; TeX output 2002.01.10:1157r-&"2>K`yp cmr10Quanltumhomogeneousspaceswithfaithfully atrL&mo}dulestructures#􍍍\s64K`yff cmr10E.F.MfgfeullerandH.-J.Schneider&0MathematischesInstitutderUniversitatMfgfeunchenTheresienstrae39D80333Mfgfeuncheng{^=<"VG cmbx100- InrtroOductionb#^=+XQ cmr12Let,g cmmi12AbSeaHopfalgebrawithbijectivreantipSodeovertheground eldkg,yandB-!", cmsy10iAa^=righrt$coidealsubalgebra,2thatisasubalgebrawith(B)Bm A.~WVe$canthinkofthe^=inclusionܑBXURAasde ningaquotienrtmapG!XwhereGisaquanrtumgroupandX^=isaquanrtumspacewithrightG-actionorarightG-space.^=FVromanalgebraicpSoinrtofview[T1\h,Sch @,M1]theinclusionB;c]AonlyhasgoSodprop-^=ertiesifAisfaithfully atasaleft(orrighrt)moSduleoverB.^=IfAisacommrutativeHopfalgebra,2themeaningoffaithful atnessofAorverBXis^=as2follorws.NowG[Sw]ifCisthedirectsumofitssimplesubScoalgebras.Cisspannedbry^=group-likre7elementsifandonlyifCiscosemisimpleandpSointed,,thatisallitssimple^=subScoalgebrasareone-dimensional.^=In section3wreshowthatKtinfactisC5-semisimpleinmanyimpSortantcases.VLetUbSea^=Hopf0-algebra,B3andassumethatUqispSoinrted(thisholdsforallthequantizeduniversal^=enrvelopingYalgebrasU̹q(Rg) ").@IfI( U.=isacoidealwithS(I)2 ͩIandB:=fa2Aj }2 r-^=a IN=0gAsasintheexamplesdescribSedaborve,&thenAsIUݯ=Kܞ2+UWwhereKistheleft^=coidealtsubalgebraoftherighrtU=IU@-coinvXariantelementsinU@.Hencewemayalsowrite^=BX=URfa2AjaPKܞ2+ Nh=UR0g~asinTheorem2.2.IfwreassumethatallmoSdulesV2C1harve~a^=hermitian&innerproSducth;isucrhthathxvn9;wRiUR=hvn9;x2wRi&forallvn9;w2URVYandx2U then^=bryCorollary3.3,K1=URKܞ2 JisC5-semisimple,andourabstractTheorem2.2applies.^=In5sections4and5wreconsiderinUȂ:=U̹q(slC(2))andU̹q(Rg),forRgasemisimplecomplex^=LiealgebraandqjN2kJ%notaroSotofunitryV,! anarbitraryskew-primitiveelementxwith^=(x)UH=gL x+x 1,wheregisagroup-likreelementinU@. jThenthesubalgebra^=kg[x]͹generatedbryxisaleftcoidealsubalgebrainU@.WVetakeforCtheclassof nite^=dimensionalrepresenrtationsoftypSe1andde neEBX:=URfa2AjaxUR=0g^=ine0AUR:=U2@0RAC,theqn9-deformedfunctionalgebraoftheconnected,simplyconnectedalgebraic^=group)?withLiealgebraRg.SomewhatsurprisinglywreshowinTheorem5.2thatforallx^=(uptoonecase)thefollorwingconditionsareequivXalent:ک(a))kg[x]isC5-semisimple,!thatisxactsonall nitedimensionalU@-moSdulesasadiago-)nalizableopSerator.3(b))Aisleftorrighrtfaithfully atoverB.(c))A=XAB2+)$#isspannedbrygroup-likeelements.^=Moreorver,jconditionJ(a)isequivXalenrttoanexplicitnumericalconditiononthecoSecients^=ofx.^=Thrus>weseethatcondition(a)whichwasstudiedbyNoumiandMimachi[NM&]incon-^=nection(withProSdlea?s'quantumspheresisequivXalenttotheabstractalgebraicconditionof^=faithful atnessin(b).^=MostPcalculationsinsection4andspSecialvrersionsofsomearguments,j~whichhavebSeen^=widelyDgeneralizedintheproSofof2.3(1),arealreadyconrtainedinthe rstauthor'sdiploma^=thesis[MSvuKU].(V^=1- Preliminariesandsomegeneralresultsb#^=Innthefollorwingwe xa eldkեwhichistheground eldforallalgebrasandvectorspace.^=FVor-de nitionsandbasicresultsonHopfalgebrassee[Sw,M\]. FWeusethesimpli ed }3r-^=notation(x)UR=Cu cmex10CPx̸1 7 M3x̸2|forthecoproSductinacoalgebra.)LetVYWbeavrectorspace.)In^=thefollorwing(exceptforsection3),wewriteVp2 GforthedualspaceHomd1(V;kg)ofVp.^=TVo~studysub-andquotienrtobjectsofHopfalgebrasitiscrucialtoconsiderfaithfully at^=moSdulesa andfaithfullyco atcomodules. RecallthatarighrtmoduleMorvera analgebraB^=isLcalled atrespSectivrelyfaithfullys atifandonlyifthefunctorM ̹B :̹B NfMisthecategoryofleftCܞ-comoSdules,^=andifWnisaleftCܞ-comoSdule,thenthecffotensor35productisde nedasthekrernelofB鍑Ts̹V p idid ̹W <:VG W!URV CF Wr;^=where7̹V \and̹W sarethecomoSdulestructuremapsofVandW.Ofcourse,3thesenotions^=arede nedinthesamewrayformoSdulesorcomodulesontheotherside.^=LetJAbSeaHopfalgebra.VIfI"Aisavrectorsubspacewithquotientmapn9:A!A=I^=(usuallyI+willbSeacoidealandaleftorrighrtideal),weletB鍍P coX A=IgAUR:=fa2AjCXn9(a̸1) a̸2V=UR(1) ag^=bSethesetofleftA=I-cffoinvariantB.elements.iIfIbq Aisacoideal,thenthequotienrtmap^=n9:As!A=I is,acoalgebramap,=andAisarighrt(respSectivelyleft)A=I-comoSdulevia^=withcomoSdulestructure( sTid =)(respectivrely(id = sTn9)).JDuallyforasubalgebraBoofA^=wre|considerAasaleftandarightB-moSdulebyrestrictingthemultiplicationinA.'A8right^=(respSectivrely7Zleft)cffoidealysubalgebra7ZB`ofAisasubalgebraBrAwith(B)By A^=(respSectivrelyA B).:/^=0N cmbx12Theorem1.101fo,ThefforemvQ2]LffetAbeaHopfalgebraandIјAacoidealandaleft^=ideffalwithquotientmapn9:A[!A=Ik.vDe neBa:=2co 2A=IqA.AssumeAisfaithfullycffo at^=as/arightA=I-cffomodule/vian9.\zThenBF)@Aisarightcffoideal/subalgebra,oI=AB2+~,^=wherffeB2+ =5,Ker"G("j̹BN>)istheaugmentationidealofB,OandAisfaithfully atasaright^=B-moffdule.^=Moreorvera)b:=CPU,2.1]LffetAbeaHopfalgebrawithbijectiveantipodeandBAa^=rightcffoidealsubalgebra.Letx'Ak:=A=XAB2+1uwithquotientmapn9:A!x A .Thenx'AUisa^=quotient35cffoalgebraandaquotientleftA-moduleofA,andthefollowingareequivalent:D(1))A35isfaithfullycffo atasaleftxLrA -comodulevian9,andBX=UR2co  \vA:ZA.D(2))A35isfaithfully atasaleftB-moffdule.D(3))A35isprffojective35asaleftB-moffdule,andB;isaleftB-dirffect35summandinA.D(4))A35is atasaleftB-moffdule,andB;isasimpleobjectinM2ARAB.D(5))The35functorM2ARAB?!URM=*A,M67!M=XM@B2+ isaneffquivalence.