH isaHopfalgebrffa.-ޠ7 &egV3. %AFFINE!ALGEBRAICGR9OUPS45YProof.@_BothWstatemenrtsareequivXalenttotheexistenceofmorphismsinK-cAlggkUR:HB4!H H N":HB4!K S):HB4!Hsucrhthatthefollowingdiagramscommute?ԠۣHԠ,H Hd:2fd?ά-ζ łǠ@feǠ??;0T 胀(coassoSciativitry)>Ǡ@fe>괟Ǡ?ԏC4 1H H AH H Hv$32fd&H`ά-/h1 AR}ԠLHԠ7ȒH Hpl:2fd٠ά-ζ Ҏ"H H"bK HPB԰[=QHPB԰[=H K,32fd<ά-mŬv" 1S,<胀(counit)nǠ@feǠ??;IJǠ@feI|Ǡ?ꬽNy1 "i1JܟܔPJܟ PJܟ?_PJܟ攴PJܟ PJܟ?^PJܟPJܟPJܟ?]P㜟 P㜟 q>g lt胀(coinrverse))"iH)"BK:2fd)@ά-Í40 .OHsT:2fd.bά-(7NǠ@fe64Ǡ??;nX7JH H!H Hӳ32fdJsά-ׁESr} idaCidK S3qǠ@fe3?`6?;8W4r!y %cffxff ̟ff ̎ ̄cffThis#PropSositionsarystwothings.FirstofalleachcommutativeHopfalgebraHde nes=afunctorK-cAlgo(HF:;-):K-cAlg#4r4!5SetMw"that=factorsthroughthecategoryofjgroupsorsimplyafunctorK-cAlgo(HF:;-)U:K-cAlg"]4!5aGrEع.&Secondlyjeacrhrepre-senrtablefunctorK-cAlgo(HF:;-)d:K-cAlg"4!3\SetJ[thatfactorsthroughthecategoryofgroupsisrepresenrtedbyacommutativeHopfalgebra.Corollary2.3.4.sWA2nalgebrffaHB2URK-35cAlg"representsananealgebraicgroupifand35onlyifH isacffommutativeHopfalgebra.ThedRcffategoryofcommutativeHopfalgebrasisdualtothecategoryofanealge-brffaic35groups.InBthefollorwinglemmasweconsiderfunctorsrepresentedbycommutativealge-bras. They\Xde nefunctorsonthecategoryK-cAlg#ǹaswrellasmoregenerallyonK-Algo.)WVes rststudythefunctorsandtherepresenrtingalgebras.ThenwreusethemtoconstructcommrutativeHopfalgebras.Lemma2.3.5.g5QTheGfunctorGa X:T1K-35Alg9-!1AbHde neffdbyGaϹ(A):=A2+x,theunderlying~additivegrffoupofthealgebraA,sisarepresentablefunctorrepresentedbythe35algebrffaK[x]thepolynomialringinonevariablex.Proof.@_Ga ݹisanunderlyingfunctorthatforgetsthemrultiplicativestructureofthealgebraandonlypreservrestheadditivegroupofthealgebra.ʂWVehavetodeterminenaturaluisomorphisms(naturalinA)GaϹ(A)PUR԰n:=K-Algo(K[x];A).EacrhelementaUR2A2+isPmappSedtothehomomorphismofalgebrasa]:K[x]3p(x)7!p(a)2A.ThisPisahomomorphismۂofalgebrassincea(p(x)+qn9(x))UR=p(a)+q(a)UR=a(p(x))+a(qn9(x)).97 &e461:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYand9a(p(x)qn9(x))R3=p(a)q(a)=a(p(x))a(q(x)).Another9reasontoseethisisthatK[x]$isthefree(commrutative)$K-algebraorver$fxgi.e.esinceeacrhmapfxgl4!AcanuTbSeuniquelyextendedtoahomomorphismofalgebrasK[x]4!A. ThemapAT3a7!a eX2K-Algo(K[x];A)isbijectivrewiththeinversemapK-Algo(K[x];A)T3fQ7!URfG(x)2A.8FinallythismapisnaturalinAsinceL@MB̌K-Algo(K[x];B)J\32fd5Ƞά-|5-VHAH{K-Algo(K[x];A),{fd6}ά-5-VHʟǠ*FfeǠ?' ~gH Ǡ*Ffe<Ǡ?`tK-Alg(K[*x]\t;gI{)`commrutesforallgË2URK-Algo(A;B).Ʉcffxff ̟ff ̎ ̄cff덍Remark2.3.6.j6SinceAA2+ ^3hasthestructureofanadditivregroupthesetsofho-momorphismsofalgebrasK-Algo(K[x];A)arealsoadditivregroups.Lemma2.3.7.g5QThefunctorGm Z=URU6:K-35Alg:N!- GrAde neffdbyGmĹ(A)UR:=U@(A),the[\underlyingmultiplicffativegroupofunitsofthealgebraA,isarepresentablefunctorrffepresentedVubythealgebrffaK[x;x21 \|]UR=K[x;yn9]=(xy.W1)VutheringofLffaurentVupolynomialsin35onevariablex.Proof.