; TeX output 2001.01.25:17037 YIN cmbx12SKEWٚDERIVATIONSOFFINITE-DIMENSIONALG`HALGEBRASٚANDACTIONSOFTHEDOUBLEOF!THEٚTAFTHOPFALGEBRAPK`y cmr10S.UUMONTGOMER*YANDH.-J.SCHNEIDER3؍61XQ cmr121.- cmcsc10IntroductionBहInthispapSer,g cmmi12kisa eldconrtainingaprimitiven-throSotofunity6!n9, wheren::>1. RWVe rststudyskrewderivXationsofcertain nite-6dimensionalikg-algebras,inparticularthealgebraA =k[u!", cmsy10ju22cmmi8n d\= O],6forW L2fkg.StudyingtheseskrewderivXationsisequivalenrttostudying6theqactionsofthen2|{Ycmr82-dimensionalTVaftHopfalgebraHB=URTnAacmr62 (!n9)[T|l]on6sucrhanalgebraA,whereQHB=URkghgn9;xjg n k۹=1;x n=0;xgË=!n9gxi;6ह(1.1)6gË2URG(HV),and(x)=x 1+g x,"(x)=0.BAs:anapplication,^wreshowthatactionsofuq(sl2)onAaredeter-6mined(brytheactionofaBorelsubalgebra.-actionsare:QB(I)GWVeGconsidernon-trivialactionsofxS~H8onann-dimensionalalgebra6AUR=kg(u)sucrhthatuisaneigenvectorundertheactionofgn9.0Allsuch6actions#Xarecompletelydeterminedbrytheactionofݑ~xonu,qandthe 营*o cmr91*7 &e62S.!MONTGOMER:YANDH.-J.SCHNEIDERY6हpSossiblevXaluesareexactly~x Qud=s2u2sn<,06=2kg;0sn1,1s6हinrvertibleRmoSdBn.BIn۴thiscase,ޱAUR=kg[uju2n= O]۴forsome 2URk,ޱand~x2n actstriviallyV,6thatisAisanHV-moSdulealgebra(seeTheorem2.2).`B(ISI)`If`Aisanarbitraryalgebrawithnonon-zeronilpotenrtelements,e6then'{foranrynon-trivialx3d~H-actiononA,valso~x2nK cmsy81!actsnon-triviallyV,6and eacrhpSower!n92i;0Aino1,TGis aneigenvXalueoftheactionof6gn9.MoreorverifAisn-dimensional,wreareinthesituationof(I)d(see6Theorem2.3).BInSection3wreextend(I)and(ISI)todescribSetheactionsofuq(sl2)D 6on A=kg[u],bforqNCa2n2th WŹroSotof1;Zhereuq(sl2)=kg[E;FS;Kܞ]isthe6 nite-dimensional?quotienrtofUq(sl2)de nedbyLusztig.7Theactions6areĸcompletelydeterminedbrytheactionofaBorelsubalgebra,Othatis6bryu=s2u2sn<.NThenD)extendstoa6n9-skewderivationofAsatisfyingDSq=88DJandDS2n 4=880.Morffeover6DS2n1-6=UR0() 6=0.BInadditionthefollowingidentitieshold,5forallpwith0URpn16:6(a)35D6u2p=URs2(P* p1 U_ i=0N 2ixi)u2p+s1.;6(b)35DS2q4u2p=ZBs22q0(P* p1 U_ iq1*=0! 2iq1)(P* (p1)+(s1) U_ iq2*=0@ 2iq2)(P* (p1)+(qI{1)(s1) U_ iq#=0V| 2iq#)u2p+qI{(s1)6(c)Lffetq0 "bethe(unique)q[Wsuchthat1q0 mݽ.iItiseasy6tocrheckbyinductionthatԍ}(u) p=URM@ p ]=qd * 2pu2p-'s2(P* p1 U_ i=0N 2ixi)u2p+s1 k0TOu2pΟq6हThrusa(u)2n=UR OIS[since 2n 1=1impliesP* n1 U_ i=0"O 2iͻ=0. EMoreorveraformula6(a)isalsoclearfromthiscomputation.7 &e64S.!MONTGOMER:YANDH.-J.SCHNEIDERYBहSincesDactsasan9-skrewderivXation,also21 ʵDS7isa-skrewderivXa-6tion.8Norwby(a),forallp, ̍-Y(n9 1 ʵDSn9)u p=UR  p 1 ʵ(D6u p])UR=  ps2(ԍp1},X ㇍i=0UV  ixi) 1u]u p+s1 0X-Y=P  p(p+s1)+is2(XUV  ixi)u p+s1߹=URD6u p5ڍ6हsince 21s=UR,Xusing(a).)TThrusn921 ʵDSË=DonallofA,Xwhicrhisthe6desiredrelationbSetrweenXandD.BTVocsee(b)and(c),~wre xpandproSceedbyinductiononqn9;notethat6(a)isthecaseqË=UR1.8Assumetheresultistrueforq1.Then6DS q4u p=URDS qI{1o(D6u p])=s2(ԍp1},X ㇍i=0UV  ixi)DS qI{1ou p+s1*@ह=T,s2(ԍp1},X ㇍i=0UV  ixi) qI{1䬹( (p+s2)󍍑zןX ㇍jq1*=0"K  jq1.8)( (p+s2)+(qI{2)(s1)󍍑 CX ㇍Tjqq7 q% cmsy61 z=0Sܝ  jqq71 )u (p+s1)+(qI{1)(s1),@ह=T,s2 q0(ԍdp1},KSX ㇍iq1*=0  iq1)( (p1)+(s1)󍍑7QX ㇍iq2*=03  iq2)( p1+(qI{1)(s1)󍍑X ㇍h2iq#=0Bq  iq#)u p+qI{(s1)!S6हwhicrhisthedesiredform. However,thedicultywiththisformal6computationMMisthatitmaryequalzeroforsomeqn9's.