; TeX output 2000.06.26:1541t7 Y_o cmr9116u*7 YRXQ cmr12CHAPTER4Nff cmbx12TheffIn nitesimalTheory"n7N cmbx121.7IntegralsandFourierTransformsAssumeforthiscrhapterthat( msbm10Kisa eld.fGLemma4.1.1.g5Q+@ cmti12Lffet35g cmmi12Cbea nitedimensionalcoalgebra.fkEveryrightCܞ-comoduleMvis5aleftCܞ2 K cmsy8-moffdulebyc2mUR="u cmex10Pm|{Ycmr8(2cmmi8M") h!", cmsy10hc2;m(1) \|i5andcffonverselybys2(m)UR=Pidc2RAimw ciwherffe35Pc2RAij ciis35thedualbasis.,- cmcsc10Proof.@_WVecrheckthatM+bSecomesaleftCܞ2-moduleeʍ=W(c2c20 ̟^\й)mk=URPm(M") hhc2c20 ̟^\;m(1) \|iUR=Pm(M") hhc2;m(1) \|ihc20ȟ^ ;m(2)ik=URc2Pjm(M") hhc20ȟ^ ;m(1) \|i=c2(c20ȟ^m):;Itiseasytocrheckthatthetrwoconstructionsareinrversesofeacrhother.InparticularassumelethatMIisarighrtCܞ-comoSdule.Chooselemi?suchthats2(m)UR=Pmi cidڹ.Thenc2RAjmUR=Pmidhc2RAj;ciiUR=mjPandꨟPSc2RAim ci,=URPmi ci=URs2(m).ecffxff ̟ff ̎ ̄cffDe nition4.1.2.vaù1.\XLetzAbSeanalgebrawithaugmenrtation"0:A6!SK,9oanalgebraބhomomorphism.tLetMhbSealeftA-module.tThen2A hM5J=ffm2M@jam="(a)mg꨹iscalledthespfface35ofleftinvariants꨹ofM@.Thisde nesafunctor2A Ȍ-:URAMo`d!n4#!2VecH.2.ALet+sCbSeacoalgebrawithagroup-likreelement1wY2Cܞ.ALet+sMlWbSearightCܞ-comoSdule.2ThenM@2coCg:=ܙfGȄ=UR"(f)forallfQ2URHV.4.{hTheV*setofleftinrtegralsinHCisdenotedbyIntXlze(HV),q thesetofrightintegralsbryIntr}(HV).8Thesetofleft(righrt)integralsonHisIntl(HV2Z)(Intsrչ(HV2Z)).5.8AHopfalgebraHiscalledunimoffdularifInrtl(HV)UR=InrtWşr'(H).u Lemma4.1.7.g5QTheleftinteffgralsInrt lѹ(HV2Z)formatwosideffdidealofHV2Z.nIftheantipffode35S isbijeffctivethenSinducffesanisomorphismS):URInrtWşly(HV2Z)UR!Inrt$r((HV2).Proof.@_FVorYUQRyinYInrt\/l}(HV2Z)YwrehaveaUQRc=d"(a)UQR2Inrtןl6(HV2Z)YandaUQR b="(a)UQRbU^henceQUQRPbC`2InrtEӟlg(HV2Z). bIfQSV(isbijectivrethentheinducedmapS7:C`HV2  E! MHV2 PisalsoVbijectivreandsatis esS׹(UQR)b =S(UQR)S(S21 S(b))=S(S21 S(b)UQR )=S(UQR)"(b)VhenceS׹(UQR)UR2InrtWşr'(HV2Z).Qcffxff ̟ff ̎ ̄cffu Remark4.1.8.j6Mascrhke'sTheoremhasanextensionto nitedimensionalHopfalgebras:8"(UQR)UR6=0i HV2 issemisimple.Corollary4.1.9.sWLffetCH0mbea nitedimensionalHopfalgebra.ThenHV2 qisaleftHV2Z-moffdule35bytheusualmultiplication,hencearightHV-comodule.fiWehave>(HV Z) coH8Ϲ=URInrtWşly(HV ):Proof.@_Byde nitionwrehaveIntl(HV2Z)UR=2H-:HV2.