@_WVe0harvetodeterminenaturalisomorphisms(naturalinA)GmĹ(A)P԰=K-Algo(K[x;x21 \|];A).tEacrhSelementa2GmĹ(A)ismappSedtothehomomorphismofalgebrasa -:=)(K[x;x21 \|]3x7!a2A).Thisde nesauniquehomomorphismofalgebrassinceeacrhhomomorphismofalgebrasffromK[x;x21 \|]_r=K[x;yn9]=(xyT1)towAiscompletelydeterminedbrytheimagesofxandofy butinadditiontheimagesharvetosatisfyfG(x)f(yn9)UR=1,-@i.e.fG(x)mrustbSeinvertibleandfG(yn9)mustbSetheinversetofG(x).8Thismappingisbijectivre.FVurthermoreitisnaturalinAsinceL@AQ}`BAQbK-Algo(K[x;x21 \|];B)Ԟ32fd*(pά-|5-Ȋ~'AȊGK-Algo(K[x;x21 \|];A)l{fd*Ѝά-5-H[BǠ*FfetǠ?'ygH Ǡ*Ffe>Ǡ?4K-v2Algp(K[*x;x-:1 ] N4;gI{)`forallgË2URK-Algo(A;B)commrute.Ccffxff ̟ff ̎ ̄cffRemark2.3.8.j6SincedU@(A)hasthestructureofa(mrultiplicative)dgroupthesetsK-Algo(K[x;x21 \|];A)arealsogroups.Lemma2.3.9.g5QThefunctorMn rg:K-35Alg!0ߗK-35Alg~withMnP(A)thealgebrffaofnn-matricffes-VwithentriesinAisrepresentablebythealgebraKhx11 ;x12;:::ʚ;xnn Рi,the35noncffommutativepolynomialringinthevariablesxijJ.Proof.@_TheMcpSolynomialringKhxijJiisfreeorverMcthesetfxijginthecategoryof(nontpcommrutative)algebras,i.e.7foreachalgebraandforeachmapfֹ:?fxijJg4!;A/7 &egV3. %AFFINE!ALGEBRAICGR9OUPS47YtheresEexistsauniquehomomorphismofalgebrasg:=Khx11 ;x12;:::ʚ;xnn Рi=4!CAsEsucrhthatthediagramD>fxijJg>4KhxijJi?{fdά-lH`}f<ׁ @< @< @< @M>@M>RHARǠ*FfeǠ?'8g commrutes.t SoSeachmatrixinMnP(A)de nesauniqueahomomorphismofalgebrasKhx11 ;x12;:::ʚ;xnn РiUR4!1A꨹andconrverselyV.cffxff ̟ff ̎ ̄cffOVExample2.3.10.uQ1.8Theanealgebraicgroupcalledadditive35grffoupq=wGaY!:URK-cAlg!4!3`Abwith|GaϹ(A)Nf:=A2+ vfromLemma2.3.5isrepresenrtedbytheHopfalgebraK[x].WVedeterminecomrultiplication,counit,andantipSode.BymCorollary2.2.7thecomrultiplicationis=1q2 2K-cAlgo(K[x];K[x] K[x])PUR԰n:=GaϹ(K[x] K[x]).8Henceq=x((x)UR=1(x)+2(x)UR=x 1+1 x:Thecounitis"UR=e(ppmsbm8K 4=02K-cAlgo(K[x];K)P԰n:=GaϹ(K)henceD"(x)UR=0:ùTheanrtipSodeisS)=URidW 1؍ K[x] 2URK-cAlgo(K[x];K[x])P԰n:=GaϹ(K[x])hencelf[S׹(x)UR=x:2.8Theanealgebraicgroupcalledmultiplicffative35groupXGm Z:URK-cAlg!4!3`AbwithXGmĹ(A):=A2 =U@(A)fromLemma2.3.7isrepresenrtedbytheHopfalgebraK[x;x21 \|]UR=K[x;yn9]=(xy1).8WVedeterminecomrultiplication,counit,andantipSode.ByCorollary2.2.7thecomrultiplicationisq=uUR=1j2V2K-cAlgo(K[x;x 1 \|];K[x;x 1] K[x;x 1])PUR԰n:=GmĹ(K[x;x 1] K[x;x 1]):Hence؍(x)UR=1(x)2(x)UR=x x: Thecounitis"UR=eK 4=12K-cAlgo(K[x;x21 \|];K)PUR԰n:=GmĹ(K)henceD"(x)UR=1:ùTheJzanrtipSodeisS_=)idWv1эvK[x;x1 ]6t2)K-cAlgo(K[x;x21 \|];K[x;x21])P)԰=GaϹ(K[x;x21])|hencebS׹(x)UR=x 1 \|: 3.8Theanealgebraicgroupcalledadditive35matrixgrffoupɸM +ڍn qʹ:URK-cAlg!4!3`AbEg;0Ġ7 &e481:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYwithjM2+RAnx(A)theadditivregroupofnn-matricesjwithcoSecientsinAisrepresentedbry2thecommutativealgebraM2@+RAn -=2K[xijJj1i;j}n]2(Lemma2.3.9).qThisalgebramrustbSeaHopfalgebra.ThecomrultiplicationisUR=12V2K-cAlgo(M2@+RAn]\;M2@+RAn M M2@+RAn)PUR԰n:=M2+RAnx(M2@+RAn M M2@+RAn).Hence0e(xijJ)UR=1(xij)+2(xij)UR=xij 1+1 xij:qbThecounitis"UR=eK 4=(0)2K-cAlgo(M2@+RAn]\;K)P԰n:=M2+RAnx(K)hence䕍"(xijJ)UR=0:TheanrtipSodeisS)=URidW 1 ,썑 Mi@"+Ln 2URK-cAlgo(M2@+RAn]\;M2@+RAn)PUR԰n:=M2+RAnx(M2@+RAn)hence+dwS׹(xijJ)UR=xij:4. ThematrixalgebraMnP(A)alsohasanoncommrutativemultiplication,thema-trixRmrultiplication, =de ningamonoidstructureM2RAnx(A).ThusK[xijJ]carriesanothercoalgebraOstructurewhicrhde nesabialgebraM2@RAn V='K[xijJ].ObviouslythereisnoanrtipSode.ThecomrultiplicationisUR=12V2K-cAlgo(M2@RAn]\;M2@RAn M M2@RAn)PUR԰n:=M2RAnx(M2@RAn M M2@RAn).Hence((xijJ))UR=1((xij))2((xij))UR=(xij) (xij)or8(xikl)UR=X ㇍ jxij xjvk :!Thecounitis"UR=eK 4=E i2K-cAlgo(M2@RAn]\;K)P԰n:=M2RAnx(K)hence䕍?"