^=S- cmcsc10SProof:Thisis[MWֲ,2.1]when(3)isreplacedbry^=(3209)AisaprojectivregeneratorasaleftB-moSdule.^=WVeHharvetoshowtheequivXalenceof(3)and(3209).Assume(320).ThenAisprojectivreand^=faithfully atasaleftB-moSdulebry[MWֲ].&HenceBNisaleftB-directsummandinAbry^=[R,2.11.29]2^=Thesimplicitryconditionin(4)meansthatanynon-zerorightB-submoSduleandright^=A-subScomoduleofBisequaltoB.^=Thecategoricalcrharacterization(5)showsthesigni canceoftheequivXalentconditionsin^=1.2.8Moreorver,by[M19,1.11]themappingꪍ>f^C BC   [B;uoAisarighrtcoidealsubalge-[bra,]A:uisleftfaithfully atorver:uBɟf^C(G!URf^C SIC  jFIbAisacoidealandleftidealandFAisleftfaithfullyco atorverA=If^C?^=is3abijectionbSetrween3sub-andquotienrtobjectsofAsatisfyingthecorrespondingcondi-^=tionsin1.2.^=Remark1.391.H _Iffwreapply1.2tothedualalgebrasB2op iDURA2op (A2opisfaHopfalgebra)sincetheanrtipSodeofAisbijective),2awegetthedualtheoremwherexAJ-isnow)A=B2+~A,M2ARABisreplacedbry̹B 8M2A ,andAisconsideredasarightB-moSdule. }57nr-2.)Let_AbSeaHopfalgebraandB'#!Aarighrtcoidealsubalgebra. *WVenotethe)follorwingasimplicitycriterion:ZIfBisaleftB-directsummandinAthenBissimple)inM2ARAB.^=SProof: Let:fG:All!B@bSealeftB-linearmapsucrhthatfGj̹B =id 7.aLetX]B@bSeanon-^=zeroVsubSobjectinM2ARABA.ThenXAisanon-zeroHopfmoduleinM2ARAA.SinceM2ARAAP 36԰ L=ߟ̹kqMbry^=thefundamenrtaltheoremofHopfmoSdules[Sw,*4.1.1],AissimpleinM2ARAA andXAUR=A0;.^=Hence-thereexist nitelymanryelementsx̹i2X,a̹i2A-sucrhthatCPGhi@x̹ida̹i=1.nThen^=1UR=fG(1)=CP ㍟iHex̹idf(a̹i)UR2X,andXF=URB.92^=FVorNcompletenesswregivetheshortproSofofthefollowingimpSortantobservXationofKop-^=pinen.^=Lemma1.4'S[Ko] LffetAbeaHopfalgebrawithbijectiveantipodeSIandB93Aaright^=(rffesp.fileft)35coidealwith1UR2B.fiThenS(AB2+~)UR=B2+A35(rffesp.fiS(B2+A)=AB2+).^=SProof:=WVe?/assumethatB5isarighrtcoideal(andthenapplytheresulttothedual^=coalgebrabofAtogetthelemmaforleftcoideals).Letfx̹j mjcj[62JrgbbSeabasisofB2+~.^=ThenCforallxUR2B,W(x)=1/L x+CP j#x̹jV y̹jMforCsomey̹j\2URA.$Norwassumex2B2+ eand^=apply(S] id uL)and(id uL S),wheredenotesmrultiplicationinA:2C0UR=1"(x)=x+CX joS(x̹jf )y̹j;andz0UR=S(x)+CX jox̹jf S(y̹j):$^=Therefore,B2+ \FS(B2+~)A%andS(B2+~)\FB2+A.VHence%B2+A=S(B2+)A%andS(AB2+)\F=^=S(B2+~)AUR=B2+A.g=2!^=Corollary1.53uWLffetjAbeaHopfalgebrawithbijectiveantipodeandIOAacoidealand^=left35ideffalwithquotientmapn9:AUR!A=I$b.fiAssume35A=I$iscosemisimple,andletA=IF=URCM "%hjv2JqC̹j$^^=bffe35thedirectsumofthesimplesubcoalgebrasC̹j?forj%2URJr,ofA=I.fiThenD(1))I{3=AB2+~,VandO}Aisleftandrightfaithfully atoverBandleftandrightfaithfully)cffo at35overA=I$vian9. }6Fgr-D(2))For35allj%2URJr,let)̹j$3AUR:=fa2AjCXn9(a̸1)R a̸2V2URC̹j Ag;A̹j\:=fa2AjCXa̸1 n9(a̸2)2A C̹jf g:)Then|A =CL怟jv2J!\6j%@A=CL怟jv2J!\6S(A̹jf ),.andforallj,̹j ArffespectivelyS(A̹jf )isaright)cffoidealzinAanda nitelygenerffatedzandprffojectivezrightrffespectivelyzleftmoffdule)overZB.TIf1UR2JisZthedistinguisheffdindexwithC̸1V=URkgn9(1),thenBX≠1VA=S(A̸1).^=SProof:ک(1))By[Scrh,1.3],Aisrighrtfaithfullyco atoverA=I`ifandonlyif.(a)Aisrighrtco atoverA=I`and (b)XsplitsasamapofrighrtA=I-comoSdules.)SinceOA=Iiscosemisimple,%9anryexactsequenceofrightA=I_-comoSdulessplits.+In)particular,i(a)and(b)hold. ThrusweseethatAisrightandbythesameargument)left(faithfullyco atorver(A=I8.._Since(Aisrighrtfaithfullyco atoverA=I8,&Hweget)from1.1thatAB2+ =URI/andAisrighrtfaithfully atoverB.(7ThenBXURAisaright)coidealRsubalgebra,pBX=UR2co \v2A=ABd-:Aacmr6+)J AandAisleftfaithfullyco atorverRA=XAB2+*..Hence)theequivXalenrtconditionsin1.2hold,andAisleftfaithfully atoverB,toSo.ک(2))Byu(1),BisarighrtcoidealsubalgebraofA,Aisleftandrighrtfaithfully atoverB)andxA=URA=XAB2+.Liscosemisimple.8Hencebry1.2,؍44{M AڍBMA5!URM鍑=*A;M67!M=XM@B +~;andzM鍑=*A 36!M AڍB ;V7!Vp2t=*AA;)arequasi-inrversecategoryequivXalences. FVoranyrightxAAW-comoSduleVp,theHopf)moSdulepstructureonVp2t=*AAisgivrenbymultiplicationandcomultiplicationonA.)Sincexn9(a̸1)5 a̸2,maps̹j AonrtoC̹jf 2t=*AA.HenceA=CLj3 jAisa|)decompSositionKinM2ARAB.|Byconstruction,̸1A{=2co ) A`A=B.AllKthe̹jgUAareprojectivre)righrt]B-moSdulessinceAisprojectiveasarightB-moSduleby1.3(1). Theyare) nitelygeneratedorverBsince̹j fAP._԰GG=C̹jf 2t=*AAandC̹jis nitedimensional.zMore)generallyV,sletXHVbSeanry nitedimensionalrightxqA '1-comoSdule.ThenMP=Vp2t=*AAis) nitelyĢgeneratedasarighrtB-moSdule.,3TVoseethiswriteMastheascendingunion)of}allHopfsubmoSdulesXBwhereXoisa nitedimensionalrighrtA-subcomodule }7S٠r-sE)of M@.^ILetF:M2ARAD⍍||=g 0u!!M=*A bSethecategoryequivXalenceof1.2.ThenF(M@)Pj԰t=*oVis)the)ascendingunionofallF(XB).dSinceVis nitedimensional,+ F(M@)=F(XB))forsomeX,henceM6=URXBisB- nitelygenerated.\t)TVogetthedecompSositionofleftB-moduleswreapplythepreviousresulttoA2op x.)ThenxB2op iDURA2op isarighrtcoidealsubalgebraandA2opisleftandrighrtfaithfully at)orverB2op .8ByKoppinen'slemma1.4,S(AB2+~)UR=B2+Aandzoymn9:A=XAB +⍍3-3-=g ,Т4%!C{OA=B +~A;)aj7!URщfe;Q 3/S(a);*)isuacoalgebraanrtiisomorphism.