`TVoseewhenthis6happSens,Ŋwre rstconsiderthesetofpossibleexponenrtsofu,Ŋnamely6fp+qn9(s1)URj0qËn1g.BSincesf1isinrvertiblemoSdnandpis xed,Cthissetcontainsn6हdistinct,elemenrtsmoSdn.Thusforexactlyoneqn9,}op +q(s1)Py԰=H0Z΍6(moSd@n),&namelyforq!=q0PQ԰jԹ=9+pt(modn).FVorthisq0,&u2p+qq0*(s1),S= O2b6हfor someb;8thrusDS2qq0 'u2pT=s22qq0\qq0*;p O2b+,whereqI{;pDistheproSductofthe6summationterms.BWVeZ?mrustalsoconsiderthecoSecientsqI{;p 8Ϲ.AZgivensummationterm0_6टP*CQ(p1)+(r++ 0,nwhereOall i,2URkg,bSetheminimrumpolynomial6ofu.8ZThenfG(n9(u))UR=f(u)=0.8ZSince isaprimitivren2th `ѹroSotof1,git6follorwsthat 1;:::ʚ; n1=UR0.8ThusfG(x)UR=x+ 0;nowset =UR 0.BPrart%(d)followsfromLemma2.1(c),tYandtheconversestatement6alsofollorwsfromLemma2.1.bcffxff ̟ff ̎ ̄cff6WVepseenextthatwhenAhasnonilpSotenrtelementsandisn-dimensional,6wremayassumethatAisoftheformA|չ=kg(u)where(vu= u,and6soTheorem2.2applies.6Theorem2.3.LffetbAbeanarbitrarykg-algebrawithnonon-zeronilpo-6tentUelementsandassumethat2A2utk#(A)suchthatn92n 1=id.΢LffetD6bffe anon-trivialn9-skewderivationofAsuchthatDSB=SD,5wherffe6lisaprimitiven2th rffootlof1. ThenDS2n1rV6=0,*andallthepffowers62id;0URin1,35arffeeigenvaluesofn9.BIf?morffeoverAisn-dimensional,Bthenthereexistu2Aand06=6 O; n2URkRsuch35that6(a)35AUR=kg(u)andu2n= 2kg;6(b)35uUR=u,andsonhasorffderninA2utk#(A);6(c)35D6uUR= 1,DS2n1-6=0,andDS2n Q0=0.@Ǡ7 &e66S.!MONTGOMER:YANDH.-J.SCHNEIDERY6In35pffarticularAiscommutative.BMorffeover8ifu20-2__Aand O20x; 20Դ2kalsosatisfy(a),9(b),and8(c),then6for35some0UR6=2kg,35wehaveu20#=URu; O20ڹ=2nj O,35and 20ʧ=UR .BहBeforeaCprorvingthetheorem,~we rstmakesomepreliminaryobser-6vXations.YSince{n92n =idandkyconrtainsaprimitiven2th 6roSotof1,pcan6bSediagonalizedinEndk#(A).8Thrus"܍aAUR=n1 i=0Aid;"6हwherezCAi:=Ifa2Ajz na=2idag.NotethatD_ nAiAi1AV,*foralli.6FVor,4ifa2Aidڹ,then̚^a(Da)=21 \|D^a(̚a)=2i1AV(D^aa);vthrus6D6aUR2Ai1AV.6Lemma2.4.Lffet=ah2AisuchthatDa6=0butDS22a=0.CThenfor6all35mwith1URm1tandm,yand6(ii)istrueform,then(ii)istrueform+1.8Norw"ʍ>DS m+1va m Z=URD6(DS m a mĹ)= mD6(Da) m:Hֹ=[^ mĹ[(D6(Da))(Da) m1+((Da))(D(Da) m1@)]H=[^ mĹ[(DS 2:a)(D6a) m1@)+((Da)) 1ڍm1@(D(DS m1va m1@))]H=[^0:"6हHere]wrehaveused(i)forminthe rstline,* (i)form.1]inthenext6to_lastline,DS229çaй=0,and_(ii)formtoseethatD5ç(DS2m1ua2m1@)=6DS2m a2m16=UR0.BWVenorwshowthatif(i)istrueformand(ii)istrueform,+1,Mthen6(i)$istrueformS+1.)TSince$D㲹isan9-derivXationwithDS7=nD,for6b;cUR2A꨹wrehave,8 DS p(bc)UR=ԍ Pp},X ㇍jv=0G܍*p v~[Tj rG% 1+n92jCDS2pjm(b)DS2j(c)R۠7 ̍@7Y376हwhereꨟG܍ jp v~ Tj G2 1istheqn9-binomialcoSecienrtwithqË=UR.8Thus&t9'DS m+1va m+16=URm+1FX ㇍$jv=0G܍6m+1 v~$t-j/ҟG5ԟ1:n92jC(DS2m+1j aa2mĹ)(DS2jd@a)ꍍC'=VKHG܍[Jm+1 v~b0n,Gs8yI(DS2m+1va2mĹ)a+G܍*m+1 v~;H1G 8%n9(DS2m a2m)(D6a)URsinceDS22:a=0a)2n1=xf0.vSinceAhasnonon-zeronilpSotenrt6elemenrts,D6aUR=0,aconrtradiction.8ThusDS2n1AUR6=0.BWVe|marythuschoSoseaUR2Aidڹ,for|somei,sucrhthatDS2n1IaUR6=0.BNow60UR6=DS2ja2Aij \,Nfor|8allj%=UR0;:::ʚ;n1,NandsoAj\6=UR0foralljӹ.This6prorvesthelemma.