(xijJ)UR=ij:5.RLet#KbSea eldofcrharacteristicp.ThealgebraH>=K[x]=(x2p])carriesthestructureofaHopfalgebrawith(x)X=xk 1+1 x,("(x)X=0,andS׹(x)=x.TVohshorwthatiswellde nedwehavetoshow(x)2p=,)0.Butthisisobviousbythe rulesforcomputingp-thpSorwers incrharacteristicp.WVehave(xj 1+1 x)2pX=x2pr 1+1 x2p=UR0.ThrusthealgebraHrepresentsananealgebraicgroup:䕍k( p](A)UR:=K-cAlgo(HF:;A)P԰n:=fa2Aja p=0g:TheKGgroupmrultiplicationistheadditionofp-nilpSotentelements. ZSowehavethegrffoup35ofp-nilpotentelements.6.The¥algebraHL=K[x]=(x2n=1)isaHopfalgebrawiththecomrultiplication(x)=x߸ x,Lthe8counit"(x)=1,and8theanrtipSode8S׹(x)=x2n1̹."Thesemapsarewrellde nedsincewehaveforexample(x)2n=UR(x+ x)2n=URx2n& x2n=UR1 1.$.ObservrethatthisHopfalgebraisisomorphictothegroupalgebraKCn 5SofthecyclicgroupCnofordern.ThrusthealgebraHrepresentsananealgebraicgroup:䕍j{nP(A)UR:=K-cAlgo(HF:;A)P԰n:=fa2Aja n=1g;1Ӡ7 &egV3. %AFFINE!ALGEBRAICGR9OUPS49Ythat)FisthegrffoupXofn-throotsofunity.The)FgroupmrultiplicationistheordinarymrultiplicationofroSotsofunityV.7.ThelineargroupsormatrixgroupsGLj(n)(A),@-SLڹ(n)(A)andothersucrhgroupsarePfurtherexamplesofanealgebraicgroups.kWVewilldiscusstheminthesectiononquanrtumgroups.Problem2.3.7.nR1.8Theconstructionofthegenerallineargroupef,GLw!(n)(A)UR=f(aijJ)2MnP(A)j(aij)inrvertible32gde nesananealgebraicgroup.8DescribSetherepresenrtingHopfalgebra.2.`\The|spSeciallineargroupSLM)(n)(A)isananealgebraicgroup.WhatistherepresenrtingHopfalgebra?3.TherealunitcircleS21(R)carrythestructureofagroupbrytheadditionofangles.!IsitpSossibletomakreS21¹withtheanealgebraK[c;s]=(s22 +c221)inrtoananealgebraicgroup?(Hinrt:{wHowcanyrouaddtwopSoints(x1;y1)and(x2;y2)ontheunitcircle,sucrhthatyougettheadditionoftheassoSciatedangles?)2N7 &e501:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSY4.FormalGroupsConsidernorwK-cCoalg.jJthecategoryofcoScommutativeK-coalgebras.3LetC5;D2K-cCoalg+z.Then74Cd D¹isagainacoScommrutative74K-coalgebrabryProblemA.11.4.Infactthisholdsalsofornon-commrutativeK-algebras,butinK-cCoalg/ePwrehaveNSProp`osition2.4.1.OThe6tensorprffoduct6inK-35cCoalg.isthe(cffategorical)6product.Proof.@_Leta!ff2K-cCoalg+z(Z5;Cܞ);g22K-cCoalg+z(Z5;DS).JThea!map(f;gn9):Z4!CW! zDode nedbry(f;gn9)(z)\:=PfG(z(1) \|) gn9(z(2))istheuniquehomomorphismofcoalgebrasRsucrhthat(13G "D)(f;gn9)(z)=fG(z)Rand("C R 3G1)(f;gn9)(z)=gn9(z)Rorsucrhthatthediagram6LtCǠCF DÔ32fdoЍ0ά!7pX.CD$32fdά-!7pX.Dꃀfi?`i?`i?`᤟᤟ uGfgڤ?`@ڤ?`@ڤ?`@b@bR .ZoǠ@feآDǠ?`R0T(fh;gI{)commrutes,where;pC(c d)UR=(1 ")(c d)UR=c"(d)andpD(c d)UR=(" 1)(c d)UR="(c)d꨹arehomomorphismsofcoalgebras.ڣcffxff ̟ff ̎ ̄cffSothecategoryK-cCoalg/ePhas niteproSductsandalsoa nalobjectK.NSDe nition2.4.2.vaùAqformalTgrffoup۹isagroupinthecategoryofK-cCoalg0ofcoScommrutativecoalgebras.Aformal.groupGde nesaconrtravXariant.representablefunctorfromK-cCoalgtoGr ,thecategoryofgroups.Prop`osition2.4.3.OLffetH6Z2IK-35cCoalg+5.5H$arepresentsaformalgroupifandonly35iftherffearegivenmorphismsinK-cCoalgeahrUR:H HB!gHF:;u:K!H;S):HB!gHsuch35thatthefollowingdiagrffamscommute?9Q*H H?IḦ́32fd?