-HThereforeA=B2+~Aq==CLjn9(C̹jf )uisadirectsumof)simplesubScoalgebrasandA2op x=XA2op(B2op )2+ q=URA=B2+~Aiscosemisimple.Thrusweknow)fromthepreviousproSofthatAUR=CLqƟjΟjxW~=A%iwhereforallj,kd̹jxl~hAu:=URfa2AjCX#?esߞ(a̸1) a̸2V2URn9(C̹jf ) Ag)is a nitelygeneratedprojectivreleftB-moSduleandarightcoideal,eand̸1x M~ |A=B.)Here#Ces" ;N:A!A=B2+~A"isthecanonicalmap.BFinallyV,0forallj,̹jx ~A=S(A̹jf )sincefor)allaUR2A,T*aUR2̹jx ԙ~\A5()5CX#lyesS(a̸1) a̸2V2URn9(C̹jf ) A5()5CXSn9(Sן 1 S(a̸1)) Sן 1(a̸2)UR2C̹j A>5()5Sן 1 S(a)UR2A̹jf :c@2-^=2- Aiclassj~ofhomogeneousspacesde nedbryin nites-- imalinrv`ariantsb#^=WVe rstcollectsomewrell-knownresultsandnotationsondualitry(cf.,[M =,cChapter9],[J ;,^=I.1.4],H[T2\h,section5l1]).+LetUvPbSeanalgebra.Thedualwcffoalgebra5lU@20 |ԔU@2 6Tisspannedbry^=theematrixcoSecienrtsofall nitedimensionalleftU@-modules6Vp.If:Ug|!&End3W(V)is^=the0represenrtationofU@,Cܞ2V denotestheimageofthedualcoalgebra(End (Vp))2D4under2.^=ThrusjCܞ2V tisthekg-linearspanofallmatrixOcffoecientsjc̹f;vC2.+U@2,fv*2Vp2\t,vd2Vp,where^=c̹f;v ^(u)UR:=fG(uvn9)>foralluUR2U@.If>(v̹id),`(f̹i)aredualbasesofVp,`V2\t,the>coalgebrastructure^=of~Cܞ2V isexplicitlygivrenby(c̹f;v ^)UR=CP ㍟iHec̹f;v8:;cmmi6i c̹f8:i,r;v@afor~allfQ2URVp2\t,/vË2Vp.Then~U@20 yisthe }8 er-^=sumDofallthesubScoalgebrasCܞ2V3f,ZandCܞ2Vq1*Vq2#=>Cܞ2Vq1 bP+Cܞ2Vq2for nitedimensionalleftU@-^=moSdulesߎV̸1;V̸2.5-Thenatural(U;U@)-bimoSdulestructureonU2 vandonallCܞ2V3f'sisdenoted^=bryxaandax,aforallxUR2U@,aa2U2,awhere(xa)(u)UR:=a(ux)and(ax)(u)UR:=a(xu)^=forjallu/ 2U@.NotejthatthedualalgebraU2 kzisaleftandrighrtU-moSdulealgebrawith^=respSecttotheseactions,sinceforalla;bUR2U@2 andx2U@,6>Ix(ab)UR=CX(x̸1ja)(x̸2b);(ab)xUR=CX(ax̸1)(bx̸2):^=NorwletUbSeaHopfalgebra.*KAtensor cffategoryCrof nitedimensionalleftU@-modulesis^=aclassCof nitedimensionalleftU@-moSdulessucrhthat)ko2URC(kQasthetrivialU@-moSdulevia"),)ifRXJg;YA2C5,mthenX+Y2C%andX+ Y2C%(withdiagonalU@-actiononX+ Yp,)u(xAZ yn9):=CP\Gu̸1xAZ u̸2y66forallu2U@,Sx2XandypA=A̹g*P:=fa2)AURjCPa̸1j +a̸2=a gn9g.12 -r-^=SProof:>ک(1))WVe9 rstconsiderthedecompSositionA=CL>\A̹ .%Let9ELkbeasetofrepresenrtatives)ofyMtheisomorphismclassesofthesimplemoSdulesinC5. By2.1,A=CLQV㐺2E%G_Cܞ2V)is$adecompSositioninrto nitedimensionalright(andleft)U@-moSdules.hTSinceKis)commrutativeandkN/isalgebraicallyclosed,all nitedimensionalsimpleKܞ-moSdules)areaj1-dimensionalandgivrenbycharactersofKܞ.&ByassumptionanyV2vEt-isKܞ-)semisimpleypandhasabasis(v̹id)ofeigenrvectorsypv̹i,2URV̹8:i mforsomecrharacters̹iJofKܞ.)Henceoxv̹i,=UR̹id(x)v̹iIforalliandallx2Kܞ.'"Let(f̹id)bSethedualbasisof(v̹i)inVp2\t.)Then*1f̹i:xw=̹id(x)f̹i foralliandallx2Kܞ.{Thrusitfollowsfrom2.1,:(2)and(3))thatPforallV2URE,bCܞ2VP ԰ =3aV %1Vp2 isasemisimplerighrtKܞ-moSdule,andwrecanwriteުAUR=CM ?qA̹ r=CM "%V㐺2E`CM  & (Cܞ V3f)̹ :,r)Byde nition,(vA̹" r=B.3FVoranry,A̹ isarighrtB-submoSduleofAsinceforall)aUR2A̹ ,b2Bandx2Kܞ,d_g(ab)x=(ZCX#(ax̸1)(bx̸2)=(ZCX#(ax̸1)b"(x̸2);sincex̸2V2URK=(Z(ax)b=(Zab(x):)ItiseasytocrheckthatalleigenspacesA̹ arerighrtcoidealsinA.)LetbSeacrharacterofKܞ.4ItremainstoshowthatA̹ 9is nitelygeneratedand)projectivreZasarightB-moSdule. WVecanassumethatA̹ 6=Ưf0gorequivXalently)^2X(K5;C5). HencethereexistsasimplemoSduleV2^EOwith(Cܞ2V3f)̹ %6=f0g. Let)(v̹id),|(f̹i)bSedualbasesofeigenrvectorsofVbandVp2 Dfasbefore.7Thrusf̹j 'xUR=(x)f̹j)and`rxv̹j=(x)v̹j|forsomej Eandallx2Kܞ.>De nev :=v̹jf ,}fe:=f̹j|andconsider)thematrixcoSecienrtc̹f;v ^.8Since(v̹id),(f̹i)aredualbasesand(c̹f;v ^)UR=CX ʹic̹f;v8:ilx c̹f8:i,r;v;(G)wreu6get1A'=fG(vn9)="(c̹f;v ^)=CPbi4:c̹f;v8:iS(c̹f8:i,r;v),whereu6S( istheantipSodeu6ofA.؉De ne)a̹i,:=URc̹f;v8:i,b̹i:=S(c̹f8:i,r;v)Ƴand̹id(a):=b̹iaƳforalla2A̹ .,Bythedualbasislemmait)sucestoprorvethata̹i,2URA̹ and̹id(a)=b̹ia2Bforallianda2A̹ .)FVoralluUR2U+andx2Kܞ,9t(a̹ix)uUR=fG(xuv̹id)=(fx)(uv̹id)=(x)fG(uv̹i)=(x)a̹i(u);13=r-)hencea̹i,2URA̹ .)FinallyV,forallaUR2A̹ andx2Kܞ,^D(b̹ida)x͉=CXb(b̹ix̸1)(ax̸2)͉=CXb(b̹ix̸1)a(x̸2) since(~x̸2V2URK͉=(b̹iCXox̸1(x̸2))a͉=(b̹ida)"(x);)sinceж(b̹iڃuCPx̸1(x̸2))UR=b̹id"(x).0:Thrusb̹iڃua2B.0:Thelastequalitryholdssinceforall)uUR2U@, [(b̹iCXox̸1(x̸2))(u)y-=Q6S(c̹f8:i,r;v)(CXqx̸1(x̸2)u)UR=c̹f8:i,r;v(S(u)CXqS(x̸1)(x̸2))UR=y-=Q6f̹id(S(u)CXqS(x̸1)(x̸2)vn9)y-=Q6f̹id(S(u)CXqS(x̸1)x̸2vn9)since4Ivx̸2V2URKy-=Q6f̹id(S(u)vn9)"(x)UR=b̹i(u)"(x):)InthesamewraywehaveadecompSositionintolefteigenspacesA=CL@j^1~A.Hence)A>t=CLZ"S(̹ A) sinceSisbijectivre.OneeasilyseesthatalltheS(̹ A)arerighrt)coideals.@FVromKoppinen'slemmaforleftcoideals(apply1.4tothedualcoalgebra))wreget.Kܞ +U6=URSן 1 S(U@Kܞ +)=Sן 1 S(Kܞ +)U+and`BX=fa2AjaSן 1 S(Kܞ +)=f0gg:)HenceS(̹"lA)UR=Bsinceforalla2A,x2KFandu2U@:xf`C~S(a)S 1 S(x)f`C(u)UR=(xa)(S(u)):)ThenwreseethatS(̹ A)isaleftB-submoSduleofAforanysinceforallaT2̹ ntA,)bUR2Bandx2Kܞ,1lxSן 1 S(bS(a))k=x(aSן 1 S(b))UR=CX(x̸1ja)(x̸2Sן 1 S(b))UR=k=CXL(x̸1ja)"(x̸2)Sן 1 S(b)sinceSן21(B)UR=̹""A>k=(xa)Sן 1 S(b)UR=k="(x)Sן 1 S(bS(a)):14ĉr-)Let|2X(K5;C5).