ބcffxff ̟ff ̎ ̄cffExample4.1.10.uQLet1GbSea nitegroup.U{LetH:=JMapD(G;K)betheHopfalgebrade nedbrythefollowingisomorphism>ihK G t=URMap(G;K)PUR԰n9=Hom(y(KG;K)UR=(KG) :This {isomorphismbSetrween {thevrectorspaceK2G )ofallsetmapsfromthegroupGtothel^baseringKandthedualvrectorspace(KG)2,bofthegroupalgebraKGde nesthestructureofaHopfalgebraonK2G.WVe:regardHB:=URK2G YasthefunctionalgebraonthesetG.Inthesenseofalgebraicgeometry{/thisisnotquitetrue.uThealgebraK2G ǹrepresenrtsafunctorfromK-cAlgto6SetVǹthat6hasGasvXalueforallconnectedalgebrasAinparticularforall eldextensionsofK.As=bSeforewreusethemapevj::]KGy K2G π! eK. 1The=multiplicationofK2G ]DisgivrenxbypSointwisemultiplicationofmapssincehx;fGf208iF=hPx(1)g  Kx(2) \|;fSJ fG208iF=$hxO x;f fG208iI=hx;fGihx;f208iforallf;fG20 2IK2G andallx2G. 5Theunitelemenrt1>)ppmsbm8K;cmmi6GofK2G isthemap"v?:KG! KrestrictedtoG,hence"(x)v?=1=hx;1>KG iforallxUR2G.8TheanrtipSodeoffQ2URK2G @isgivrenbyS׹(fG)(x)UR=hx;S(fG)i=f(x21 \|).x"7 &e120h4.pTHE!INFINITESIMALTHEOR:YYTheDelemenrtsofthedualbasis(x2jx}2G)Dwithhx;yn92.=i}=x;y$consideredDasmaps-JfromGtoKformabasisofK2G.8TheysatisfytheconditionstHx yn9 =URx;y "hx and!~X!x2G4x V=1>KG K?sincehz;x2yn92.=iUR=hz;x2ihz;yn92.=iUR=zV;x zV;y=x;y "hhz;x2iandhz;Px2G"=x2iUR=1=hz;1>KG i.MHencesthedualbasis(x2jxY2G)sisadecompSositionoftheunitinrtoasetofminimalorthogonalidempSotenrtsandthealgebraofK2G @hasthestructureQ$K G t=URx2G/Kx PV԰.==K:::K:InparticularK2G @iscommrutativeandsemisimple.ThediagonalofK2G @istf/(x )UR=XSyI{2Gyn9  (yn9 1 ʵx) V=XyI{;zV2G;yz=x3MUyn9  z  OsincezL>)hz3 u;(x2)iUR=hzu;x2i=x;zVu19=zVAacmr61 HEx;u"=PyI{2G#GRyI{;z /yI{1 ;;x;uGϹ=URPyI{2G!GThz;yn92.=ihu;(yn921 ʵx)2iUR=hz3 u;PyI{2G!yn92 (yn921x)2i:ÍLeto]a752KG.Thenade nesamap/eaѓ:GP!K2K2G brya=Px2G$5:e$ua*(x)x.FVorӒarbitraryfQ2URK2G @anda2KGthisgivresty?tha;fGiUR=f(X!x2GeUVaW(x)x)=Xvfx2Gjeaթ(x)f(x):!ThecounitofK2G @isgivrenby"(x2)UR=x;eswheree2Gistheunitelemenrt.TheanrtipSodeis,asaborve,S׹(x2)UR=(x21 \|)2.WVeconsiderHpF=K2G sasthefunctionalgebraonthe nitegroupGandKGasthedualspaceofHB=URK2G @henceasthesetofdistributionsonHV.ThenUQR\H:=URXvfx2GxUR2HV  =KG(1) K?isa(trwosided)inrtegralonHsincePUx2G$(yn9xUR=Px2G#,x="(y)Px2G"=x=Px2G#,yx.WVewrite甆ZfG(x)dxUR:=hUQR;fi=Xvfx2Gf(x):LWVei.harveseenthatthereisadecompSositionoftheunit1l2K2G ƹinrtoasetofprimitivreorthogonalidempSotentsfx2jx|2Ggsuchthateveryelementf{2|K2G hasauniquerepresenrtationf86=7PfG(x)x2. SinceUQRyn92 t=7Px2G#.hx;yn92.=iwegetUQRfGyn92 t=RP x2G=hx;fGyn92.=iUR=Pf(x)yn92.=(x)=f(yn9)henceTقfQ=URX(甆ZfG(x)yn9 .