ά- rEǠ@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@(g)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 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@cmbx8Nq0*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~i84!5GrJljde nedbrySL(n)(A),Lthe8groupofn߮n-matrices8withcoSecienrtsinthecommutativealgebraA¹andwithdeterminanrt1,YHisrepresentedbythealgebraOUV(SL(n))%=SL(n)=K[xijJ]=(detQ(xij)1)i.e.eb~SLqz+(n)(A)PUR԰n:=K-cAlgo(K[xijJ]=(detQ(xij)1);A):SincebSL(n)(A)bhasagroupstructurebrythemultiplicationofmatrices,1therepresent-ingcommrutativealgebrahasaHopfalgebrastructurewiththediagonal=12hence(xikl)UR=Xxij xjvk ; эthecounit"(xijJ)UR=ij 'andtheanrtipSodeS׹(xij)UR=adjӹ(X)ij 'whereadj(X)istheadjoinrtmatrixofXFչ=UR(xijJ).8WVelearvethevreri cationofthesefactstothereader.fvWVenconsiderSLU(n)URMn=A2n-:2 Aasnasubspaceofthen22-dimensionalanespace.6Example2.5.5.oQLetKMq(2)o=Kq qʍWa$dbcc$Xd*Zq2q==IןqI-detZq`qʍi=a20N920Zb20N920jvc20N920~̰d20N920$qz:(2)bHInparticularwrehave(detQq溹)\2=detqB detQqjand"(detQq)=1.DThequanrtumdetermi-nanrtisagrouplikeelement(see2.1.6).Norwwede neanalgebrai-SLq(2)UR:=Mq(2)=(detQqb1):ThealgebraSLq(2)represenrtsthefunctorݍHZSLWq[(2)(A)UR=fqʍXa20\b20 cc20Βd20$͟q0w2Mq(2)(A)jdetQq渟qʍa203CVb20 JIc202Jd20;qG/=1g:aٍThereisasurjectivrehomomorphismofalgebrasMq(2)4!-SLq(2)andSLq(2)isasubfunctorofMq(2).LetICXJg;Y峹bSecommrutingquantummatricessatisfyingdetq/(X)=1=detԟqҹ(Yp).SincedetK[qY(X)detQq渹(Yp) ==detq$}(XYp)forcommrutingquantummatricesweget7?7 &e5.pQUANTUM!GR9OUPS^55YdetQq溹(XYp)=1,henceO{SL(q&(2)O{isaquanrtumsubmonoidofMq(2)andSLq(2)isabialgebrawithdiagonalt"qʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq- qʍ a9 b 9cd!q,Y;tand8K"qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,:~TVoshorwthatSLq(2)hasanantipSodewe rstde neahomomorphismofalgebrasT:URMq(2)4!1Mq(2)2op ŹbryypTğqʍ wa0)b Ucd"q.ɹ:=URqʍMd5qn9b *qn921 ʵc$ Mq(2)>xMq(2)[t]t{fd@ά->>MGq(2)񥔟{fdЍά-H0A`ğׁ @ğ @ğ @ğ @D>@D>RH#rǠ*FfeVǠ?H`Մׁ Մ Մ Մ >> 0withtUt?7!deteq&aqʍ#a207b20$c206d20?.qH<1V:ThrusGLɭq^(2)(A)isasubsetofMq(2)(A).ObservethatfpMq(2)UR4!1GLq(2)isnotsurjectivre.Since mthequanrtumdeterminantpreservesproSductsandtheproductofinrvertibleelemenrtsAisagaininvertiblewegetGL(q(2)isaquantumsubmonoidofMq(2),hencetUR:GLq(2)4!1GLq(2) GLq(2)gwithqʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq+ qʍ \a"b Ncdq+U׹and(t)UR=t t.WVeconstructtheanrtipSodeforGLq(2).8Wede neT:URMq(2)[t]4!1Mq(2)[t]2op Źbryq퍑 {Tğqʍ wa0)b Ucd"q.ɹ:=URtqʍyd49qn9b Vqn921 ʵc; 1aH1şqnqandwTƹ(t):=detq< qʍ#ba3ob#c3 K K`0=(1 )(1 E> +E Kܞ)=(1 )(E). Similarly2wreget(b 1)(Fƹ)UR=(1b )(F).FVorNK[theclaimisobrvious.Thecounitaxiomiseasilycrheckedonthegenerators.Norw&weshowthatSٹisanantipSodeforUq(slC(2)).SFirstde neS):URKhE;F;K5;Kܞ21 9i4!nUq(slC(2))2op Źbrythede nitionofSonthegenerators.8WVehave6I$K4S׹(KܞK21 9)UR=1=S(Kܞ21 9Kܞ);P2S׹(KܞEK21 9)UR=KܞEK21 9K21l=URqn922.=EK21l=URS׹(qn922.=E);US׹(KܞFK21 9)UR=KܞKFK21l=URqn922 ʵKF=S׹(qn922 ʵFƹ);6fS׹(EFLnFE)UR=KܞFEK21EK21 9KF=URKFK21KEKFEFd=ōKܞ21K[z/ ΍7qqn918ϙ=URS՟qō `KFKܞ21 `[z/ ΍7qqn91="qHx:6j SoESde nesahomomorphismofalgebrasS):URUq(slC(2))4!1Uq(slC(2)).SinceESsatis esPS׹(x(1) \|)x(2)H="(x)forallgivrengenerators,uSisaleftantipSodeby2.1.3.Symmet-ricallySѱisarighrtantipSode.wThusthebialgebraUq(slC(2))isaHopfalgebraoraquanrtumgroup.This0quanrtumgroupisofcentralinterestintheoreticalphysics.ItsrepresentationtheoryXiswrellunderstoSod. !IfXqisnotaroSotofunitythenthe nitedimensionalUq(slC(2))-moSdulesaresemisimple.oManrymorepropertiescanbefoundin[Kassel:QuanrtumGroups].; 7 &e{6.pQUANTUM!A9UTOMORPHISMGROUPSoL59Yn'6.gQuantumAutomorphismGroupsLemma2.6.1.g5QThe35cffategoryK-Alg1ofK-algebrashasarbitrarycoproducts.T̍Proof.