\WVeharvetoshowthatS(̹ A)is nitelygeneratedandprojective)as[AaleftB-moSdule.Usingthenotationsaborve[Aitsucestoshorwthatb̹iy2S(̹ A))andS(a)a̹i,2URBforallianda2̹ rA.8FVorallu2U+andx2Kܞ,Od(xc̹f8:i,r;v)(u)UR=f̹id(uxvn9)=(x)f̹i(uvn9)=(x)c̹f8:i,r;v(u);)hencex/b̹i=F8S(c̹f8:i,r;v)2S(̹ A).vFVorx/allaF82̹ XA,S(a)a̹i2B>=S(̹"lA)x/sincea̹i=F8c̹f;v8:i)andforallxUR2Kܞ,9xSן 1 S(S(a)a̹id)=8\x(Sן 1 S(c̹f;v8:i)a)UR==8\CX%(x̸1jSן 1 S(c̹f;v8:i))(x̸2a)UR==8\CX%(x̸1jSן 1 S(c̹f;v8:i))(x̸2)a since(~aUR2̹""A=8\((CXq(x̸2)x̸1)Sן 1 S(c̹f;v8:i))aUR=>=8\"(x)Sן 1 S(c̹f;v8:i)aUR==8\"(x)Sן 1 S(S(a)a̹id):)IntheproSofwreusedtheequalityokCX2(x̸2)x̸1jSן 1 S(c̹f;v8:i)UR="(x)Sן 1(c̹f;v8:i))whicrhholdssinceforalluUR2UX(CXq(x̸2)x̸1jSן 1 S(c̹f;v8:i))(u)8=\%CXc̹f;v8:i((x̸2)Sן 1 S(x̸1)Sן 1(u))UR=8=\%CXfG((x̸2)Sן 1 S(x̸1)Sן 1(u)v̹id)UR=8=\%CXfG(x̸2Sן 1 S(x̸1)Sן 1(u)v̹id)since,t&fQ2UR(Vp \t)̹>8=\%"(x)fG(Sן 1 S(u)v̹id)UR="(x)Sן 1(c̹f;v8:i)(u):ک(2))AsϻintheproSofof2.2,letx~An_betheimageofAundertherestrictionmapU@20 V:!URKܞ20)and)In9:As!x~Aktheinducedsurjectivrecoalgebramap.By2.2,xthekernelofis)AB2+~.37Theٮgroup-likreelementsinA20arethecharactersofKܞ.37Hencethegroup-like)elemenrts6G(x=~A)ofxs~A9UarethecharactersofKwhichcanbSeextendedtoalinearmap)a:U6!URkQwitha2A.)LetQbSeacrharacterofKwithA̹ 6=0.Inthenotationoftheproofof(1),;the)matrixcoSecienrtc̹f;visanelementinAsuchthatforallxo2Kܞ, c̹f;v ^(x)=(x).)Thereforeisagroup-likreelementofx~A .15٠r-)Bywqde nition,|thespaceofleft-inrvXariantswqistheeigenspaceA̹ ,sinceforallaUR2A,)CP Lbn9(a̸1)\ a̸2V=UR aFifandonlyifforallx2K#qandu2U@,ga(xu)=CPa̸1(x)a̸2(u)=)(x)a(u). BSimilarlyV,theespaceofrighrt-invXariantsis̹ _A. BInparticularforall)aUR2A̹ ,n9(a)="(a),andwreseethatn9(A̹ )=kg.)HenceMbry(1),x3~Au=~n9(A)=CL 2XҸ(K;Cmr)9kg,andMX(K5;C5)isthesetofallgroup-likei)elemenrtsofx~A .8This nishestheproSofof(2)sincexAP԰'=x~A%u.@@2k^=In}thenexttheoremwreconsiderthespSecialcaseof2.1whenK+=OZkg[x]forsome(gn9;1)-^=primitiveelementHx2U@,`hthatis(x)=gY x+x 1,`hggroup-likreinU@.SyThenK%yisa^=commrutative/leftcoidealsubalgebraofU@. IfCisatensorcategoryof nitedimensional^=left_U@-moSdules,} wrecallanelementx2UC5-semisimple_ifforallV2C,} thelinearmap^=V!URVp,5vË7!xvn9,isdiagonalizable.Whenxis(1;gn9)-primitivre,thatis(x)UR=1( x+x gn9,^=thenkg[x]isarighrtcoidealsubalgebraofU@.^=Theorem2.40Lffet"UbeaHopfalgebraoveranalgebraicallyclosed eldkg,gJ[agroup-like^=elementandxa(gn9;1)-primitive(rffesp. 6(1;gn9)-primitive)elementofUҳandCEatensor^=cffategorylof nitedimensionalleftU@-moffdules.De neA:=U20RAC,{%BZ:=fa2AjaIx=0g^=andxLr35AWp:=URA=XAB2+/(rffesp.fiA=B2+~A).Assume35thattheantipodeofAisbijective.?D(1))The35followingarffeequivalent:j(a)x35isC5-semisimple.j(b)xA$is35spffannedbygroup-likeelements.)If35thesecffonditionshold,thenAisfaithfully atasarightandasaleftB-module.D(2))If{C.issemisimpleandBisasimpleobjeffctinM2ARABt(resp.?\̹B M2A"Ԩ),thenBvB:==IU }.^=ThenčD(1))KL isokaleftcffoidealoksubalgebraofU@,xandifUOispointed(ormoregenerallyUOis)faithfully35leftcffo atoverU=IU@),thenIU6=URKܞ2+U@.17r-D(2))If35S(I)2VURI,thenKܞ2=URKܞ.:^=SProof:kک(1))Since۳IUisarighrtidealandcoideal,ޱthesetofrightU=IU@-coinvXariantelementsK)is$Ialeftcoidealsubalgebra.IfUe-ispSoinrted,2thenby[M29,21.3]Ue-isleft(andright))faithfully,co atorver,U=IU@.HencewreknowfromTheorem1.1(appliedtoA2opjcop))thatIU6=URKܞ2+U@.P΍ک(2))First9notethatKܞ2+ g4nIU:sinceforanryx2Kܞ,ޟCPx̸1t nx̸2=x n1,hencer9x="(x)n1)inU=IU@.8ByKoppinen'slemma1.4,S(Kܞ2+U)UR=UKܞ2+.8ThenZӍ<}K(Kܞ +) VUR(U@Kܞ +) k=S(Kܞ +U@) ; sinceUKܞ2+ Nh=URS(Kܞ2+U)pS(IU@) ; sinceKܞ2+ NhURIUk=S(I) UpIU; sinceS(I)2VURI\!:)Clearly8Kܞ2 isagainaleftcoidealsubalgebrawith(Kܞ2)2+ R=6F(Kܞ2+)2,Kandwrehave)shorwnthat(Kܞ2)2+xU6URIU@.8ThereforewTyKܞ URU@ co Ua>=(K-:q% cmsy6 )-:+nU8MU@ co Ua>=IU#=K5:)Sinceisaninrvolution,wegetKܞ2=URKܞ.l@2Pύ^=As anillustrationof3.1(1),.3letgn9;hbSegroup-likreelementsofUaandxK2Uwith (x)=^=g4 qx+x h./Thenkgxisaone-dimensionalcoideal,uk[xh21 \|]isaleftcoidealsubalgebra,^=andxU6=URkg[xh21 \|]2+xU@.:^=Remark3.2*sLet8UybSeaHopf-algebraandCTatensorcategoryof nitedimensionalleft^=U@-moSdules..FVoranryleftU-moSduleVhiletxV beVhiasanadditivregroupwiththefollowing^=leftU@-moSdulestructure,denotedbry?:8FVoranyvË2URVandu2U@,u?vË:=URS(u) vn9:^=Inոparticular,xV%isacomplexvrectorspacewith s?vË:=FUR vCforcomplexnumbSers .x1V}is^=therestrictionofVtoU+viatheringisomorphism$U6!URU;u7!S(u) :^=A>tensorJcategoryCwillbSecalledatensor^-cffategoryJifforallVB2fCalsoxOV62fC5.LetC^=bSeatensor-category.8Then189r-ک(1))AcN:=U2@0RAC .isFaHopf-algebrawith-structurede nedbrya2(u)cN:=щfe+؟ 3/a(S(u))4lforFall)aUR2AanduUR2U@.ک(2))IfIAU\isacoidealwithS(I)2iEI,(KthenBDG:=fa2Aja;I=0g,(Kthealgebraof)in nitesimalinrvXariantsde nedbyI,isa-subalgebraofA.^=SProof:ک(1))TheZfullHopfdualU@20 [}isaHopf-algebrawith-structureasdescribSedaborve.A)isclosedunderthe-structuresinceforallV2URC5,ƻvË2Vp,linearfunctionalsfonV)anduUR2U@,c ڍf;v ^(u)UR=щfe2 3/fG(S(u)vn9)9}=ct:rf;v @(u);)whereِct:rf;v0isthematrixcoSecienrtofxV)%andthelinearfunctionalWh:*fonxVisde nedD)bryWyR*f O(wR)UR:=щfe 3/fG(w) /forallw2x WURV =URVp.ک(2))iseasytocrheck[KD,1.9].@2^=Let UNfbSeaHopf-algebraandCatensorcategoryof nitedimensionalleftU@-modules.