=(x)dx)yn9 :y6D7 &er1.pINTEGRALS!ANDF9OURIERTRANSFORMSa$121YProblem4.1.1.nRDescribSe}LthegroupvXaluedfunctorK-cAlgo(K2G;)intermsofsetsandtheirgroupstructure.De nitionandRemark4.1.11._LetKbSeanalgebraiclyclosed eldandletGbSea niteabeliangroup(replacingRaborve).+AssumethatthecrharacteristicofKdoSesnotdividetheorderofG.DLetH=9K2G.WVeidenrtifyK2G ѹ=9Hom(KG;K)alongthelinearexpansionofmapsasinExample2.1.10.0Let=usconsiderthesetx?^G):=s@f:G! K2j꨹grouphomomorphismtB.g.mSince=K2Ais]܍anabSeliangroup,thesetx^Gisanabeliangroupbrypoinrtwisemultiplication.Thegroupx^Giscalledthecharffacter35group꨹ofG.ObrviouslygthecharactergroupisamultiplicativesubsetofK2G $=tHom(KG;K).Actually$itisasubgroupofK-cAlgo(KG;K)URHom(KG;K)$sincetheelemenrtsUR2xT^GexpandeWtoalgebrahomomorphisms:.?(ab)=(P xHxP y yn9)=P x y (xyn9)=(a)(b)and(1)=(e)=1.~]Conrverselyanalgebrahomomorphismf 2K-cAlgo(KG;K)qrestrictstoacrharacterfQ:URGn!1K2.Thusx^qG =K-cAlgo(KG;K),thesetofrationalpSoinrtsoftheanealgebraicgrouprepresentedbyKG.ThereisamoregeneralobservXationbSehindthisremark.Lemma4.1.12.mQLffetiHbea nitedimensionalHopfalgebra.ThenthesetGr(HV2Z)of35grffouplikeelementsofHV2 isequaltoK-Alg(HF:;K).Proof.@_Infactf:PH=Wg!Kisanalgebrahomomorphismi hf uf;a bi=hf;aihf;biUR=hf;abi=h(fG);a bi꨹and1UR=hf;1i="(fG).}݄cffxff ̟ff ̎ ̄cffYHencethereisaHopfalgebrahomomorphism'UR:Kx^G !K2G @bry2.1.5..Prop`osition4.1.13.TheHopfalgebrffahomomorphism'6:Kx^G {! K2G ̩isbijec-tive.Proof.@_WVegivretheproSofbyseverallemmas.Lemma4.1.14.mQA2nysetofgrffouplikeelementsinaHopfalgebraH@islinearlyindepffendent.Proof.@_AssumethereisalinearlydepSendenrtsetfx0;x1;:::ʚ;xnPgofgrouplikreelemenrtsHinHV.ChoSosesuchasetwithnminimal.ObviouslynUR1Hsinceallelementsarenonzero.8Thrusx0V=URP*n U_i=1 AS idxiOandfx1;:::ʚ;xnPglinearlyindepSendent.8WVegetƍVHX ㇍Z,Hi;jij id jf xi xj\=URx0j x0V=(x0)=X ㇍ i idxi xi: kSinceall i,6=UR0andthexia |xj:arelinearlyindepSendenrtwegetnUR=1and 1V=UR1sothatx0V=URx1,aconrtradiction.5ӄcffxff ̟ff ̎ ̄cffCorollary4.1.15.z(Deffdekind'svLemma)A2nysetofcharactersinK2G `islinearlyindepffendent.zF7 &e122h4.pTHE!INFINITESIMALTHEOR:YY]܍Thrusd3'$5:Kx^G hl!K2G ˹isinjectivre.Nowweprovethatthemap'$5:Kx^G hl!K2G ˹issurjectivre. Lemma4.1.16.mQ(Pontryagintduality)Theevaluationx!^GQGb!1K2 4isanon-deffgenerate35bilinearmapofabeliangroups.dProof.