ącffxff ̟ff ̎ ̄cff8BTVo. nishtheproSofofTheorem2.3wrecannowassumethatAis6n-dimensional.6Prffoof.ZByzzLemma2.4,nAjis1-dimensionalforalljӹ.VChoSoseanry0J6=6uwZ2A1.8Then u=u,andu2i4=2idu2i.8Allu2i46=wZ0sinceAhas6no&nilpSotenrtelements,andthustheelementsf1;u;:::ʚ;u2n1g&areakg-6basisdforA.InparticularA=kg(u)dandA0o=k1.Norwu2n Wd2An=A06हand#sou2n ^= `2kg.vNote#herethatanrysuchumustbSeinA1,2'asitis6a-eigenrvectorforn9,andsoisuniqueuptoascalarmultiple.BItzremainsonlytoshorwpart(c).SinceA0V=URkg1;D93u= 1zforsome6 n2URkg,andDS2n Q0=0.8ByLemma2.4,DS2n1-6=0.q%cffxff ̟ff ̎ ̄cffBWVenorwrephrasetheabSovetheoremsintermsofactionsoftheTVaftT6algebras6onA.tRecallthede nitionsofH=Tn2 (!n9)andx~Hin(1.1)B,6(1.2)O.r6Theorem2.5.LffetAbeann-dimensionalkg-algebrawithnonon-zero6nilpffotentcQelements,XandassumethatAisanxo:~H -modulealgebrasuch6thatX~x]/A|6=0.'Thentherffeexistsu|2Aand0|6= O; I2kN6isnon-zeroifandonlyifgn921 ʵaisnotcenrtralinA.BAssume>thatn9(a)]!=!21 ʵa.FThen>DSZ=!DS,dand>ifnisodd,d]whas6ordernanda2n iscenrtral,thenDS2n Q0=UR0.v6Prffoof.ZTheequalitryDS=Z!n9DAofollowseasilyV.BWriteD]=ZLaRa26हEndA,where#La;Ra denoteleftandrighrtmultiplicationbya.QThen6!n9(Ra)La Ĺ=La(Ra),Pandŕwreseefromtheqn9-bimomialformulathat6forallbUR2A,%EgDS n޹(b)UR=L nڍaP(b)+(Ran9) n(b)UR=a nb+(1) n!n9 33n(n1)338R\)uF eP2"f]n9 n(b)a n=UR0:G,bcffxff ̟ff ̎ ̄cffBह(2))Alltheassumptionsin(1)canbSerealizedformatrixringsA=6Mrb(kg).1Assumen6>2isoSdd.1LeteijJ,K1i;jrS,bethematrix6unitsofA.TVakrege=!n9e11 Q+QP*Vr U_Vi=2 =eiiI,Eanda=e1rf.Thentheinner6automorphism&#de nedbryghasordern,Nn9(a)UR=!a,Na2n=0&iscenrtral,6andihgn921 ʵa-=!n921aihisnotcenrtral.ThusihTn2 (!n9)actsonAwithnon-zero6skrew-derivXation3x.HoweverAќ6=kg(u)whenevrerr%*>1.Hencewresee6(with_nUR=rS22)thatTheorem2.5isnottrueifAhasnon-zeronilpSotenrt6elemenrts.LX3.ghAnapplica32tiontoFrobenius-LusztigkernelsBहWVe considerthespSecialcaseoftheFrobSenius-Lusztigkrernelasso-6ciatedtoUq(sl2),whenqPisaprimitivre2n2th YroSotof1,forn>2;]ait6is@a nite-dimensionalquotienrtofUq(sl2).;See[K "w,VIV.5.6].ThisHopf6algebra isalsocalledtherffestricted quanrtumenvelopingalgebraofsl2.6SpSeci callyV,`ǍluqP=URuq(sl2):=kghE;FS;Kܞi rؠ7 ̍@9Y6हwhereE;FS;KFsatisfytherelationsE2n =UR0;UPFƟ2n h=0;Kܞ2n @=1,andgb KܞE i=URqn9 2.=EK5;UPKF=qn9 2 ʵFK5;UPEFLnFE i=ōKFKܞ21[z/ ΍7qqn916हwithIcoactionEC=Es (\K+1 E,qF1Z=F" 1+Kܞ21av Fƹ,qand6K1=URKF Kܞ.BNotethatuqQisgeneratedbrytwodi erentcopiesoftheTVaftalgebra6as_in(1.1),|althoughwithtrwo_di erentchoicesfor!n9.NamelyV,|H1Y:=6kghKܞ21 9;FiPf?԰'=!Tn2 (qn922 ʵ)andH2 &C:=f?khKܞ21 9;EKܞ21iPf?԰'=!Tn2 (qn922.=).Onecan6thinkofH1iasb2x, the\Borel"subalgebraofuq,andsimilarlyofH2ias6b2+x.8WVecannorwapplyourresultsfromSection2touq.BInthespiritofTheorem2.2,bwregivetheformofallpSossibleactions6ofuq(sl2)onAUR=kg(u)sothattheactionofKzstabilizesthespaceku.Ow6Theorem3.1.LffetA4=kg(u)bffeann-dimensionalk-algebrffa,{forn4>6ह2.AssumeJ thatAisauq(sl2)-moffdulealgebrasuchthatK1u= 21 p u,6forsome w2kg,UandthatFVu6=0orE Vu6=0.Thentherffeexist6 O; ;2vkjwithEM ;6=v0EMand0s;t;lcn1EMwitht(1s)P԰=B1EM(moffd6n)35ands+lP԰ɹ=C235(moffdn)suchthatB(a)35u2n=UR 2kg, h=qn922tandKhasorffderninA2ut(A);B(b)35FLnuUR= u2sqandE^u=s2u2l!;B(c)SIfs.2=0Sandlq=.22,Śors=2andlq=0,Śthens2 N=qn9.Ifbffoth6s;lUR2,35thens+l=URn+2,35 6=0,andis2 n=ō 2s2^G(1 )[zhʡ m͍ O(qqn91 ʵ)(P* s2 U_ jv=0c jy)q۫=ō$.1 21[zf 3 O(qqn91 ʵ)(P* lK2 U_ i=0 ixi)l{:E6Conversely,5anyguchoicffeof O; ;ڧandgus;t;lsatisfyingtheabffoverelations6de nes35anuq(sl2)-moffdulealgebrastructureonA.Ow6Prffoof.ZWVeЍassumethatFuUAUR6=0;@theЍcaseE)lAUR6=0Ѝissimilar.0,Apply6Theorem2.2toH1 pwith5=YKܞ21 9,"D=Fƹ,and=qn922{Nto nd6 O; 2kBwith 6=0andt;switht(1Ns)P԰=41(moSdn)toseethat6u2n=UR O, h=qn922t'andFLnu= u2sn<.8Thisprorves(a).j.BNorwisinceEFZFE,=FuKK-:1ٟz ꍑqI{q1)1and isaprimitiven2th w$roSotof1,it6follorwsthat(EFLnFE)uUR6=0,andsoalsoE^AUR6=0.BNext,[wreaapplyTheorem2.2toH2,with2=JKܞ21J{butnorwwith6DS20=EKܞ21hand/=qn922.