@_Thisisawrellknownfactfromuniversalalgebra.Infactallequationallyde ned-algebraiccategoriesarecompleteandcoScomplete.6bWVeindicatetheconstruc-tionofthecoproSductofafamily(AidjiUR2I)ofK-algebras.De nepi`i2I$Aiֹ:=8Tƹ(L UXi2I-Aidڹ)=LpiwhereT/denotesthetensoralgebraandwhereRL꨹isthetrwosidedidealinTƹ(L UXi2I-Aidڹ)generatedbrytheset|zJqĹ:=URfjk#(xkyk)(jk#(xk))(jk(yk));1T.:("a6cmex8L |A8:i,r)$O jk(1Ai?k t)jxk;ykx2URAk;ko2URIg:zʍThenonecrheckseasilyforafamilyofalgebrahomomorphisms(fk:YAk4!ABjkv2I)thatthefollorwingdiagramgivestherequireduniversalpropSerty^M`T.Ak`~LAie}ޒfd0ά-(y^ji?k``Tƹ(LUVAidڹ)<ޒfd ά-f#Ǒ``"RTƹ(LUVAidڹ)=L ŌޒfdC0ά-f#S`;D8Bհfi?ks9P}ŽjPP9P̎iPP9P֎hPóPͳ9P׳gP᳌P볌9PfPP 9PeP_,ǑP_,Ǒq`հ8f̟H̟H̟H̟H̟H̟H̟H̟H̟H̟H̟H%ǗH%Ǘj`" 6fǟ-:0 @ @ @ @# @-Ǘ@-ǗR`?Ǡ;fe@LǠ?h-㎍FrDfzʍ %cffxff ̟ff ̎ ̄cffT̍Corollary2.6.2.sWThe35cffategoryofbialgebrashas nitecoproducts.Proof.@_ThexcoproSduct`BiYofbialgebras(BidjiUR2I)xinK-Algisanalgebra.FVorthediagonalandthecounitwreobtainthefollowingcommutativediagramsL>'Bk: Bk^`سBi `BisD32fd `ά-荒pji?k ji?kHBkHZ`Bi{fdC+ά-e˓ji?kHZҟǠ*FfeǠ?|/i?kHLҟǠ*FfeǠ?D29q1*T/H"BkH`cBi|{fdά-e̓nji?kH㨤ú"i?kcLׁ @cL @cL @cL @ީ̟>@ީ̟>RH""7KǠ*Ffe,Ǡ?D9q1*"#̍since`inbSothcases`BilisacoproductinK-Algo. Thenitiseasytoshorwthatthesehomomorphismsde neabialgebrastructureon`Biuandthat`Biusatis estheunivrersalpropSertyforbialgebras.bɄcffxff ̟ff ̎ ̄cffT̍Theorem2.6.3.pLffetBbeabialgebra.PThenthereexistsaHopfalgebraHV(B)andDahomomorphismofbialgebrffasP):B/!HV(B)DsuchthatforeveryHopfalge-brffaH andforeveryhomomorphismofbialgebrasfRϹ: B!H thereisaunique<7 &e601:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYhomomorphism35ofHopfalgebrffasgË:URHV(B)!H such35thatthediagramRHBH⑹HV(B)|{fd i@ά- ޏH`ņ5f|ׁ @Ƚ| @ҽ| @ܽ| @>@>RHH *Ǡ*Ffe<\Ǡ? gcffommutes.{Proof.@_De neasequenceofbialgebras(BidjiUR2N)bry!ʍB0V:=URB;Bi+1:=URBOopcopái;i2N:LetB20bSethecoproductofthefamily(Bidji 2N)withinjectionsi p˹: Bi4!IB20i?.BecauseB20yisacoproSductofbialgebrasthereisauniquehomomorphismofbialgebrasSן20b:URB204!-pB20$џ^opcop=c,sucrhthatthediagramsQ˵BOopcopái+12B20^opcopnj,32fdά-͍ϫ8:i+1ȥ>Biȥ*B20\{fd33@ά-8:iH Ǡ*Ffe<Ǡ?k}`idH"Ǡ*FfeUǠ?n+Pidڹ(x(1) \|)Sן20i(x(3)) u"idڹ(x(2))u"idڹ(x)qYN=URPidڹ(x(1) \|)Sן20i(x(3)) (1Sן20+u")idڹ(x(2))>+Pidڹ(x(1) \|)Sן20i(x(2)) 1Bd0 u"idڹ(x) 1Bd0qYN=URPidڹ(x(1) \|)Sן20i(x(3)) (1Sן20+u")idڹ(x(2))>+(1Sן20+u")idڹ(x) 1Bd0qYN2URB20 I++I B20i?:RFThrusI+isacoidealandabiidealofB20i?.NorwletHV(B):=B20i?=I{andlet~ǹ:B20&?4!HV(B)bSetheresidueclasshomomor-phism.>WVedshorwthatHV(B)isabialgebraand&isahomomorphismofbialgebras.HV(B)CisanalgebraandQ isahomomorphismofalgebrassinceIƹisatrwoCsidedideal.SinceIFURKerBm(")thereisauniquefactorizationRyHIB20H B20i?=ID{fd!wЍά- {&H ; "20ׁ @ʭ @ԭ @ޭ @>@>RH""K2Ǡ*Ffe,dǠ?9"'whereM"UR:B20i?=IF4!Kisahomomorphismofalgebras.Since(I)B20K IE+I B20Ker( F:3B20' B20r4!2B20i?=IR B20=I)andthrusIv3KerrN(( ǹ))wehaveauniquefactorizationIHB20 B20ⵠB20i?=I+ B20=I̞32fd#ά-*i~ HDLB20H68B20i?=I{fdDά-H@>RHB20i?=IǠ*FfeDǠ?S>n7 &e621:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYThisholdsifIFURKerBm(Sן20).8SinceKer()UR=I+itsucestoshorwSן20(I)URI.8WVeharve8c7NP gSן20((Sן20+1)idڹ(x))UR=􍍒 g=URrW(Sן20 ^2{i Sן20idڹ)i(x) g=URrW(Sן20+ 1)(i+1 i+1AV)idڹ(x) g=URr(1 Sן20)(i+1 i+1AV)Widڹ(x) g=URr(1 Sן20)(i+1 i+1AV)i+1(x) g=UR(1Sן20)i+1AV(x)7andXL^Sן 0(u"idڹ(x))UR=Sן 0(1)"idڹ(x)=Sן 0(1)"i+1AV(x)=Sן 0(u"i+1AV(x))qhencewreget^8Sן 0((Sן 0+1u")idڹ(x))UR=(1Sן 0u")i+1AV(x)UR2I:ThisPlshorwsSן20(I)URI.wSoPlthereisauniquehomomorphismofbialgebrasS):URHV(B)4!HV(B)2opcop[sucrhthatthediagramK- ٍB20+^opcop ٍޛHV(B)2opcop`32fdά-H $B20H1HV(B)t{fd*ά- H"Ǡ*FfeKTǠ?nI+Sr}-:0HǠ*FfeԟǠ?