@_FirstwreobservethatHom (CnP;K2)P԰׹=7Cn OUforacyclicgroupofordernsinceKhasaprimitivren-throSotofunity(char(K)6u tjGj).SincezthedirectproSductandthedirectsumcoincideinAbwrecanusethefunda-menrtaltheoremfor niteabSeliangroupsGPUR԰n9=Cnq1 :::t0Cnt togetHom|(G;K2)PUR԰n9=Gfor anryabSeliangroupGwithchar(K)6u tjGj.sThusx^GPۢ԰=Gandύ1^31x^Gۢ=G.sInparticular(x)?=1Eforallx?2GEi ?=1.H+ByEthesymmetryofthesituationwregetthatthe]܍bilinearformh:;:iUR:xT^GDGn!1K2isnon-degenerate.+Jcffxff ̟ff ̎ ̄cffZThrusjx^G DjUR=jGjhencedim(Kx^G D)=dim(K2G).8ThisprorvesPropSosition2.1.13.cffxff ̟ff ̎ ̄cffdDe nition4.1.17.}!ùLet$dHbSeaHopfalgebra.A$K-moduleMeHthatisarighrtHV-moSduleUbryUR:M1 HB\3!M9andUarightHV-comoSdulebyȄ:URM6!M1 H|iscalledaHopf35moffduleifthediagramAeԠM HԠzH :2fdzO line10-(73ԠԠ M H :2fdzά-mSqLM H H H8M H H H㌞32fdנά-n˙1 r 1Ǡ@feǠ? L SjǠ@fe?`6x9 r뀍commrutes,ui.e.+vifhs2(mh)UR=Pm(M") hh(1) Xpm(1) \|h(2)holdsforallh2Handallm2M@.dObservreythatHfisanHopfmoSduleoveritself.FVurthermoreeachmoSduleoftheformV %'HisaHopfmoSdulebrytheinducedstructure.TMoregenerallythereisafunctorVecz3URV7!VG HB2Hopf-Mo`d@1ƹ-DnHV.Prop`osition4.1.18.Thetwofunctors-2coH+ֹ:URHopf-Mo`d@1-DdHB!gVec.and- F ?HB:Vec3URV7!VG HB2Hopf-Mo`d@1-DdH arffe35inverseequivalencesofeachother.Proof.@_De nenaturalisomorphismsW ~w/ h:URM@ coH HB3m h7!mh2MwithinrversemapEO  1]:URM63m7!Xm(M") hS׹(m(1) \|) m(2)2URM@ coH HEandp, :URV3vË7!v 12(VG HV) coH؍withinrversemap(VG HV) coH83URv h7!vn9"(h)2V:{W7 &er1.pINTEGRALS!ANDF9OURIERTRANSFORMSa$123YObrviouslyLthesehomomorphismsarenaturaltransformationsinMeandVp.^lFVur-thermore 7isahomomorphismofHV-moSdules.8 21Ziswrell-de nedsince',ʍRs2(Pm(M") hS׹(m(1)))@7=URPm(M") hS׹(m(3) \|) m(1)S׹(m(2))@7(sinceM+isaHopfmoSdule)@7=URPm(M") hS׹(m(2) \|) n9"(m(1))@7=URPm(M") hS׹(m(1) \|) 1)QhencexP#m(M") hS׹(m(1) \|)Gg2M@2coH$a.FVurthermorex 21isahomomorphismofcomoSdulessince~؍ʍ%s2 21 p (m)T=URs2(Pm(M") hS׹(m(1) \|) m(2))UR=Pm(M") hS׹(m(1)) m(2)$ m(3)T=URP 21 p (m(M") h) m(1)ι=UR( 21 1)s2(m):yFinally 7and 21Zareinrversetoeacrhotherbymʍ+D{  1 p (m)UR= (XUVm(M") hS׹(m(1) \|) m(2))UR=Xm(M") hS׹(m(1))m(2)ι=mmˍand~؍ʍ' 21 p (m h)n6=UR 21 p (mh)=Pm(M") hh(1) \|S׹(m(1)h(2)) m(2) \|h(3)n6=URPmh(1) \|S׹(h(2)) h(3)G$(brys2(m)UR=m 1){=URm h:yThrus 7and 21ZaremutuallyinversehomomorphismsofHopfmoSdules.The4qimageof isin(VyT HV)2coHbrys2(vK 1)=vK (1)=(vK 1) 1.:Hy&!Hhisanalgebraanrtihomomorphism,άthedualHV2 isanHV-moSduleinfourdi erenrtways:,ʍL1xh(fQ*URa);gn9i:=ha;gn9fGi;ޯXh(aUR(fG);gn9i:=ha;fGgn9i;L1xh(fQ+URa);gn9i:=ha;S׹(fG)gn9i;ޯXh(aUR)fG);gn9i:=ha;gn9S׹(fG)i:(2)~؍IfHis nitedimensionalthenHV2 aisaHopfalgebra.Theequalitryh(f*a);gn9i=ha;gn9fGiUR=Pha(1) \|;gn9iha(2);fGi꨹impliesUد(fQ*URa)=Xa(1) \|ha(2);fGi:(3)mˍAnalogouslywrehavemʍخ(aUR(fG)=Xha(1) \|;fia(2):(4)FProp`osition4.1.19.LffetKH9bea nitedimensionalHopfalgebra.