; TeX output 1999.09.21:07257 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)d;7  -%n eufm10,@ cmti12+- cmcsc10*o cmr9' msbm10!u cmex10K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10l