=;Rxthisgivresus06=s220 2kand0rr;ln16with v=c.qn922randDS20-fu=s220Aku2l!ȹ. SinceEE=DS20!Kܞ,(itfollorwsthat6EJu=s2u2l!ȹ,whereY1:= 21 p 20Ak.Moreorverqn922t~=qn922rimpliesrP9԰Ru=tt6ह(moSdn)landthrusthats51P2԰K=1lF(modln);equivXalenrtlyV,:s+lPv@԰(=H26(moSdn).8Thrus(b)isproved.BIt8remainsonlytocrheck8therelationbSetrween8  ands2.TVoseethiswre6usethefactthattheactionsofEFqFEHandFu4KK-:14z ꍑqI{q1)$mrustagreeonu. 7 &e610S.!MONTGOMER:YANDH.-J.SCHNEIDERYj.6हTVriviallyFu=KK-:1=z ꍑqI{q1)7Zu=Fu -:1 Ÿzyfꍑ`qI{q1!k[u,}so wreconsidertheactionofEFb FE.446ApplyingLemma2.1withË=URKܞ21#¹andD=Fƹ,wreseethatⳍɳFLnu lw=UR (lK1X ㇍i=0UV  ixi)u lK+s1: eJ6हSimilarlyV,useLemma2.1withË=URKܞ21#¹andDS20w=EKܞ21#¹toseethat|M2E^u sÎ=UR  1s^Gs2(EPs1X ㇍Iijv=0UV  jy)u s+lK1: 66हComrbiningthese,wehaveG(EFLnFE)yË=UR s2[  1ss1^EX ㇍jv=0$  j$A2lK1X ㇍ti=0  ixi]u s+lK1*cG=j s2  1s^G[EPs1X ㇍Iijv=0UV  j$A  s1IlK1^EX ㇍(Fi=0$  ixi]u s+lK1*G=j s2  1s^G[EPs2X ㇍Iijv=0UV  j$A (#n1{YX ㇍i=s1L ixi)]u s+lK1G=j s2  1s^G(1+ )(EPs2X ㇍Iijv=0UV jy)u s+lK1;6हassumingɫforthemomenrtthats;ly2,lhencesB~+l=n=2ɫandso6u2s+lK1J=UR Ou.8Thruswehave|MōǪ 21 Ǫ[z) ΍qqn91p=UR s2 O  1s^G(1+ )(EPs2X ㇍Iijv=0UV jy):!󃍑6हSince' 21" L= 21 p (1 22ӓ),Gthe rstformof Yin(c)norwfollows6bry,cancelling1+ ?from,bSothsides(notethat 22 L 6=xw1impliesthatc6 6=0).{The^secondformthenfollorwsbynotingthat 2s1^EP*lK2 U_i=0.JF 2i=}26P*s2 U_jv=0a 2jy.EBNorwmifs=0,andmsot=1mandlw=2,u2s+lK1=umandso doSesnot6appSear.4jIn+thiscasethesecondformoftheformrulafor s2,(without6the O,makressenseandso,usingthatnorw h=URqn922t1ѹ=qn922 ʵ,wreseethat6 ,=qn9. Thecases=2;lU=0issimilar,musing U=qn922 Pinthe rst6formoftheformrulafor s2.BConrverselyV,suppSose9i O; ;ands;t;l|aregivrenasinthetheorem.6Theyrde neactionsofH1 2andH2onA<=kg[uju2n C= O]rbryTheorem62.2. It remainstoshorwthatAisinfactanuq(sl2)-moSdule,Qthatis d7 ̍11Yj.6ह(EF}&FE)a =(Fu33KK-:133z ꍑqI{q1"l")a3Yforalla 2A.This3Yistruefora =u3Ybry獑6thepreviousargumenrt.ZHenceitistrueforallah|2A,sinceEFTRFE6हandFuKK-:1۟z ꍑqI{q1*ArbSothactas(;n921 ʵ)-skrewderivXations.b لcffxff ̟ff ̎ ̄cffg6Corollary3.2.LffetAbeann-dimensionalkg-algebrawithnonon-zero6nilpffotentelements,andassumethatAisauq(sl2)-modulealgebrasuch6that{F?A6=0(orthatER)A6=0). @DThentherffeexistsu2Aand6ह0UR6= O; ;Ȅ2URkRsuch35thatB(a)35AUR=kg(u),u2n= O,andKFu=qn922.=u;B(b)35FLnuUR= 1andE^u=s2u22;B(c)35 Ȅ=URqn9.BMorffeover35uisuniqueuptoascffalarmultiple.`6Prffoof.ZAssbSeforewremayassumethatFAUR6=0.yWVesapplyTheorem2.56toEH1 with!]s=:qn922 ʵ;g=Kܞ21 9,[andEx=Fڹto ndu2Aand O; V2k6हsucrhCthatAUR=kg(u),$Kܞ21`;'!u=qn922 ʵu,andCF'!u= 1.#iTheorem3.1norw6applies,aswreareinthecasesUR=0;l=2.bcffxff ̟ff ̎ ̄cff6Remark3.3.ڹTheorem_3.1givresaquantumanalogofclassicalwork6of Jacobson[J ;]ontheWittalgebra.VSeealso[Z.4].Thatis,assumethat6khasjcrharacteristicp/6=0;2,andjletA/=kg[vn9jv2p e=/0].pWVejmaywrite6A =kg[uju2pOg=1]tbrysettingu =v-+1.DThentDerk#(a)isspannedbyall6eidڹ,{iUR=0;:::ʚ;p1,where_eiĚisdeterminedbrysettingeiuUR=u2i+1AV. One6marybScheckthat[eid;ejf ]!