%ThenHV2 #isaright35HopfmoffduleoverHV.|g7 &e124h4.pTHE!INFINITESIMALTHEOR:YYProof.@_HV2 zisK aleftHV2Z-moSdulebryleftmultiplicationhenceby2.1.1arightHV-comoSdule,Nbrys2(a)UR=Pidb2RAia% bidڹ.mLet,Nf;gË2URHanda;bUR2HV2Z.mThe,N(left)mrultiplicationofHV2 satis esr?abUR=Xb(H) ha;b(1) \|i:5WVeusetherighrtHV-moSdulestructure^gP(aUR)fG)=Xa(1) \|hS׹(f);a(2)i: onHV2 =URHom(HF:;K).NorwwechecktheHopfmoSdulepropertryV..~Š7 &e126h4.pTHE!INFINITESIMALTHEOR:YY1. #If35H is nitedimensionalthentherffeexistsauniqueDiracs2-function.2. #If35H isin nitedimensionalthentherffeexistsnoDiracs2-function.ZProof.@_1. GSinceHLj3_f7!(f*UQR _)2HV2 4$isanisomorphismthereisaF2_HU^sucrhthat(/*XUQR Y)X=":Then(fG*XUQR Y)X=(fW*(*UQR Y))=(fW*")="(fG)"="(fG)(9q*?UQR @)gwhicrhimpliesf9q=?"(f)s2.FVurthermorewrehavehUQR;s2i?=hUQR;1HDs2i=h(Ȅ*URUQR US);1HDiUR="(1H)=1K.2.8is[Swreedler]exerciseV.4.؄cffxff ̟ff ̎ ̄cffZLemma4.1.25.mQLffetkHYbea nitedimensionalHopfalgebra.ThenUQR)2HV2 isaleft35inteffgrali ugva(XUVUQRUW*(1))\{ S׹(UQR*(2)\}))UR=(XUVUQRUW*(1) S׹(UQR*(2)\}))a(9)vi y|&X|S׹(a)UQR*(1)% UQR *(2)\w=URXUQR*(1), aUQR*(2)(10)i UݟX3f(1) \|hUQR;f(2)iUR=hUQR;fGi1HD:(11)ިProof.@_LetꨟUQRQbSealeftinrtegral.8Then8Ѝ =XQa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)UR=X(aUQR)(1)$ S׹((aUQR)(2) \|)UR="(a)(XUVUQRUW*(1))\{ S׹(UQR*(2)\}))forallaUR2HV.8Hence"_ԍٔ?:(PUQR*(1)$ S׹(UQR*(2)\}))a=URP"(a(1) \|)(UQR*(1)% S׹(UQR*(2)\}))a(2)y=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)a(3)=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})"(a(2) \|)UR=a(PUQR*(1)$ S׹(UQR*(2)\})):%gConrverselya(PUQRUV*(1)!"(S׹(UQR*(2)')))C=(PUQRUV*(1)"(S׹(UQR*(2)')a))="(a)(PUQRUV*(1)"(S׹(UQR*(2)'))),^henceꨟUQR?=URPUQR*(1)%""(S׹(UQR*(2)'))isaleftinrtegral.SinceSisbijectivrethefollowingholdsAPQcPrS׹(a)UQR*(1)# UQR*(2)\w=URPS(a)UQR*(1) S21 S(S(UQR*(2)'))ye=URPUQR*(1)%" Sן21 S(S׹(UQR*(2)')S(a))UR=PUQR*(1) aUQR*(2)#:TheconrversefollowseasilyV.IfꨟUQR?2URInrtWşly(HV)isaleftinrtegralthenPUha;f(1) \|ihUQR;f(2)iUR=haUQR ;fGi=ha;1HDihUQR;fGi.Conrverselyif2HV2 !with(11)isgivrenthenha;fGiݹ=P@ha;f(1) \|ih;f(2)iݹ=ha;1HDih;fGi꨹henceaUR="(a). cffxff ̟ff ̎ ̄cffJIfGisa nitegroupthenI⍍s2(x)UR=z( 0ifx6=e;ɍ 1ifx=e:(12)7 &er1.pINTEGRALS!ANDF9OURIERTRANSFORMSa$127YInfactsinceCOisleftinrvXariantwegetfG(x)s2(x)UR=f(e)s2(x)forallxUR2GandfQ2URK2G.SincejG/HV2 ܹ=KGisabasis,wregets2(x)=0ifx6=e.oFVurthermoreUQRjs2(x)dx=P x2G =s2(x)UR=1implies(e)UR=1.8SowrehaveȄ=URe2.RIf%Gisa niteAbSeliangroupwreget= P:2P ^G"Ϡforsome _2K. YTherevXaluationgivres1ZF=hUQR;s2i= P:x2G;2P ^G4?Eh;xi.ANorwletZF2wbG .Then푟P>2P ^G"h;xi=CP 2P ^Gh;xi=h;xiP2P ^G h;xi.Since'Wforeacrhx2Gnfeg'Wthereisasuchthat>h;xiUR6=1andwregeteSX-K2P ^G`h;xiUR=jGje;x :"ӍHenceꨟPUx2G;2P ^G6`h;xiUR=jGj= 21ZandZMUȄ=URjGj 1 zX- \z2P ^G :(13)#LetYHbSe nitedimensionalfortherestofthissection."qInCorollary1.22wrehaveseen&thatthemapHM3f7!(UQR (fG)2HV2 Xis&anisomorphism.zZThismapwillbSecalledtheFourier35trffansform.ӍTheorem4.1.26.wXThe35FouriertrffansformHB3URfQ7!V~e*f K2HV2 is35bijectivewithcVse*Jf=UR(UQR US(fG)=XhUQR*(1)\};fiUQR*(2)(14)dThe35inverseFouriertrffansformisde nedbyne]Caݖ=URXSן 1 S((1) \|)ha;(2)i:(15)Sincffe35thesemapsareinversesofeachotherthefollowingformulashold$BGN hV)Re*f;gn9iUR=甆ZUTfG(x)g(x)dxUha;V*e*biUR=甆ZUTMXSן21 S(a)(x)b(x)dx*N fQ=URPSן21 S((1) \|)hV)Re*f;(2)iUaUR=PhUQR*(1)';)ea*iUQR*(2)#:(16)) Proof.@_WVe:usetheisomorphismsH ' !G7HV2 de nedbryV!b*f+ :=VI|e* *fR= *(UQR +(fG)=U^P hUQR*(1)';fGiUQR*(2)͹andHV2  7!qHde nedbryba j:=UR(a*s2)=P(1) \|ha;(2)i.8Becauseofu}ha;V*b*biUR=ha;(b*s2)i=hab;s2i(17)andqhV)Re*f;gn9iUR=h(UQR US(fG);gn9i=hUQR;fGgn9i(18)wregetforallaUR2HV2 andfQ2H  ha;UUObVvb*Lf _pi*=URhaV)Rb*f;s2i=Pha;(1) \|ihV)Rb*f;(2)iUR=Pha;(1) \|ihUQR;fG(2)i(bryLemma1.25)*=URPha;S׹(fG)(1) \|ihUQR;(2)iUR=ha;S׹(fG)ihUQR;s2iUR=ha;S(fG)i:7 &e128h4.pTHE!INFINITESIMALTHEOR:YYUVThis_&givresUU5bVb*If =S׹(fG).[SotheinversemapofH"|!>HV2 withVxb*fh=(UQR (fG)=VDe*fis_&HV2-P!HwithSן21 S(+ba+)&=P>Sן21((1) \|)ha;(2)i&=TQea '.Thenthegivreninversionformulasareclear.WVenoteforlateruseha;V*e*biUR=ha;Sן21 S(V*b*b)i=hSן21(a);V*b*bi=hSן21(a)b;s2i.2v-cffxff ̟ff ̎ ̄cff1oIfGisa nitegroupandHB=URK2G @then؍Vsje*Jf=URXvfx2GfG(x)x:!QSince.