`sTSCtcommrutes.NorwnweshowthatHV(B)isaHopfalgebrawithantipSodeS׹.ByProposition2.1.3it5sucestotestongeneratorsofHV(B)henceonimagesidڹ(x)ofelemenrtsx82Bi.WVeharve位ʍ:j(1S׹)idڹ(x)UR=r(lo Sǹ)idڹ(x)=r(lo )(1 Sן20)idڹ(x)UR=uG=URǹ(1Sן20)idڹ(x)=u"i(x)=u"i(x):位ByPropSosition2.1.3SisananrtipodeforHV(B).WVe(prorvenowthatHV(B)togetherwithrW:=0 2[:B ]4!AHV(B)isafreeHopfalgebra orverB..LetH`bSeaHopfalgebraandletfQ:URBX4!_7HbSeahomomorphismofbialgebras.4MWVewillshorwthatthereisauniquehomomorphismWk*fD:URHV(B)4!1HDsuchthatCȍHBH⑹HV(B)|{fd i@ά-pH`ǡf|ׁ @Ƚ| @ҽ| @ܽ| @>@>RHH *Ǡ*Ffe<\Ǡ?6ݍ㎍rfcommrutes.WVede neafamilyofhomomorphismsofbialgebrasfi,:URBi4!) Hbry:ʍf0V:=URf;fi+1:=URSHDfid;i2N:?37 &e{6.pQUANTUM!A9UTOMORPHISMGROUPSoL63YWVeharveinparticularfi,=URS2ibHDfȹforalli2N.-ThrusthereisauniquehomomorphismofbialgebrasfG20k:URB20=`Bi,4!) HsucrhthatfG208i,=URfiOforalli2N.WVeshorwthatfG208(I)UR=0.8Letx2Bidڹ.ThenF5ʍ[fG208((1Sן20)idڹ(x))=URfG208(r(1 Sן20)(i idڹ)i(x))=URPfG208idڹ(x(1) \|)fG20Sן20i(x(2) \|)=URPfG208idڹ(x(1) \|)fG20i+1AV(x(2))=URPfidڹ(x(1) \|)fi+1AV(x(2))=URPfidڹ(x(1) \|)Sfi(x(2))=UR(1S׹)fidڹ(x)=u"fi(x)=u"i(x)=URfG208(u"idڹ(x)):F5ThistogetherwiththesymmetricstatemenrtgivesfG208(I)UR=0. Hencethereisauniquefactorization xthroughahomomorphismofalgebrasW"*f q:URHV(B)4!1H ιsucrh xthatfG20k=W*URf gǹ.TheohomomorphismW3*f:7~HV(B)4!H\߹isoahomomorphismofbialgebrassincethediagramQ8B20i?=Iğ:2fd-6ά-־ɯ ɯ ğ{fdz{0ά-zr9fǟ-:0Rt:2fd.ά-퍍㎍^r$fKK`32fd`K@ά-h2rfǟ-:0 fǟ-:0AAHB20i?=I+ B20=I`tfd 6ά-zf AAԟtfd `ά-^͍㎍?cry)fpH ㎍:rfAy3Hy'*H H8r耄@fe9)耬?=$X.HA `fe$ `?nf-:0ώ+tB20+B20 B20 r耄@fe>耬?`v 썹commruteswiththepSossibleexceptionoftherighthandsquareW*f and(W*fg WCj*f)209.ButgisXsurjectivresoalsothelastsquarecommutes.!Similarlyweget"HW *Df==UR"H(Bd).!ThusϢW*f ccisPahomomorphismofbialgebrasandhenceahomomorphismofHopfalgebras. dcffxff ̟ff ̎ ̄cffQRemark2.6.4.j6InҢcrhapter1wehaveconstructeduniversalbialgebrasM@(A)withcoactionE:]A4!bGM@(A) AforcertainalgebrasA(see1.3.12). JThisinducesahomomorphismofalgebras5s2 0:URA4!1HV(M@(A)) AsucrhQthatAisacomoSdule-algebraovertheHopfalgebraHV(M@(A)).nIfH?BisaHopfalgebra@andAisanHV-comoSdulealgebrabry@L:0A4!Hҽ gA@thenthereisauniquehomomorphismofbialgebrasfQ:URM@(A)4!1HsucrhthatT|H@AH~M@(A) A{fdPЍά--pHk}?=@Wׁ @W @W @W @؝>@؝>RHaH AŸǠ*FfeǠ?`tf 1@ێ7 &e641:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYcommrutes.SinceythefQ:URM@(A)4!1HftfactorizesyuniquelythroughW*f :URHV(M(A))4!1HwregetacommutativediagramKgyHȠAHPHV(M@(A)) AD{fd sά-LL̟-:0Hk}P@htׁ @ht @ht @ht @Ӯ>@Ӯ>RHH A"Ǡ*FfeTǠ?6ݍ㎍`rf 1lwithauniquehomomorphismofHopfalgebrasWyR*fR:URHV(M@(A))4!1H.ThisEcproSofdependsonlyontheexistenceofaunivrersalalgebraM@(A)forthealgebraA.8HencewrehaveCorollary2.6.5.sWLffeteX&beaquantumspacewithuniversalquantumspace(andquantumFmonoid)M(Xӹ).ZThentherffeisaunique(uptoisomorphism)quantumgroupH(M(Xӹ))35actinguniversallyonX.This2TquanrtumgroupH(M(Xӹ))canbSeconsideredasthe\quantumsubgroupofinrvertibleelements"ofM(Xӹ)orthequantumgroupof\quantumautomorphisms"ofXӹ.A슠7 &e&7.pDUALITY!OFHOPFALGEBRAS~Q65Y7.DualityofHopfAlgebrasInށ2.4.8wrehaveseenthatthedualHopfalgebraHV2 ۹ofa nitedimensionalHopfalgebraHPsatis escertainrelationsw.r.t.theevXaluationmap.ThemrultiplicationoffHV2 isderivredfromthecomultiplicationofHandthecomultiplicationofHV2 isderivredfromthemultiplicationofHV.Thiskindofdualitryisrestrictedtothe nite-dimensionalsituation.