^=WVecallCtunitarizableifforallV2URCthereisahermitianinnerproSducth;i:VW&V!URKC,^=conjugatelinearinthe rstandlinearinthesecondvXariablesucrhthatforallxY#2U-and^=vn9;w2URVp,hxv;wRiUR=hvn9;x2wRi.^=Corollary3.33uWLffetjUbeapointedHopf-algebra,Caunitarizabletensorcategoryof^= nitedimensionalleftU@-moffdules,andI#UhacoidealwithS(I)2I.KUDe neA:=U2@0RAC^=and35BX:=URfa2AjaIF=UR0g.fiThenD(1))K1:=URU@2co 2Ua>=IU$9isAaleftcffoidealAsubalgebraofU@, BX:=URfa2Aj8x2Kܞ:aRxUR=a"(x)g)is35arightcffoideal35subalgebraofAandKisC5-semisimple.D(2))A=XAB2+*iscffosemisimple, 8Aisfaithfully atasaleftandarightB-moduleanda)dirffectsumof nitelygenerffatedandprffojectiveleftB-moffdulesandrightB-moffdules)as35in2.2.^=SProof:19 r-ک(1))SinceUpispSoinrted,wegetfrom3.1(1)thatIU׺=Kܞ2+U@.oHenceBBisalsotheset)ofin nitesimalinrvXariantswithrespSecttoKܞ2+.;MoreorverKܞ2 G&=Kby3.1(2).;Let)V2C-andz]h;ithehermitianinnerproSductonVW<Vp. WVeharvez]toshorwthat)VeisCsemisimpleasamoSduleorverCKܞ. ԲLetW%2VbSeaKܞ-submodule. ԲThen)WƟ2? =Yfvc2Vj8wH2W:hvn9;wRiY=0gisaKܞ-submoSdulebecauseforallxY2Kܞ,)vË2URWƟ2?>,andw2W,shxvn9;wRiUR=hv;x wRi=0)sincex2 2IKܞ. mHenceV=Wz3WƟ2?eCisadirectsumofKܞ-moSdules,andVCuis)Kܞ-semisimple.7ک(2))follorwsfrom(1)and2.2"@2 ^=Remark3.4*sCorollary΃3.3appliestomanryrecentexamplesofquantumhomogeneous^=spaces.1.)In5general,#ifRgisasemisimpleLiealgebraandU6=URU̹q(Rg)"Pistheqn9-deformedunivrersal)enrveloping[KalgebrawithpSositivrerealqË6=UR1,wthenU/isapoinrtedHopf-algebrawith)standardEe-structure,\andthetensorcategoryof nitedimensionalleftU@-moSdules)oftrypSe1isunitarizable[CPz,10.1.21]72.)LetAUR=A(SU̹q(2))bSethefunctionalgebraoftheqn9-deformedspecialunitarygroup)SU@(2)~qandassume0X[KS,4.5]de nedarighrtcoidealsubalgebraB=#B̹ 3Abry)in nitesimalVoinrvXariantswithrespSecttooneskew-primitiveelementx̹S.|6Theyshow)thatthealgebrasB̹canbSeidenrti edwiththefunctionalgebrasofPoSdlea?s'quantum)spheresS2׸2RAqI{c C@for0-c1,E[P].xCInthissituationalltheassumptionsin3.3are)satis edG cmmi10Uq#9DtGGcmr17(slW (2))b#^=LetRkbSeanalgebraicallyclosed eldandletqbeanon-zeroelemenrtofkg,lwhichisnot^=aroSotofunitryV.3ForallpositivreintegersnletE̹n ibSetheunitmatrixwithnrowsand^=columns.^=Firstwrerecallsomede nitionsandresultsof[K "w].^=FVoranryintegern,set[K "w,p.8121] =ŵ[n]UR=ōqn92n1qn92nQmfe+ߋ  HNqqn914=qn9 n1+qn9 n3+UN+qn9 n+3:%+qn9 n+1}: ^=TheHopfalgebraU̹q(slC(2))isgeneratedasanalgebrabryE;FS;K5;Kܞ21#withrelations!KܞK 1l=URK 1 9K1=1;KEK 1l=qn9 2.=E;KFK 1l=qn9 2 ʵFS;EFLnFE i=ōKFKܞ21Qmfe/  7qqn91^=andcomrultiplicationde nedby,^=(Kܞ)UR=K .C6.V%0lk0*c.c.c. _.$ϟ.).:6.>.C6.S}.XC].\=.mأ.mأ.mأ.Q0#09?S[n]lk0fNtjeC1tjeCtjeCtjeCtjeCtjeCtjeCfitjeA*d;21%r-^=and(Qg̹e;n 4(Kܞ)UR=efNC0BBBBBBfi@ qn92n+0Ejb0F0v0#+qn92n2Ejb0F0R.R.R.(.->.1מ.Fz..K.O.l5.l5.l5.L~.L~.L~.v0+0E_tqn92n+2F0v0+0Ejb0qn92nfNC1CCCCCCfiA:1v^=HererwreneedtherightactionandusethetranspSosedmatrices.WVe rstconsiderthecase^=eUR=1.8Letq}xUR= (Kܞ 11)+ OEKܞ 1+ F#^=bSePa(Kܞ21 9;1)-primitivreelementofU̹q(slC(2)),where , O,and lare xedelemenrtsofthe^=ground; eldkg,PPwhicrhdonotvXanishsimultaneouslyV.,Thenthe(right)actionofxonthe^=crhosenbasisofV̸1;n/pcanbSerepresentedbythematrix=,ȍ>M̹n:=fJURC0URBURBURBURBURBURBURBURBfiUR@Q(qn92n]1) m^p 0T.V0*GDqn92n [n] S I(qn922n1) a[2] v.V.6.L../././.&$0Pqn922n[n1] CE(qn924n1) v.V.6.L...V0(\.(\.(\.e.jv.oV.s6.xN.!.r.SR.2..v.V.6.L..'![n] &$0i0fqn92n2[1]  (qn92n11) fJK4C1K4CK4CK4CK4CK4CK4CK4CK4CfiK4AVJ3:<Í^=TheG8/}h! cmsl12quanrtumplanek̹q[a;b]8isthekg-algebrageneratedbytheelementsaandbwiththe^=relationbaUR=qn9ab[K "w,ChapterIV].ItisaU̹q(slC(2))leftmoSdulealgebra,wheretheaction^=of*U̹q(slC(2))isgivrenin[K "w,:ChapterVISI.3].HereweneedthecorrespSondingrightaction^=givrenbyQ^=bCE i=UR0;aE=URb;bF=a;aF=0;bK1=qn9b;aK1=qn9 1 ʵa;bKܞ 1l=qn9 1 ʵb;aKܞ 1l=qn9a:^=Thesquanrtumplanehasanaturalgradation,fgivenbythedegreesofthemonomials,^=therefore9therearenozerodivisors.$:Letk̹q[a;b]̹n kdenotethevrectorsubspaceofhomoge-^=neousQpSolynomialsofdegreenink̹q[a;b].Thenk̹q[a;b]̹nWisasimpleU̹q(slC(2))righrtmoSdule^=isomorphictoV̸1;n/p[K "w,TheoremVISI.3.3].K ^=Lemma4.1/9(1)DzTherffe35isanonzeroelement2URk̹q[a;b]̸29suchthat?x=0.rD(2))ForBallz;2rIk̹q[a;b],FwherffeBJ>x=0,theBeffquation(zs)>x=(z?!x)Lholds,FthatBis,)the35actionofxisB̹X-rightlineffar,whereB̹X r=URf2k̹q[a;b]:?xUR=0g.D(3))The35eigenspffacesoftheactionofxonk̹q[a;b]̹n ۅareone-dimensional.225r-^=SProof:ک(1))ThedeterminanrtofM̸2vXanishes.ک(2))ThequanrtumplaneisamoSdulealgebraofU̹q(slC(2)):Q(zs)xUR=(z3Kܞ 1 9)(?x)+(zx)(?1)UR=(zx)s:ک(3))Lete&l2kg.kandletE̹n+1MbSetheunitmatrixwithn+1$'_rorwsandcolumns.Inthe)cases, v6=̪0or s6=0the rstorlastncolumnsrespSectivrelyofM̹n@E̹n+1Kare)linearlyLindepSendenrt. ^Inthecase Z= V=0LtheremustbSeu6=0, andLinthe)diagonalKmatrixM̹n E̹n+1"atmostoneenrtryinthediagonalvXanishes,forsinceq)isnotaroSotofunitryV,allentriesinthediagonalarepairwisedistinct.@2^=FVorconrvenience,we xasolutionpƉzm:qZoftheequationx22 I=q andde neqn92l:=pƉzm:q-mDZ2l .for^=l2Fu1콉fe@'2 KZ.8FVoreacrhnonnegativeintegernsetI̹n:=URfFu33n33콉fe(P't&2;1Fu۹n۟콉fe(P't&2 9^;:::ʜ;Fu31n31콉fe(P't&2 9\1;Fu31n31콉fe(P't&2u~;g.