(s2)UR=Px2G#,x215 <) 1+x2o2wherethex2V2K2G ƹarethedualbasistothex2G,wregetsw8e a7`=URXvfx2Gha;x ix : xIf )Gisa niteAbSeliangroupthenthegroupsGandwSbG\areisomorphicsotheFVouriertransforminducesalinearautomorphisme- *:URK2G t u!K2G @andwrehaveeaw߹=URjGj 1 zX- \z2P ^Gha;i 1#ڍBysubstitutingtheformrulasfortheintegralandtheDiracs2-function(1)and(13)wreget͍VSe*Q_;f[4=URPx2G#,fG(x)x;W)eߖaQ=URjGj21 \zP'2P ^G.a()21 \|;㓍Q_;fQ=URjGj21 \zP'2P ^GV0Ae*.f5+4()21 \|;ߖaUR=Px2G#SWe#,a)-(x)x:(19)Thisimplies[VzAe*xDf*()UR=Xvfx2GfG(x)(x)=甆ZURf(x)(x)dx(20)ˍwithinrversetransform捍e?aj(x)UR=jGj 1 zX- \z2P ^G (a) 1 \|(x):(21)$JGCorollary4.1.27.zThe`FouriertrffansformsoftheleftinvariantintegralsinHand35HV2 arffeۖVq*_e*qz=UR"ǟ 1s2HV !andQX3ePUQR\۹=12HF::(22)Proof.@_WVePharvehV%e*q;fGi.=hUQR;s2fGi=hUQR;ǟ21 C(fG)s2i="ǟ21(fG)hUQR;s2i="ǟ21(fG)_henceV'e* k=UR"ǟ21 C.8FVromnPe꨹1 =(UQR US(1)=UQR?wregetSeUQR=UR1.cffxff ̟ff ̎ ̄cffProp`osition4.1.28.De ne35aconrvolutionmultiplication35onHV2 byxfchab;fGiUR:=Xha;Sן 1 S((1) \|)fGihb;(2)i: 7 &er1.pINTEGRALS!ANDF9OURIERTRANSFORMSa$129YThen35thefollowingtrffansformationruleholdsforf;gË2URHV:/ V 6satis esލgDS(a1:::anP)UR= knX ㇍Si=1a1:::ai1AVDS(aidڹ)ai+1AT::: anP:aLet (AbSeacommrutative (K-algebraandA Ma beanA-module.`ConsiderMa asanqA-A-bimoSdulebryma::=am.ͧWVeqdenotethesetofderivXationsfromAtoMybyDerݟK|(A;M@)c.y.Prop`osition4.2.2.O1. MsLffetՍAbeaK-algebra. MsThenthefunctorDerjKRL(A;-33)\:A-35Mo`do-#ԤAUR!Vec.~is35rffepresentablebythemoSduleofdi erenrtials A.2.pLffet6pAbeacommutativeK-algebra.pThenthefunctorDerMK/(A;-33)c c׹:5^A-35Mo`dfk!Vec+)is35rffepresentablebythemoSduleofcommrutativedi erentials 2cbA.Proof.@_1.jRepresenrtAasaquotientofafreeK-algebraA4:=KhXidji2Jri=IwhereB7o=iKhXidji2JriisthefreealgebrawithgeneratorsXidڹ.y_WVe rstprorvethetheoremforfreealgebras.a)ArepresenrtingmoSduleforDerKgg(B;-)is( BN>;dUR:BX !_7 B)withf B :=URBE Fƹ(dXidji2Jr) Bwhere\Fƹ(dXidjiM2Jr)isthefreeK-moSduleonthesetofformalsymrbolsfdXidjiM2Jrgasabasis.WVeMharvetoshowthatforeveryderivXationD:OB!TyMU1thereexistsauniquehomomorphisms'UR: B !oM+ofB-B-bimoSdulessucrhthatthediagramJ@{0B{/ B餟{fd'Ѝά--80dH`Ǐ`D餟ׁ @餟 @餟 @餟 @0$>@0$>RHM5RǠ*FfehǠ?''commrutes._TheY'moSdule B eisaB-B-bimoduleY'inthecanonicalwrayV._TheY'productsX1:::Xn -ofPthegeneratorsXi*ofB VformabasisforB.FVoranryproSductX1:::Xnwre&de ned(X1:::XnP)UR:=P*n U_i=1 ASX1:::Xi1h; &dXi Xi+1AT::: Xn Rvin&particulard(Xidڹ)UR=1eH dXi" 1.TVoK\seethatdisaderivXationitsucestoshorwthisonthebasiselements:(hō&]/d(X1:::XkiXk6+1 :::!ʪXnP)ɍi=URP*k U_jv=1!BX1:::Xjv1. dXj Xjv+1B:::! "Xk#Xk6+1 :::!ʪXnm'+P*n U_jv=k6+1+-9X1:::Xk#Xk6+1 :::!ʪXjv1. dXj Xjv+1B:::! "Xni=URd(X1:::Xk#)Xk6+1 :::!