=(i"jӹ)ei+j \.Thrus,>settingh!:=e0,>itfollorws6that<[eid;h]UR=ieiandthat[eid;ei R]=2ih.ThrusDerk#(A)containsacopy6ofsl2foreacrhiUR6=0moSd/p.6Remark3.4.ڹCorollary3.2alsogivresananalogof[MSm,mG3.8],in6whicrhitwasshownthatwhenqt2isnotaroSotof1,Lthereareessen-6tiallytrwomoSdule-algebraactionsofUq(sl2)onthepolynomialring6C[X].*TVosstatethisresultprecisely,wrerecalltheDrinfel'd-JimbSofor-e6mrulationԭofx"~Ueȹ=URUq(sl2)[Ji L],[D9].1Assumeko=Candthat06=1~q isnot6aroSotof1.8Then)x<~U:=URChx ~E f;x%~F <;x$~K ۴;x$~Kܞ 180i6हwherex~E Qd;x%~F <;x$~K\satisfytherelationsoxb>~_Kxl~ivEv=1~URqn9 2xwF~ ,EH:=x~Kܞ21xι~E$3,Kq0:=x~Kܞ22Ĺ,6andqË:=1~URqn922U.8Itisthenstraighrtforwardtocrheckthat7b KܞE i=URqn9 2.=EK5;UPKF=qn9 2 ʵFK5;UPEFLnFE i=ōKFKܞ21[z/ ΍7qqn91:6हand,9that(E)x=E9 Kb6+1 EPand,9(Fƹ)x=F'^ 1+Kܞ21 F,6asinuq.Afterdoingthis,:wreseethatpart(b)ofTheorem3.5says6thatNK}YK=qn922.=Yp,gEY=qn9Yp22 \O=s2Yp22\t,gandNFY=1= 1.erAsNin6Corollary3.2,s2 n=URqn9.BItֻisinrterestingtonotethatinthegenericcase,therearetwoes-6senrtially(distinctactions,x\whereasintheroSotofunitycase,x\thereis6onlyone.ThecasecorrespSondingtopart(a)ofTheorem3.5inour6Corollary+3.2isobtainedfrom(b)brysettingu20V:=u2n1̹.%Ofcoursein6thegenericcase,onecannotreplaceYbryYp2n1!<.7BWVeYclosethissectionwithanapplicationtoidealsinsmashproSducts.6WhenH=Tn2 (!n9)actsonAasinTheorem2.5andAisa eld,2fit6wrasashownin[CFM]thatA#Hissemisimple(infact,theproSofthere6assumed>that =UR!wandthat n=1,Sbuttheargumenrtsworksimilarly6forD2themoregeneralconstanrts).EHeretheactionisouterinthesense6that4theactionofnonon-trivialHopfsubalgebraisinner.Hencethis6resultmarybSeviewedasaweakformofclassicalresultsonouteractions,6asforexampleAzumarya'stheorem,whichsaysthatA#Hissimpleif6agroupalgebraHPactsonasimplealgebraAandthegroupactionis6outer.BOnemighrthopSethatthesameistruefortheactionofuq(sl2)ona6 eldQAasinCorollary3.2.Theactionisagainouterbutthesmash6proSductA#uq(sl2)isnotevrensemisimple,aswewillsee.X2BFirst,Huq1nispunimoSdular,withnon-zeroinrtegralUR=E2n18FƟ2n1&(P* n1 U_ i=0/wKܞ2iAx).6ThiscanbSevreri eddirectly;ΝalternativelyV,itcanbSeseenasaconse-6quenceofthefactthatuq gisaHopfimageoftheDrinfelddoubleof6Tn2 ,asin4.7.6Corollary3.6.LffetAbeauq(sl2)-modulealgebraasinCorollary3.2.6Then35A#uq(sl2)hasanon-zerffonilpotentidealIFչ:=URAA. 7 ̍13Y6Prffoof.ZFirst,forwanryaUR2A;h2HV,itwiseasytoseethathaUR=(h(a)6andxahUR=(\-zȟ ӍSȹ(h)a).ItfollorwsthatIjisanidealofA#uq.WVeclaim6thatAUR=0.BTVoseethis,(notethatLemma2.1(b)impliesthatFƟ2n1[AURkg,since6s׹=0.1ButthenEFƟ2n1^<7A׹=0andsocertainlyA׹=0.1Itnorw6follorwsthatI22 ٹ=URAAA=0.cffxff ̟ff ̎ ̄cffɍm!b4.|AfctionsoftheDrinfel32'ddoubleDS(HV)BFVoranry nite-dimensionalHopfalgebraHV,werecallthattheDrin-6fel'dPdoubleDS(HV)=(H2Z)2copq./H=isPgivrenasfollows:asacoalgebra,6ith\issimplyHV2cop_u HV. qThealgebrastructureismorecomplicated,kand6for|ourpurpSoseswreuseaformulafrom[R];seealso[M =,10.3.11].PThat6is,forf;fG20k2URHV2 andh;h20#2URHV,㍑kʀ(fQ./URh)(fG 0k./h 09)=XfG(h1V*f 0k(\-zȟ ӍS Bh3)./h2h 09:䍑6हInparticularwrehavev%("UR./h)(fQ./1)=Xhf3;h1ihf1;\-zȟ ӍS h3if2V./URh2:6ह(4.1)/BFVorsimplicitrywewritehUR=("./h)andfQ=UR(f./UR1).BWVednorwspSecializetothen22-dimensionalTaftalgebraHB=URTn2 (!n9)as6in1.1.8InthiscaseitisknorwnthatHV2P ԰ =UHV;thuswemaywrite񍍍yCuHV  =URkghG;XjG n=";X n %=0;XG=!n9GXi6ह(4.2)I'6wherev(G)3=G5 G,ي(X)3=X 5"+G X,يhG;1i3=1,and6hXJg;1iUR="H n(X)=0.BThedualpairingbSetrweenHandHV2 isdeterminedbrynhG;gn9iUR=! 