*Neverthelessone1wrantstohaveaproScessthatisclosetothe nite-dimensionalsituation.I{ThisshortsectionisdevrotedtoseveralapproachesofdualityforHopfalgebras.Firstwreusetherelationsofthe nite-dimensionalsituationtogiveageneralde nition.yDe nition2.7.1.vaùLetHandLbSeHopfalgebras.8Letc(evW:URL HB3a h7!ha;hi2KbSeabilinearformsatisfyingU`ha b;XUTh(1)$ h(2) \|iUR=hab;hi; h1;hiUR="(h)(3)`ohXUVa(1)$ a(2) \|;h jiUR=ha;hji; ha;1i="(a)(4)ePha;S׹(h)iUR=hS(a);hi(5)Sucrhamapiscalledaweffak@dualityofHopfalgebras.eThebilinearformiscalledleft(right$D)nondeffgenerateifha;HVi̹=0impliesa=0 ߔ(hL;hi=0impliesh=0).HEAduality35ofHopfalgebrffas꨹isawreakdualitythatisleftandrightnondegenerate.Remark2.7.2.j6IfHisa nitedimensionalHopfalgebrathentheusualevXalua-tionev+:URHV2X HB4!K꨹de nesadualitryofHopfalgebras.Remark2.7.3.j6AssumethatevJ:URL HB4!Kǹde nesawreakdualityV.ByA.4.15wre(haveisomorphismsHomAq(L HF:;K)PA>԰A&=Homy(L;Homy(HF:;K))$andHomq(L H;K)Pl6԰=-Hom,L(H;Homy(L;K)).Denote$theho-momorphisms{assoSciatedwithev:bKL K>4!K{bry':L4!#Hom3W(HF:;K)resp. Ë:URHB4!Hom.+(L;K).8Theysatisfy'(a)(h)=ev(a h)= n9(h)(a).ev#:!RL K4!8KOisleftnondegeneratei ':L4!\1Hom2պ(HF:;K)isinjectivre.ev:URL K14!K꨹isrighrtnondegeneratei Ë:HB4!Hom.+(L;K)isinjectivre.Lemma2.7.4.g5Q1.TheAbilineffarformevF8:nL H\N!1KAsatis es(3)ifandonlyif'UR:L!Hom-(HF:;K)35isahomomorphismofalgebrffas.2. NThebilineffarformevՋ:L H!;Ksatis es(4)ifandonlyif ,:H!Homy(L;K)35isahomomorphismofalgebrffas.Proof.@_evOK:URL0 HB4!K -satis estherighrtequationof(3)i '(ab)(h)=hab;hi=ha= b;Ph(1) N h(2) \|iUR=Pha;h(1)ihb;h(2)iUR=P'(a)(h(1))'(b)(h(2))=('(a)='(b))(h)brythede nitionofthealgebrastructureonHomd1(HF:;K).BX7 &e661:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYev :URL HB4!K꨹satis estheleftequationof(3)i '(1)(h)=h1;hi="(h).ThesecondpartoftheLemmafollorwsbysymmetryV.`cffxff ̟ff ̎ ̄cff+荍Example2.7.5.oQThere/isawreakdualitybSetweenthequantumgroupsSLܟqQڹ(2)andUq(slC(2)).8(Kassel:ChapterVISI.4).Prop`osition2.7.6.OLffetꅹev:URL ! HB!gKbeaweakdualityofHopfalgebras.N.LetI:=tKer('t:L!eUHom.(HF:;K))andJ:=tKer( m:tH!RHom/4(L;K)).zLffet\-z ӍL:=L=Iand~\-z ӍHX:=HF:=Jr. HThen~\-z ӍLand~\-z ӍHarffe~Hopfalgebrasandtheinducedbilinearformdz 1Kev:UR\-z ӍL \-z ӍH!"eOK35isaduality.Proof.@_FirstobservrethatIWandJFaretwosidedidealshence\-z ӍL7and\-z ӍHO>arealgebras.%Then9ev:Lw H/4!K9canbSefactoredthroughdz 1Kev:ٟ\-z ӍL w\-z ӍHe4!#[LKandtheequations(3)and(4)arestillsatis edfortheresidueclasses.TheidealsIandJxarebiideals.Infact,&letxUR2Ithenh(x);a, biUR=hx;abi=0hence(x)52Ker"('j '5ֹ:Lj L54!9Hom0(HX3 HF:;K)5=I\` jL+L Iy(thelastequalitry isaneasyexerciseinlinearalgebra)and"(x)H=hx;1i=0.Hence asintheproSofofTheorem2.6.3wregetthat\-z ӍL&=ۡL=Iyand\-z ӍHU-=HF:=Jharebialgebras.SincehS׹(x);ai¹=hx;S׹(a)i=0Zwrehaveaninducedhomomorphism\-zȟ ӍS :Ÿ\-z ӍL4!'lb\-z ӍL/j. 7Theidenrtitiessatis edinLholdalsofortheresidueclassesin\-z ӍL(+sothatLandsimilarly\-z ӍHbSecomeHopfalgebras.‚Finallywrehavebyde nitionofI thathdzRKxR;dz+Ka*ii=hx;ai=0forPalla*2H>!i Pa2IBNordz+Ka~=0.kHThrusthebilinearformdz 1Kev&:\-z ӍL 2\-z ӍH4!#K-ide nesadualitryV.?cffxff ̟ff ̎ ̄cff+荍Problem2.7.8.nR(inLinearAlgebra)1. #FVor4U_{V7de neU@2? 