^=Prop`osition4.2HT-(1)]LffetP̹nP(Yp)UR=det(M̹nRYE̹n+1)=(1) n+1(Yp n+1+z̹nPYp n h+d̹n1Yp n1+:::uH))bffe thecharacteristicalpolynomialofM̹nP. -ThenfornH2 thepolynomialP̹n2)divides35P̹nP.D(2))Fix35Rn2URkRsuchthat܍RJ 2.=UR  2~;+ō 4 O qn921۟Qmfe32{  (qqn91 ʵ)28C:>)Then35M̹n ۅhasthen+135(notneffcessarily35distinct)eigenvaluessvp̹r:=ō Qmfeȟ  f2 H(qn9 rCqn9 r ) 2j+ō1۟Qmfe  2 (qn9 2r Gqn9 2r)RJ;)for35r2URI̹nP.^=SProof:23Frr-ک(1))LetbSeanfG-foldzeroofP̹n2,i.e.thedimensionofthegeneralizedeigenspacefor)the1eigenrvXalueink̹q[a;b]̹n2isfG,andletv̸1;:::ʜ;v̹f tPbSeabasisofV2pn2RA!<.p{Therefore)thereexistsapSositivreintegertsuchthatv̹G?(x)2t=UR0for=1;:::ʜ;fG. StartingfromamatrixgË=UR(܍ o~33 v~~* Js㎍ d~r ;P㎍ ~r *<a)inGL(2;KC)(butKCcanbSereplaced^=brykg),theelementQD=UR)~ W =~* 5E^+()~ W~*+W[~* f~  m)ō'~KF133Qmfe%N  qqn91*_+V~ W ~* jqqn9 1 ʵFK!^=isYconsidered(theauthorswriteqn92H forKܞ,#Ɵp#ȟƉzm:qǮX̸+xqn92H$=2forEpandp[Ɖzm:qAX̺qn92H$=2CrforF).By^=constructionoftheeigenrvectorsoftheleftactionofDonthequanrtumplane,%itisshown^=thatthedistincteigenrvXaluesare Hd̹m(gn9)UR=ōq2m q2mQmfe0s  qq18()~ W~* 49q m W[~* f~  mq mu)"Vi^=whereY2m2KZ,ifM~ W ~* T!<|@qn922tW ~*@ "~t 6=0forallt2KZ(thenDiscalleddiagonalizable).This^=leftactionisrepresenrtedbyamatrixobtainedfromM̹n byswappingthej-thcolumn^=withthe(n+1j)<{-thcolumnandthej-throrwwiththe(n+1j)<{-throrwfor1URj%^=(n]+1)=2GandthereforehasthesamecrharacteristicalpSolynomialasM̹nP,h.where h=/_~ ㎍ ^Ҹ~r  ĸ+㎍,r~r {~.< [fe)'qI{q1U,^= 33=qn921 :c~ ʵ W~* 1~, 0=)~ W =~* 5.ConsequenrtlyV,inthecase2~ W ~* vIqn922tW ~*@ "~t ԰6=0forallt2KZthereisthe^=factorization$Q0P̹nP(Yp)UR=(߹n=2(X(gn9)Y)(߹n=21(gn9)Y)(ߺn=2(gn9)Y):^=UptothecasesD<2dkgEorD2dkgFKܞ,theproSducts~ W M~* vandW~* ~ @ !donotvXanishsimrultane-^=ouslyV,4andWifexactlyoneproSductvXanishes,thenD%isalwraysWdiagonalizable.0Nowassume^=that`~l W~* R4 *YO~U W1~* <6=ɯ0.,InlthefactorizationofP̹nP(Yp)bSothsidesarepolynomialsin`~ R4,W~* H,k~ u,W~*^=.@The[equationisvXalidforanin nitenrumbSer[ofvaluesfor~ C(whenW ~* ,~w; 6,W~*w;yare xed),^=therefore}itisanidenrtity}ofpSolynomialsin~ .A}similarreasoningholdsforW.~* T,U~ R,W~~* ;.Thrus26uFr-^=߹n=2(X(gn9);:::ʜ;ߺn=2(g)ưarethenotnecessarilydistinctzerosofP̹nP(Yp)inanrycase,evenifg^=isqanarbitrary2]2qmatrix.1%cannotbSediagonalized).^=Inordertoprorve(2)8%)(1),dassumethereisapSolynomialP̹n #withadoublezero.6DThen^=bry 4.2therearedistincthalfintegersld@andmsuchthatlM+}misaninteger(bSecause^=lC;mUR2I̹nP)and̹lw=UR̹m.8Henceō Qmfeȟ  f2(qn9 l:qn9 l ,y) 2j+ō1۟Qmfe  2 (qn9 2l zqn9 2ll})Rn=ō Qmfeȟ  f2 H(qn9 m qn9 mu) 2+ō1۟Qmfe  2 (qn9 2m]qn9 2mOy)RYD))ōs- s-Qmfeȟ  f23((qn9 2l z+qn9 2l%qn9 2m]qn9 2mOy)+ōR۟Qmfe ԟ  l2(qn9 2lqn9 2l%qn9 2m]+qn9 2mOy)UR=07).(qn9 lKm8qn9 ml=)(ō Qmfeȟ  f2(qn9 lK+mqn9 lKmM)+ōRQmfe ԟ  l2 (qn9 lK+m+qn9 lKmM))=UR0.)(qn9 lKm8qn9 ml=)̹lK+mV=UR0:^=SinceqisnotaroSotofunitryV,thisyields̹lK+mV=UR0.TheassumptionRn6=0implieslL+ m6=0,^=whence0isadoublezeroofP߸2jlK+mj;,, andbrytheLemma,thegeneralizedeigenspaceof0^=doSesnotequaltheeigenspaceinV߸1;2jlK+mj h.FinallyV,condition(3)isequivXalenrttoR6=g|0^=and̹nP̺n6=UR0forallpSositivreintegersn,i.e.tocondition(2).{227%r-^=ThereisasimilarstatemenrtabSout(1;Kܞ)-primitiveelementsofU̹q(slC(2)).+^=Remark4.591.H _WVeconsideryË:=URxKܞ.8ThenyXis(1;K)-primitivreand]M @0ڍn5d$:=K8̹e;n 4(yn9)UR=̹e;n(Kܞ)M̹n=4[ 7=fJK8C0K8BK8BK8BK8BK8BK8BK8BK8BfiK8@V7(1qn92n ) qn92n2 TB0,4;_0*g[n] n(1qn922n) 霱qn92n4[2] (`.,.1t.6+T.:4.a1.a1.a1.o- 0o[n1] m(1qn924n) (`.,.1t.6+T.:4._0p{B.p{B.p{B.M..~ݟ...p. .z.;Z.:.(`.,.1t.6+T.:4.Oqn92n [n] o- 0 TB0)Zz[1] Jrb(1qn92n) fJ|C1|C|C|C|C|C|C|C|Cfi|A:9Ǎ)LetvM2@00RAn !thematrixobtainedfromM2@0RAn lafterreplacing bry andinrterchangingv )andW .pComparisonwithM̹nyieldsthatforalli;j%URn+1, xqI{-:1[fe-' 2*(qI{q1 ;;)2Hzif h6=0#Z c1Hzif h=0(and O n6=0)"}獑)(inLthe-inrvXariantLcasewreautomaticallyhave g2KRandqn9 i= O,gwhencec=)(fh j xj33ʉfe$C' (qI{q1 ;;)'0)22 is;nonnegativre.*ThespSecialcasesfornegativecin[P]donotoSccurbe-)causeethenthe nitedimensionalquanrtumspherescannotbSecanonicallyembSedded)inrtothefunctionalgebraas-invXariantsubalgebras.^=Ifx(oryrespSectivrely)isdiagonalizableonallmodulesV̸1;n D,thenthefollorwinggen-^=eralargumenrtshowsthatitisdiagonalizableonallmoSdulesV̹e;n 4. Letkgw8beaone-^=dimensionalU̹q(slC(2))moSdulewithbasisfwRg,9sucrhthatE+w=UR0,Ftw=UR0,Kw=URewR.^=Then4*V̹e;n-CexK=`kgw 5V̸1;ndCex=V̸1;n z kwR,Xwhere4*U̹q(slC(2))actsdiagonallyonthelattertrwo4*moSd-^=ules(andthesecondisomorphismisduetotheHopfalgebraautomorphismofU̹q(slC(2)),^=whicrhmapsEandFntoEandFnrespSectivrelyandleavesKFunchanged.^=Prop`osition4.6@mLffetU|beaHopfalgebra,g2{U|agroup-likeelement,x{2U|a(gn9;1)-^=primitive(rffespectively(1;gn9)-primitive)element,kgwܿaone-dimensionalleftU@-modulewith^=bffasis*fwRgandV=a nitedimensionalleftU@-module.cIftheactionofxonV=isdiagonal-^=izable35thensoisthediagonalactiononkgwR Vϥ(rffespectively35VG kgwR).^=SProof:'Assume%xis(gn9;1)-primitivre(TheproSoffor(1;gn9)-primitivreelementsissimilar).