; TeX output 1999.11.03:0759@7 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr964A*7 &e&7.pDUALITY!OFHOPFALGEBRAS~Q65YN cmbx127.DualityofHopfAlgebrasInށ2.4.8wrehaveseenthatthedualHopfalgebrag cmmi12HV2 K cmsy8 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!", cmsy10 HB3a h7!ha;hi2( msbm10KbSeabilinearformsatisfyingU`ha b;"u cmex10XUTh|{Ycmr8(1)$ h(2) \|iUR=hab;hi; h1;hiUR="(h)(1)`ohXUVa(1)$ a(2) \|;h jiUR=ha;hji; ha;1i="(a)(2)ePha;S׹(h)iUR=hS(a);hi(3)Sucrhamapiscalleda,@ cmti12weffak@dualityofHopfalgebras.eThebilinearformiscalledleft(right$D)nondeffgenerateifha;HVi̹=0impliesa=0 ߔ(hL;hi=0impliesh=0).HEAduality35ofHopfalgebrffas꨹isawreakdualitythatisleftandrightnondegenerate.yRemark2.7.2.j6IfHisa nitedimensionalHopfalgebrathentheusualevXalua-tionev+:URHV2X HB4!K꨹de nesadualitryofHopfalgebras.Remark2.7.3.j6AssumethatevJ:URL HB4!Kǹde nesawreakdualityV.ByA.4.15wre(haveisomorphismsHomAq(L HF:;K)PA>԰A&=Homy(L;Homy(HF:;K))$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(1)ifandonlyif'UR:L!Hom-(HF:;K)35isahomomorphismofalgebrffas.2. NThebilineffarformevՋ:L H!;Ksatis es(2)ifandonlyif ,:H!Homy(L;K)35isahomomorphismofalgebrffas.+- cmcsc10Proof.@_evOK:URL0 HB4!K -satis estherighrtequationof(1)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).B7 &e661:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYev :URL HB4!K꨹satis estheleftequationof(1)i '(1)(h)=h1;hi="(h).ThesecondpartoftheLemmafollorwsbysymmetryV.`cffxff ̟ff ̎ ̄cff+荍Example2.7.5.oQThere/isawreakdualitybSetweenthequantumgroupsSLܟ2cmmi8qQڹ(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(1)and(2)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Problem2.7.1.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)$wisacoal-gebra.?XWVeB%shorwnowthatB2o !=B2 /isasubalgebra.?XLetf;gXv2B2o xwithdimm(BfG)<1Ianddim)(Bgn9)^<1. VLetIa2B. VThenIwrehave(a(fGgn9))(b)^=(fgn9)(ba)=PfG(b(1) \|a(1))gn9(b(2)a(2))X{=P((a(1) \|f)(b(1))(a(2)gn9)(b(2))X{=P(((a(1) \|f)(a(2)gn9))(b)܄hencea(fGgn9) 0=Pݹ(a(1) \|f)(a(2)gn9) 02(Bf)(Bgn9).Sincebdimm(Bf)(Bgn9) 0<1bwrehavedimH(B(fGgn9))<1sothatfg2B2o6ƹ. FVurthermorewrehave"2B2o6ƹ,msinceKerv(")hascoSdimension1.2]ThrusB2oURB2 2)isasubalgebra.ItisnorweasytoseethatB2o isabialgebra.Norw+letS7bSetheantipSode+ofHV.jWVeshowSן2r۹(HV2o)ZHV2o.jLet+a2HV,f2H2o.ThenhaSן2r۹(fG);bi=hSן2(fG);bai=hf;S׹(ba)i=hf;S׹(a)S(b)i=hfGS׹(a);S(b)i=hSן2r۹(fGS׹(a));bi. yThisQimpliesaS2r۹(fG)=S2r۹(fGS(a))QandHVS2r۹(fG)=S2r۹(fGS(HV))Sן2r۹(fGHV). Sinceδf92:H2o Wʹwregetdimc(fGH):<1sothatdimc(Sן2r۹(fGHV))<1anddimH(HVSן2r۹(fG))UR<1.*YThisshorwsSן2(fG)UR2HV2o.*YTherestoftheproSofisnorwtrivial. dcffxff ̟ff ̎ ̄cffD&7 &e681:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYDe nition2.7.11.}!ùLet+G=K-cAlgo(HF:;-)bSeananegroupandR=2K-cAlgo.WVeM&de neGh K!Rn:=URGj33R -^cAlg((tobSetherestrictiontocommrutativeM&RJ-algebras._TheۍfunctorG K cRisrepresenrtedbyH Rn2URRJ-cAlg:ePGj33R -^cAlg%(A)UR=K-cAlgo(HF:;A)P԰n:=RJ-cAlg(H RJ;A):Theorem2.7.12.wX(TheVCartierdual:)ULffetHrbea nitedimensionalcommu-tative"cffocommutativeHopfalgebra.3LetG=K-35cAlg(HF:;-33)bffetheassociatedanegrffoup35andletDS(G)UR:=K-35cAlg(HV2Z;-33)35bethedualgroup.fiThenwehaverDS(G)UR=G.r(G;GmĹ)wherffeԍG.rMA(G;GmĹ)(RJ)=GrJ(G"+ K R;Gm & KR)ԍisthesetofgrffoup(-functor)homo-morphisms35andGm 7isthemultiplicffativegroup.Proof.@_WVepharveG.r$(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 ̎ ̄cffBK;7  ,@ cmti12+- cmcsc10)ppmsbm8( msbm10"u cmex10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12M