1 ʵ;UPhG;xi=0;hXJg;gn9i=0;andhXJg;xi=1:6ह(4.3)΍6Lemma4.4.DS(HV)NisgenerffatedNasanalgebrffabyfx;gn9;XJg;Gg.The6rffelationsGamongthesegenerators,ɪinadditiontotherelationsinHand6HV2Z,35arffeasfollows:G gn9GUR=Gg;UPxG=! 1 ʵGx;UPXgË=! 1 ʵgXJg;UPandxX+Xx=Ggn9:6Prffoof.ZItisclearthatthegivrenelementsgenerateDS(HV).FTVocheckthe6relations,(@wreuse(4.1).3First,gn9Gɹ=hG;gihG;g21 ʵiGgt=hG;1iGg=6Ggn9.BFVorTJthenextrelation,r]wreuse2(x)UR=xw 1 1+g x 1+g g x6हandthefactthathG;\-zȟ ӍS xiUR=0.8Then/<%xGUR=hG;xihG;1iG1+hG;gn9ihG;1iGx+hG;gihG;\-zȟ ӍS xiGgË=UR! 1 ʵGx:6हThethirdrelationissimilar.Q7 &e614S.!MONTGOMER:YANDH.-J.SCHNEIDERYBहFVor thefourthrelation,owreagainuse2(x)aswellas2(X).3]Then6withhUR=x;fQ=X+in(4.1),wrehaveuR3xXFչ=URh";xihXJg;1i"1+h";gn9ihX;1i"x+h";gn9ihX;\-zȟ ӍS xi"go޻+h";xihG;1iX1+h";gn9ihG;1iXx+h";gn9ihG;\-zȟ ӍS xiXgo޻+hXJg;xihG;1iG1+hX;gn9ihG;1iGx+hX;gn9ihG;\-zȟ ӍS xiGg\3=o޻0+0g+0+Xx+0+G+0+0\3=o޻Xx+Ggn9:⍍bcffxff ̟ff ̎ ̄cffӍBहWVeۦnorwapplytheresultsofSection2toDS(HV)-actionsonthealgebra6A.sAsZinthelastsection,vzwreapplyTheorem2.2andTheorem2.5to6trwodi erentcopiesoftheTVaftalgebrainDS(HV).؍6Theorem4.5.LffetAbeann-dimensionalkg-algebrawithnonon-zero6nilpffotent8Welements,9andletH%betheTaftalgebraTn2 (!n9).uAssumethat6A jisaDS(HV)-moffdulealgebrasuchthatxJA6=0 jorthatX;JA6=0.6Then35therffeexistsuUR2A35and0UR6= O; n2kRsuch35that6(a)35AUR=kg(u)andu2n= 2kg;6(b).Og uUR=!n9uandGuUR=!n921 ʵu,/JandsobffothgandGhaveordernin6A2utk#(A);6(c)35xuUR= 1andX+u=u22,wherffe =!n921u]1.BInpffarticular,ذanyDS(HV)-modulealgebraactiononAisdetermined6by35theactionofHBURDS(HV)onA.BConverselygivenAasin(a)and006= O; ;vb20k6suchthat(b)and6(c)arffesatis ed,thenAbecomesaleftDS(HV)-modulealgebrawiththe6given35actions.6Prffoof.ZFirstٻassumethatxMnAA6=0.WVeٻapplyTheorem2.5toHٗ6DS(HV)to nduys2Aand O;  2ysk*3sucrhthatA=kg(u),91u2n !ù= O,6xUuUR=!n9u,e"andCxUuUR= 1.>Thisprorves(a)andthe rstpartsof(b)and6(c).LNorwbyLemma4.4,gn9(Gku)UR=G(gcku)=G(!u)=!(Gku).LThrus6Gju542A1,Kwhicrhisone-dimensionalbyLemma2.4;itfollowsthat6Gju7= u,pfor some 27kg.'Butwrealsohavex(Gju)7=!n921 ʵG(xju),6or xuUR= 1=!n921 ʵ 1.8Thrus h=!n921 ʵ, nishing(b).BNorw 2considertheactionofHV2copc@sDS(HV).~Since2cop (X)="o 6Xr+X G,`theNelemenrtD:=URXG212ʹisan(";G21 \|)-primitiveelement6andsatis esDSG21ι=UR!n921 ʵG21 \|D.7ThrusHV2copB=kghG21 \|;DiP԰n:=Tn2 (!n921 ʵ).6ApplyTheorem2.2toHV2cop,wreplacingwithG21 \|,with!n921 ʵ,and6using"G21{5uw}=!n9u.8MThent=n1andsosw}=2;^thrusthereexists6s220jj2(k!sucrhgthatXG21[Tu=s220Aku22.Setting:=!n921 ʵs220Ak,itfollorwsthat6X+uUR=u22.7 ̍15YBहItUgremainsonlytocrheckUgtherelationbSetrweenUg and.yTVodothis,6wreapplythefourthrelationinLemma4.4tou.8Thustxx(X+u)X(xu)UR=Gugu;6हorxg(u22) XY'1UR=(!n9212Yg!n9)u.