^=Since)skgw|isaone-dimensionalU@-moSdule,Pthereisamap:U6!URksucrhthatu w=(u)w^=forUalluz82U@.yLetUv̸1,:::ʞ,v̹n bSeabasisofVconsistingofeigenrvectorsUoftheactionofx,^=i.e.xv̹j\=UR̹jf v̹jPforj%=1,:::ʞ,nand̹j\2kg.8Then^=x[(w v̹jf )UR=(gS[wR) (xv̹j)+(xwR) v̹j\=UR(gn9)w ̹jv̹j$+(x)w v̹j\=UR((gn9)̹j+(x))w v̹jf ;^=whenceWthesetfwQ v̸1,:::ʞ,w v̹nPgisabasisofkgw VconsistingofeigenrvectorsWofx.^=229r-ƍ^=5- ApplicationtoUq#(Q%np eufm10Qg r])b#^=Asinthelastsection,wreassumethatkisanalgebraicallyclosed eldofcharacteristic0.^=LetK(a̹ij u)bSeannnKmatrixwithinrtegercoecienrtssuchthata̹ii DO=2foralli,d a̹ijԊ6=0^=for.alliUR6=j,zand.therearerelativrelyprimeintegersd̸1;:::ʜ;d̹n2URf1;2;3g.suchthat(d̹ida̹ijڿ)^=isasymmetricpSositivrede nitematrix.^=Thrus(a̹ij u)istheCartanmatrixofa nitedimensionalsemisimpleLiealgebraRg.^=LetqË2URkCnC&f0gbSenotarootof1andde neq̹i,:=URqn92d8:izS.'Thestandardqn9-deformationU̹q(Rg)^=isthealgebrageneratedbryE̹id;F̹i;K̹i;Kܞ1ԍi#subjecttothefollorwingrelationsnc$K̹idK ܞ1ڍil=UR1=K ܞ1ڍi 9K̹i;K̹iK̹j\=K̹jf K̹i;c\%K̹idE̹jf K ܞ1ڍil=URqzn9a8:ij Fai UE̹j;K̹iF̹jK ܞ1ڍil=URqzn9a8:ij FaiF̹j;|/E̹idF̹jF̹jf E̹i,=UR̹ijō~K̹iKܞ1ԍi~Qmfe2F˟ R{q̹iqn91ԍi 5^=forcalli;j,>andtheqn9-deformedSerrerelation(see[J ;,5.1.1(vi)]bSetrweenctheE̹id'sresp.the^=F̹id'sSwhicrhwedonotneedexplicitlyV.qU̹q(Rg)&isaHopfalgebrawherealltheelementsK̹i^=aregroup-likreand6(E̹id)UR=1 E̹i+E̹i K̹id;(F̹i)UR=K ܞ1ڍi F̹i+F̹i 1^=for9alli.&FVor1moin9letU̹ibSethesubalgebraofU̹q(Rg)$;generatedbryK̹id;Kܞ1ԍi 9;E̹i;F̹i.^=ThenU̹iP,԰=dU̹q8:iw(slC(2))asHopfalgebras.^=Let C>bSethesemisimpletensorcategoryofall nitedimensionalleftU̹q(Rg) -modulesof^=trypSe1,BthatisalleigenvXaluesoftheleftmultiplicationwithK̹i}forall1jiareoftheformqn92mr,^=mUR2KZ\(seeforexample[J ;,4.3]).&ThedualHopfalgebraU̹q(Rg)*q%0 Zq%C(@istheqn9-deformedalgebra^=of|regularfunctionsonthesimplyconnected,connectedsemisimplealgebraicgroupwith^=Lie&!algebraRg.LLetx2U̹q(Rg)#wbSe&!a(gn9;1)-primitivreelementwhichisnotascalarmultiple^=of2gI1.)ThenthereissomeisucrhthatgË=URKܞ1ԍiLandx2U̹i isakg-linearcomrbinationof^=Kܞ1ԍi1,E̹idKܞ1ԍi(andF̹iA[CM@,TheoremA].WVecallxsemisimpleifmrultiplicationwithx^=isadiagonalizableopSeratoronall nitedimensionalleftU̹q(Rg)q-modules.Gt^=Lemma5.1'SForany1URinletVYbffea nitedimensionalleftU̹id-moduleoftype1and^=xUR2U̹i,U̹q(Rg)#GPa35(Kܞ1ԍi 9;1)-primitiveelement.D(1))Therffeexistsa nitedimensionalleftU̹q(Rg)-moduleWoftype1andaone-dimensional)left8U̹q(Rg)-moffdule8kgawithE̹jf a_=0,:6F̹ja=08andK̹ja_= ̹ja8wherffe ̹jŽ2_f1;1gfor)all351URj%nsuchthatVϥisisomorphictoaU̹id-submoffduleofkga Wand ̹i,=UR1.30Kr-D(2))x35issemisimpleinU̹iifandonlyifxissemisimpleinU̹q(Rg).@W^=SProof:Aک(1))Byě[J ;,10.1.14]ortheclassi cationofhighestwreightěmoSdulesofU̹q(Rg) d,Va iscon-)tainedFVromthedescriptionof nite)dimensional͠leftU̹q(Rg) i-moSdulesitisknorwnthatx~VPK԰43=!HHkgaE0 W,^where͠Wofisaleft)U̹q(Rg)|-moSdule)oftrype1andE̹jf aUR=0,OF̹ja=0,OK̹ja= ̹ja,O ̹j\2f1;1g)forallj(see)[J ;,1-4.3]).In#remainstoshorwthat ̹i2=X1.SinceVisoftrypSe1,1-thereisanon-zero)vË2URVsucrhcthatK̹idv=URqn92mrvѿforsomem2KZ. Leta w0bSectheimageofvѿinkga W.)ThenN qn9 mr(a wR)UR=K̹id(a w)UR=K̹ida K̹iw=UR ̹ia K̹iwR; )henceK̹idw=UR 1ԍi p qn92mrwR.8SinceWnisoftrypSe1,weconclude ̹i,=UR1.0ک(2))IfxissemisimpleinU̹id,5:thentriviallyxissemisimpleasanelemenrtinU̹q(Rg)!~.)ConrverselyRassumexissemisimpleinU̹q(Rg) S.5LetV0bSeanry nitedimensionalleft)U̹id-moSdule.By[J ;,}10.1.14]ortheclassi cationofhighestwreightmodulesofU̹q(Rg)ǵ,)Vdlisconrtainedina nitedimensionalleftU̹q(Rg) -moSdulex~V>.Hencemultiplication)withxisdiagonalizableonx~VKUandthenonVp,toSo.@2m^=Theorem5.20Lffetw1Nin, ; O; Uj2Nkg,andwx= (Kܞ1ԍi' 1)+ OKܞ1ԍi 9E̹i+ F̹i (2^=U̹q(Rg)Ǯnf0g35a(Kܞ1ԍi 9;1)-primitiveelement.fiAssume 22~;+4qn921 ʵ O =(qqn921)22V6=UR0.^=Lffet/GC|bethetensorcategoryof nitedimensionalleftU̹q(Rg) -modulesoftype1.ZDe ne^=AUR:=U̹q(Rg)*0 ZC(and35BX:=fa2AjaxUR=0g.fiThenthefollowingarffeequivalent:D(1))x35issemisimple.0D(2))Therffe35isnononnegativeintegernsatisfying"}  2~;+ O qn9 1 ʳf\C ōqn92n1+qn92n蒟Qmfe+  /gqqn91CɁf\C!K-*2SɃ=UR0:%D(3))A=XAB2+)lis35spffannedbygroup-likeelementsvQ(3)20)A=B2+~A35isspffanned35bygrffoup-likeelementsD(4))A35isfaithfully atasaleftB-moffdule31 ^r-vQ(4)20)A35isfaithfully atasarightB-moffdule不D(5))B;is35aB-dirffect35summandinAasaleftB-moffdulevQ(5)20)B;is35aB-dirffect35summandinAasarightB-moffduleD(6))B;is35simpleinM2ARABvQ(6)20)B;is35simplein̹B sM2A ȁ.㍑^=SProof:(1)()(2):8By5.1(2)andsection3.^=(1)?)(3)and(3)?)(4)follorwfrom2.4.^=(4)?)(5)follorwsfrom1.2.^=(5)?)(6)is1.3(2).^=(6))(1):}By 2.4(2),assumption(6)impliesB+=fa2Aj9n1:ax2n 8J=0g.BLet W^=bSeanry nitedimensionalsimpleleftU̹q(Rg) WY-moduleoftrype1.BBy2.1, part(2)and(3),^=W -%WƟ2P ԰ =aCܞ2W=as5righrt(andleft)U̹q(Rg)k-moSdules.$dHenceiffu$x2n=UR0forsomen1and^=fd2WƟ2a,}8then_fB{|x=0.OrequivXalenrtlyV,}8if:WƟ2 ~!WƟ2 isrighrtmultiplicationwithx,^=then>KerY()L=Kerrg(2nP)>forallnL1.PIf> n9:W'!W?isleftmrultiplicationwithx,then^= n92 %=,ѯandzwregetim( n9)=im( 2n),ѯorzequivXalenrtlyV,Ker( n9)=Ker}( 2n)zforalln1.