0׹:=ffz2Vp2\tjfG(U@)=0g.s A)2.WVeúwranttoshowthatr2(fG)o2A2u, (A2.g%Letg1;:::ʚ;gn dbSeabasisofAfG.g%Thenthereexisth1;:::ʚ;hn 2oA2sucrhLthatbf=pPhidڹ(b)gi.,LetLa;bp2A.ThenLhr2(fG);a bip=hf;abi=hbf;ai=Phidڹ(b)gi(a)UR=hPgi hid;a bi꨹sothatr2(fG)UR=Pgi hi,2A2j A2.5. I=)$O+3.^LetZr2(fG)j=Pgi' Mhi2A2sQ A2.^ThenZbf=Phidڹ(b)gi\4forallb2AasbSefore.8ThrusAf2isgeneratedbytheg1;:::ʚ;gnP.,cffxff ̟ff ̎ ̄cffProp`osition2.7.9.OLffetS(A;m;u)bffeanalgebra.Thenwehavem2(A2o)URA2o (A2o.Furthermorffe35(A2o;;")isacffoalgebrawithUR=m29and"=u2.Proof.@_LetfQ2URA2oandletg1;:::ʚ;gn 4bSeabasisforAfG.6Thenwrehavem2(fG)UR=Pgi Rhi)ѹforsuitablehi,2URA2asintheproSofofthepreviousproposition.Sincegi,2URAfwreϟgetAgi,URAfanddimd(Agidڹ)<1andhencegi,2A2o./ChoSosea1;:::ʚ;an2AsucrhthatDgidڹ(ajf )j=ijJ.^Then(fGajf )(a)=f(ajf a)=hm2(f);ajH >ai=Psgidڹ(ajf )hi(a)j=hj(a)implies_fGaj=hj2fGA.GObservre_thatdim(fA)<1_ʹhencedim(hjf A)<1,}so_thathj\2URA2o.8Thisprorvesm2(fG)UR2A2oFh A2o.OnecrheckseasilythatcounitlawandcoassoSciativityhold.hNcffxff ̟ff ̎ ̄cffTheorem2.7.10.wX(The|Sweedlerdual:) '2Lffet(B;m;u;;")bffeabialgebra.Then笹(B2o6;2;"2;m2;u2)againisabialgebrffa. IfB=}HisaHopfalgebrawithantipffode35S,thenS2 isanantipffode35forB2o=URHV2o.Proof.@_WVe$wknorwthat(B2[ ;2;"2)$wisanalgebraandthat(B2o6;m2;u2)$wisacoal-gebra.?XWVeB%shorwnowthatB2o !=B2 /isasubalgebra.?XLetf;gXv2B2o xwithdimm(BfG)<1Ianddim)(Bgn9)^<1. VLetIa2B. VThenIwrehave(a(fGgn9))(b)^=(fgn9)(ba)=PfG(b(1) \|a(1))gn9(b(2)a(2))X{=P((a(1) \|f)(b(1))(a(2)gn9)(b(2))X{=P(((a(1) \|f)(a(2)gn9))(b)܄hencea(fGgn9) 0=Pݹ(a(1) \|f)(a(2)gn9) 02(Bf)(Bgn9).Sincebdimm(Bf)(Bgn9) 0<1bwrehavedimH(B(fGgn9))<1sothatfg2B2o6ƹ. FVurthermorewrehave"2B2o6ƹ,msinceKerv(")hascoSdimension1.2]ThrusB2oURB2 2)isasubalgebra.ItisnorweasytoseethatB2o isabialgebra.Norw+letS7bSetheantipSode+ofHV.jWVeshowSן2r۹(HV2o)ZHV2o.jLet+a2HV,f2H2o.ThenhaSן2r۹(fG);bi=hSן2(fG);bai=hf;S׹(ba)i=hf;S׹(a)S(b)i=hfGS׹(a);S(b)i=hSן2r۹(fGS׹(a));bi. yThisQimpliesaS2r۹(fG)=S2r۹(fGS(a))QandHVS2r۹(fG)=S2r۹(fGS(HV))Sן2r۹(fGHV). Sinceδf92:H2o Wʹwregetdimc(fGH):<1sothatdimc(Sן2r۹(fGHV))<1anddimH(HVSן2r۹(fG))UR<1.*YThisshorwsSן2(fG)UR2HV2o.*YTherestoftheproSofisnorwtrivial. dcffxff ̟ff ̎ ̄cffDҠ7 &e681:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYDe nition2.7.11.}!ùLet+G=K-cAlgo(HF:;-)bSeananegroupandR=2K-cAlgo.WVeM&de neGh K!Rn:=URGj33R -^cAlg((tobSetherestrictiontocommrutativeM&RJ-algebras._TheۍfunctorG K cRisrepresenrtedbyH Rn2URRJ-cAlg:ePGj33R -^cAlg%(A)UR=K-cAlgo(HF:;A)P԰n:=RJ-cAlg(H RJ;A):Theorem2.7.12.wX(TheVCartierdual:)ULffetHrbea nitedimensionalcommu-tative"cffocommutativeHopfalgebra.3LetG=K-35cAlg(HF:;-33)bffetheassociatedanegrffoup35andletDS(G)UR:=K-35cAlg(HV2Z;-33)35bethedualgroup.fiThenwehaverDS(G)UR=G.r(G;GmĹ)wherffeԍG.rMA(G;GmĹ)(RJ)=GrJ(G"+ K R;Gm & KR)ԍisthesetofgrffoup(-functor)homo-morphisms35andGm 7isthemultiplicffativegroup.Proof.@_WVepharveG.r$(G;GmĹ)(RJ)UR=Gr>(G KR;GM KR)PUR԰n:=R-Hopf-Alg2"(K[t;t21 \|] RJ;H5^ HR)PUR԰n:=R-Hopf-Alg2"(R[t;t21 \|];H5^ HR)PUR԰n:=fxUR2U@(H HRJ)j(x)=x x;"(x)=1g,since(x)UR=x xand"(x)UR=1implyxS׹(x)="(x)=1.Considerdx%V2HomߟR#r((H3 RJ)2;R)%V=HomߟR(HV27 RJ;R).Thend(x)%V=x xd޹i x(vn92.=wR2)L=hx;vn92wR2i=h(x);vn924 swR2i=x(vn92.=)x(wR2)Tand"(x)L=1Ti hx;"iL=1.HencexUR2RJ-cAlg((H RJ)2;R)PUR԰n:=K-cAlgo(HV2Z;R)UR=DS(G)(RJ).`cffxff ̟ff ̎ ̄cff2;7 &02@cmbx8-%n eufm10,@ cmti12+- cmcsc10*o cmr9(ppmsbm8' msbm10"a6cmex8!u cmex10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10=Q