-Norwxu22V=UR (1+!n9)ubryLemma62.1,andso (1+!n9)uUR=(!21u]!)u.8Thrus n=!21u]1.BThis nishesthecasexU.AUR6=0;notewrehaveshownthatXFU.AUR6=0.6Assuming&X/ˬA6=0,'FwrewouldobtainsimilarlythatxˬA6=0,'Fsothe6previousargumenrtswouldapplyV.BIt isclearfromtheproSofthattheD(HV)-actionisdeterminedbrythe6HV-action.oThe)VconrversefollowsfromLemma2.1,PasAwillbSeamodule6algebraforbSothHandHV2cop;MbryconstructiontherelationsinLemma64.4aresatis ed,andsoAwillbSeaD(HV)-module.Xcffxff ̟ff ̎ ̄cffōBNorw,analogouslyytoCorollary3.6,wreshowthatA#DS(HV)isalso6notsemisimple.BFirst,bry,aresultofRadford[R]forany nite-dimensionalHV,DS(H)6isbunimoSdularwithinrtegral0" 6==./t,wherebisaleftintegral6ofHV2 andtisarighrtintegralofHV.8ThuswhenHB=URTn2 asabSove, ҍ8UR=(Fn1X ㇍i=0UVG idڹ)X n1ˡ./x n1̹(Fn1X ㇍Iijv=0gn9 jC):#O֍6Corollary4.6.Lffet{AbeaDS(HV)-modulealgebraasinTheorem4.5.6Then35A#DS(HV)hasanon-zerffonilpotentidealIFչ:=URAA.㍍6Prffoof.ZFirst,forvanryaB2A,wM2DS(HV),itviseasytoseethatwRaB=6(wu"la)2A;similarlyV,Ɗawf2A.HItfollorwsthatIisanidealof6A#DS(HV).8WVeclaimthatAUR=0.BNorw:-sincexB/uUR= 1,]xit:-followsfromLemma2.1(b)thatx2n1B/AURkg.6ButthenXx2n1yAUR=0andsocertainlyAUR=0.4Itnorwfollowsthat6I22 ٹ=URAAA=A(A)AUR=0.bcffxff ̟ff ̎ ̄cff6Remark4.7.ڹAnalternateapproacrhtotheresultsinSection3can6bSegivrenusingtheresultsofthissection.BFirst,yitiswrell-knownthatuq(sl2)isaHopfhomomorphicimageof6DS(HV),+UforHB=URTn2 (!n9),brysetting!Ë=URqn9225andde ning:DS(HV)!uq6हasfollorws:t6GUR./17!K5;UP"./gË7!Kܞ 1 9;UP"./x7!FS;UPandXF./17!\-z f ӍE`:=(qn9q 1 ʵ)E6हMoreorver^Ker()=DS(HV)kgG.2+Ҧ,zwhereG޹=fG2i^ 4gn92ij0i=1(moffdn)ands+lPiM԰5=b2(modn)such6thatB(a)u2n `= O,:J =!n92t,GFu= 21 p u,andgsandGhaveorffdernin6Aut(A);B(b)35xuUR= u2sqandX+u=u2l!;B(c)S=(q1$qn921 ʵ)s2,\&wherffe!2=qn922and (isgivenasinThefforem63.1.BConversely,OQanyIchoicffeof O; ;andIs;t;lAsatisfyingtheabffoverela-6tions35de nesaDS(HV)-moffdulealgebrastructureonA.6Prffoof.ZWVeusethepreviouscorollarytogetthecorrespSondinginduced6actionofuq(sl2)onA.8NorwapplyTheorem3.1.h0cffxff ̟ff ̎ ̄cffxBWVecanalsogivreanalternateproSofofCorollary3.6,asfollows.HGiven6an>actionofDS(HV)onA,Randthecorrespondingactionofuq(sl2),Rthe6homomorphism.inRemark4.7inducesacorrespSondinghomomorphism6of7thesmashproSductA#D(HV)onrtoA#uq(sl2).+Inparticular,J2ifis6theinrtegralinDS(HV),then()istheintegralinuq(sl2).#Itfollows6thatpthenilpSotenrtidealofA#D(HV)constructedinCorollary4.6has6ashQanon-zeroimagetheidealofA#uq(sl2)consideredinCorollary3.6,6whicrhmustthereforebSenilpotenrt.ፍt5.kQYetter-Drinfel32'dModuleAlgebrasBहIn$ thissectionwregivesomegeneralresultsabSoutYVetter-Drinfeld6moSdule algebrasforanryHopfalgebraHV,Tandthenspecializetothe6TVaftalgebras.7 ̍17YBहLetp,MbSealeftHV-moduleandaleftHV-comoduleviaR:M-6!6H M@.8TheusualYVetter-Drinfel'dconditionisZr(hm)UR=X(hm)1$ (hm)0V=URXh1m1 \|Sh3j h2m06ह(5.1)c6forrallm2M;h2HV.?WVerwilljustrefertothisastheYpDS-condition,6andthecategoryof(left,left)HV-YVetter-Drinfel'dmoSdulesas2HbH YD.BTheYlleftHV-comoSdulestructuredualizesasusualtoarighrtH2Z-6moSduleFVrom=(5.2),Qthegivren(u)=PYu1?I u0"is=thecorrectcoactionif6andonlyif,Uforallf"2a#HV2Z,fabu=P h\-zȟ ӍSf;u1 \|iu0.MSinceSpisbijectivre,6wremaycheckinsteadthat΍X()&SfuUR=Xhf;u1 \|iu0:_{6हItsucestocrheckthisforallfQ2URfX2qG2rbg,abasisofHV2Z.BFirstassumeqË=UR0.8Then,usingLemma5.4,andfQ=G2rb,6G r .uUR=!n9 tru=a0hG rb;gn9 tiu+X ҍm>0a>amhG r;x mgn9 m+tiu m(s1)+1+¹=URa0!n9 tru:%}6हThrusa0V=UR1.8WVemaythereforeassumethatqËUR1.BNorwgassumesUR2. FVorgfQ=X2qG2rb,>tnotethatSfQ=G2r .ڹ(G21 \|X)2qP=6G2r .