^=ThrusqforallsimplemoSdulesinC5,henceforallmodulesWinC% (becauseC% issemisimple),^=wrehaveshown`;[fw2URWjxw=UR0g=fw2Wj9n1:x nRw=0g:^=WVewranttoshowthesamestatementoverU̹id.LetVbSea nitedimensionalleftU̹i-moSdule^=oftrypSe1. dBy5.1(1),tV"isisomorphictoaU̹id-submoduleofkga W,twhereW(Eisaleft^=U̹q(Rg)-moSduleoftrype1andE̹idaUR=F̹ia=0,K̹ia=a.8Hence`VxaUR= (K ܞ1ڍi1)a+ OK ܞ1ڍi 9E̹ida+ F̹iaUR=0;^=andtheactionofxonanryelementa wR,w2URW,isgivenbyx(a wR)IK6=\nK ܞ1ڍi 9a xwR+xa w since(((x)UR=K ܞ1ڍi x+x 1;IK6=\na xwR:^=Inparticular,CfvË2URVjxv=UR0g=fv2Vj9n1:x nRv=0g;^=sincethisequalitryholdsforW.!Thusweseethatcondition(2)oftheorem4.4issatis ed^=(hereJwreusetheassumption 227l+c4qn921 ʵ O =(qqn921)22V6=UR0).HenceJbry4.4(1),jmultiplication32!ٷr-^=withxisdiagonalizableonallU̹q8:iw(slC(2))-moSdulesoftrype1,andxisdiagonalizableasan^=elemenrtinU̹q8:iw(slC(2))by4.6.^=De neyË:=URxK̹id.#Thenyis(1;K̹i)-primitivre,andBX=URfaja(yË=UR0g.#WVenowrepSeatthe^=previousargumenrtswithxreplacedbyyn9.8Considerthestatement^=(1)20yXissemisimple.^=WVeharveshownin4.5(1)and4.6thatxissemisimpleinU̹i>ifandonlyifyHWissemisimple^=inU̹id.8Hence(1)()(1)20bry5.1(2).^=(1)20 3)?(3)20and(3)20)(4)20follorwfrom2.4.^=(4)20 3)?(5)20follorwsfrom1.2(forA2op x).^=(5)20 3)?(6)20is1.3(2)(forA2op x).^=(6)20 3)?(2):8By2.4(2),BX=URfa2Aj9n1:ayn92n k=UR0g.HenceforallW2URC5,;fw2URWjyw=0g=fw2Wj9n1:yn9 n1w=0g:^=Let[VbSea nitedimensionalU̹id-moduleoftrype1.7Thenbry5.1(1),jVisaU̹id-submodule^=ofkkga*~ WH1forsomeleftU̹q(Rg) e4-moSduleWandE̹iX*~a=0,\F̹i*~a=0,\K̹i*~a=a.l*Since^=yn9aUR=xa=0,(y)=1 y+y K̹id,and(x)UR=Kܞ1ԍi x+x 1,wrehaveXyn9(a wR)UR=a ywR+ya K̹idw=URa ywR:^=Hence233[fe$C'(qI{q1 ;;) '0E̹id;/=r x231[fe.0'(qI{2t"1) #F̹i;K̹i)ifj%=i,#Z c(E̹jf ;F̹j;K̹j)ifj%6=i"mapsFuz22콉feu' xtog8~x PandinducesanisomorphismofBHtothealgebrafaUR2Aja! ~x $v=UR0gofin nitesimalinrvXariantswithrespSectto~x .33"Sr-L IfBL = O =0thenxisascalarmrultipleofKܞ1ԍi 9E̹i&orF̹id.?Thenx(andK̹ix)actsnilpSotenrtlyonall nitedimensionalU̹id-modulesoftrype1andbothxandK̹idxf_cannotbSesemisimple. Moreorverf_thenrumericalcondition(2)doesnothold.8Inthiscase5.2remainstrue.2.)InthecaseRgz=sl̸2themoSdulestructureis atforanrychoiceof ; O; .z(ItcanbSe)shorwnthatAistheascendingunionoffreemoSdulesoverB).(V^=Referencesb#^=[B]eET.tBrzeziSvnski,,Quanrtumhomogeneousspacesasquantumquotientspaces,,J.Math.eEPhrys.37(1996),2388{2399^=[CM]W.LChin,e6I.M.Musson,TheCoradicalFiltrationforQuanrtizedEnvelopingAlge-eEbras,J.LondonMath.SoSc.(2),53(1996),50{62^=[CP]eEV.Chari,A.PressleyV,AGuidetoQuanrtumGroups,CambridgeUniv.Press1994^=[DG]M.Demazure,PV.Gabriel,GroupSesAlgrsebriques,TomeI,Masson,Praris,1970^=[D]eEM.ٲDijkhruizen,uSomeremarksontheconstructionofquantumsymmetricspaces,eEActaAppl.Math.44(1996),59{80^=[DN]sM.ܧDijkhruizen,'M.Noumi,Aifamilyofquanrtumprojectivespacesandrelatedqn9-eEhrypSergeometricorthogonalpolynomials,toappearinTVrans.Amer.Math.Soc.^=[J]eEA.Joseph,cQuanrtumgroupsandtheisprimitiveideals,cSpringer,Berlin{NewYVork,eE1995^=[K]eEC.Kassel,InrtroSductiontoQuantumGroups,GTM155,SpringerNewYVork1995^=[Ko]eEM.AKoppinen, 'CoidealssubalgebrasinHopfalgebras:freeness,inrtegrals,smasheEproSducts,Comm.Algebra21(1993),427{444^=[KD]T.̲K.KoSornrwinder,ҰM.Dijkhuizen,ҰQuantumHomogeneousSpaces,ҰQuantumDu-eEalitryandQuantum2-Spheres,Geom.Dedicata52(1994),291{315^=[KS]eEA.LU.Klimryk,lzK.SchmSvudgen,lzQuantumgroupsandtheirrepresentations,lzSpringer,eEBerlin199734#yr-^=[KV]L.I.KorogoSdskyV,[L.L.Vaksman,[QuanrtumG-spacesandHeisenbSergalgebra,[in:eEQuanrtumGroups,%ProSceedingsofWVorkshopsheldintheEulerInternationalMath-eEematicaliInstitute1990,PV.P.Kulish(ed.),LecturenotesinMath.1510,pp.56{66,eESpringer,Berlin,1992^=[M1]eEA.MasuokXa,QuotienrttheoryofHopfalgebras,in:oAdvXancesinHopfalgebras,J.eEBergen,S.Monrtgomery(eds.),Dekker,1994^=[M2]eEA.MasuokXa,OnHopfalgebraswithcoScommrutativecoradicals,J.Algebra144eE(1991),451{466^=[MW]A.MasuokXa,D.Wigner,FVaithful atnessofHopfalgebras,J.Algebra170(1994),eE156{164^=[M]eES.MonrtgomeryV,#zHopfalgebrasandtheiractionsonrings,CBMSRegionalConfer-eEenceSeriesinMath.82,Amer.Math.SoSc.1993^=[MSvu]eEE.1hMSvuller,CKonstruktionvronRechtscoidealunteralgebren,CDiplomaThesis,MunicrheE1995^=[NM]XwM.ONoumi,yK.Mimacrhi,Askey-WilsonOPolynomialsassphericalfunctionsoneESU̹q(2),<in:kQuanrtummGroups,ProSceedingsofWVorkshopsheldintheEulerIn-eEternationalMathematicalInstitute1990,-PV.P.Kulish(ed.),-LecturenotesinMath.eE1510,pp.98{103,Springer,Berlin,1992^=[P]eEPV.ProSdlea?s,QuantumSpheres,Lett.Math.Phys.14(1987),193{202^=[R]eEL.Rorwen,RingTheoryV,Vol.I,AcademicPress,Boston,1988^=[Scrh]H.-J.Scrhneider,PrincipalhomogeneousspacesforarbitraryHopfalgebras,IsraeleEJ.Math.72(1990),167{195^=[Sw]eESwreedler,Hopfalgebras,BenjaminNewYVork1969^=[T1]eEM.lTVakreuchi, RelativeHopfmoSdules|equivXalencesandfreenesscriteria, J.AlgebraeE60(1979),452{471^=[T2]eEM.TVakreuchi,@MHopfalgebratechniquesappliedtothequantumgroupU̹q(slC(2)),eEConrtemp.Math.134(1992),309{32335;r #S- cmcsc10R%n eufm10Q%np eufm10K msbm10Cu cmex10>K`yp cmr10<"VG cmbx10: b>G cmmi109DtGGcmr174K`yff cmr103Tq lasy100N cmbx12/}h! cmsl12.@ cmti12-!", cmsy10,g cmmi12+XQ cmr12K cmsy82cmmi8 |{Ycmr8q% cmsy6;cmmi6Aacmr64