ڹ(!n9DS)2q >=8@(!)2qG2rDS2q茹. O?UsingD%:u8@=s220Aku2s |=!n92t {s2u2sn<, Oapply6Lemmae22.1(b)withp%=1e2and 9t=%!n92ttoseethattheleft-handside 7 ̍19Y6हof(*)isRH6SfuUR=(!n9) qG r .DS q4u&@=T(!n9) q(! trݹ) qI{(s1)+1$ت(! t {s2) q(s1KSX ㇍iq2*=0! tiq2)( (qI{1)(s1)󍍑 X ㇍ iq#=0, ! tiq)u 1+qI{(s1),㋍@=T() qs2 q0!n9 qI{(1t)!n9 rt.FVortherighrt-hand6sideof(*),usingLemma5.4,itfollorwsthatR;՟XM$-hX qG rb;u1 \|iu0V=n1FX ҍURm=0amhX qG r;x mgn9 m+tiu m(s1)+1'+Eչ=X]aqԍ 5)q},Y ҍzV=1Rqō zYy ǹ1&ߟq/7 p!I{1?y!n9 (qI{+t)r&Cu qI{(s1)+1(-=URaq!n9 rWVemaryZ΍6alsocancel!n92r ku+a1x 1.FWVe"harvealreadyseenthat6a0V=UR1,usingqË=0.8WhenqË=1,6SXuUR=!n9Du=!n9(! 1 ʵs2)1UR=a0hXJg;gn9iu+a1hXJg;xi1UR=0+a1V=a1:6हThrusa1V=URs2, nishingtheproSof.cffxff ̟ff ̎ ̄cff=6Example5.6.uConsiderL"thespSecial(\generic")casearisinginTheo-6rem4.5,&thatis,G#uI=!n921 ʵuandX[#uI=s2u22.NInthatcase,sI=26andDt킹=n1.EThrustheleftHV-coactioncorrespSondingtothisaction6isgivrenbyA45(u)UR=n1FX ҍm=0amx mgn9 (m+1)#3 u m+1*6हwheream Z=UR(s2)2m!n92mr!lm(m+1)l8R\).) A2ȹ.BFVor,TinPthiscase,eacrhfactorinthenumeratorofam Uoftheformt6टP*CQi U_CQj8:i,r=0W?!n92j8:i ispairedwiththeterm1K+!n921 +:::M6+!n92i jinthedenominator,6formiUR=1;:::ʚ;m1,togivreaquotientof!n92i.DSubstitutingintothefor-Md6mrulaforam inPropSosition5.5,weseethatam fb=a(s2)2m!n922m !lm(m1)l8R\)Xԟ W2#=6(s2)2m!n92mr!lm(m+1)l8R\).) A2ȹ.2J7 &e620S.!MONTGOMER:YANDH.-J.SCHNEIDERYBहWVecannorwdeterminetheleftYpDS-structuresonourringA.z6Theorem5.7.LffetAbeann-dimensionalkg-algebrawithnonon-zero6nilpffotentelements,andletHN=Tn2 (!n9).zAssumethatAisaleftHV-6moffdulealgebrasuchthatxA/@6=0.xThenforeffachsuchaction,Pthere6existsauniqueleftHV-cffomodulealgebrastructureonAsuchthatA8226Hb6H>YDR.BInppffarticular,assumetheHV-actionisasdescribedasinTheorem62.5;u that is,?hA⥹=kg(u)foru2n = 2kg,?handgIu=!n9u,?hxu= 1.6ThentheuniqueleftHV-cffomodulealgebrastructureonA,;suchthatAUR226Hb6H>YDR,35isgivenby!h(u)UR=n1FX ҍm=0  mHX(!1) m![lm(m+1)l8R\).) A2x mgn9 (m+1)#3 u m+1@:!6Prffoof.ZApplyTheorem4.5togetauniqueleftDS(HV)-modulealgebra6structure]onA;bryPropSosition5.5theleftHV2cop-modulealgebrastruc-6turegivresauniqueleftHV-coactiononA.83ThatthisimpliesA22HbH nYD6हfollorwsfromLemma5.3.BNorwconsidertheparticular\generic"caseinthetheorem.%ByThe-6orem4.5,nwremusthaveGʫu%Y=!n921 ʵuandX.ʫu%Y=s2u22 Qsuchthat6s2 Ȏ=!r!n921-jb1.cThrus()2m &6= 2mHX(1b!n921 ʵ)2mĹ.cSincewreareinthe6situationofExample5.6,wreusethesimpli edformofam ӹfoundthere6toseethatNLam Z=UR  mHX((1!n9 1 ʵ)!n9) m![lm(m+1)l8R\).) A2;CI6हwhicrhMisthedesiredcoSecient. _Theformof(u)alsofollowsfrom6Example5.6.oxcffxff ̟ff ̎ ̄cffqBThewYtheoremgivresananswertothequestionraisedin[CFM,2.2];6thecoactionpropSosedthereisinfactacoaction,asitisthespecial6caseofTheorem5.7with 3͹=1.]WVenotethatthisfactseemsquite6dicult]toprorve]directlyV.For,Jonehastovrerifythecommutativityof6the>comoSdulediagram;wthatis,athat(idKW )UR=(KW id).TVo>crheck6thisonu{2A, rstnotethatbSecomesanalgebramapbryde ning6(u2p])UR:=(u)2p"for-allp;lthisisshorwnin[CFM,S2.2].Butthenonemust6stillcrheckthatXHn1W;X ҍXp=0jap]x pgn9 (p+1) (u) p+1+=n1FX ҍURm=0amĹ(x mgn9 (m+1) i) u m+1@:!6हThisYYistheproblem.pTheindirectargumenrtviapassingtoHV2Z-moSdules6as{inTheorem5.7abSorve{avoidshavingtocheckthisidentityV. 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