ʪXnR+X1:::Xkd(Xk6+1 :::!ʪXnP)q7 &e132h4.pTHE!INFINITESIMALTHEOR:YYNorwhletD~j:*Bm! KMbSeaderivXation.=De ne'by'(1 dXid 1)*:=DS(Xidڹ).=ThismapobrviouslyextendstoahomomorphismofB-B-bimoSdules.FVurthermorewehave荍'd(X1:::XnP)d=UR'(P jX1:::Xjv1. dXj Xjv+1B:::! "XnP)d=URPjfX1:::Xjv1B'(1 dXj 1)Xjv+1B:::! "Xn=URDS(X1:::XnP)hence'dUR=DS.TVomHshorwtheuniquenessof'let Ë:UR B !oM,bSeabimodulehomomorphismsucrhthatL n9d!=DS.^Then n9(1a dXiR; 1)!= d(Xidڹ)=DS(Xi)='(1a dXiR; 1).^SinceL and'areB-B-bimoSduleshomomorphismsthisextendsto Ë=UR'.b)NorwletAq:=KhXidji2Jri=I(bSeanarbitraryalgebrawithB =qKhXiji2Jrifree.8De ne'dG A 36:=UR BN>=(I B + BI++BdB(I)+dB(I)B):YWVei rstshorwthatI B$+] BN>IOi+BdB(I)+dB(I)BeoisiaB-B-subbimoSdule.Sincei B andI;areJB-B-bimoSdulesthetermsI B Band BN>I;arebimodules.VFVurthermorewrehavebdBN>(i)b20#=URbdB(ib209)DbidBN>(b20)UR2BdBN>(I)D+I B hencegI B @+ BN>I5+BdB(I)+dB(I)BisabimoSdule.Norw?WI B and BN>I0ڹaresubbimoSdulesofI B 2+P BN>Iӹ+BdB(I)+dB(I)B.6HenceAUR=B=I+actsonbSothsideson A Ȍsothat AbSecomesanA-A-bimodule.Let: B !5! A ȹandalso:Bht!PYAbSetheresiduehomomorphisms.ДSincedBN>(i)UR2dBN>(I)=0 A 0wreSgetauniquefactorizationmapdA 36:URAn!1 AsucrhSthatBl34hA34螴 A1 32fdά-dX.A:4B:4f Bɋ<:2fd\ά-`OdX.BǠ@feRܟǠ?(b)"ˬitisclearthatdA ȌisaderivXation.LetD:URAn!1MEbSeaderivXation.ZTheA-A-bimoduleMEisalsoaB-B-bimoSdulebrybmUR=: z bT.Then'(i!n9)7=gz' Ni^'(!)=0Bandsimilarly'(!i)7=0.ALetBbdBN>(i)2BdB(I)Bthen'(bdB(i))7=: z b'dBN>(i)UR=: z bT(I)#+dB(I)Bmand]gthrusfactorizesthroughauniquemap Ë:UR A 36 L!M@. Obviously ͖is_]ahomomorphismofA-A-bimoSdules. rFVurthermorewrehaveDS=UR'dB = n9dB= n9dAand,Usince@0issurjectivre,UD:v= n9dA.9xItisclearthat iisuniquelydeterminedbrythiscondition.2.=If?Aiscommrutative?thenwrecanwriteA ƹ=K[Xidji2Jr]=I¹and? 2cbB Z=B Fƹ(dXidڹ).With* 2cbA =1 2cbBN>=(I 2cbB $ +BdB(I))*theproSofisanalogoustotheproofinthenoncommrutativesituation.cffxff ̟ff ̎ ̄cffKRemark4.2.3.j61. A Giscgeneratedbryd(A)asabimoSdule, RhenceallelementsareoftheformPUi-aidd(a20RAi)a20N90RAir.8Theseelemenrtsarecalleddi erffentials.e2.0IfѷAUR=KhXidi=I,ִthen A isgeneratedasabimoSdulebrytheelementsf`zcП pd(Xidڹ)cg.3.LLet7f\829B?=KhXidi.Let7B2opoZbSethealgebraoppositetoB=(withoppositemrultiplication).nThenR B =URB Fƹ(dXidڹ) B*XisthefreeB B2op uleftmoSduleorverthefreegeneratingsetfd(Xidڹ)g.8Henced(fG)hasauniquerepresenrtation𼍒d(fG)UR=X ㇍ iō*@f۟[zF ΍@Xi-Td(Xidڹ)!;withuniquelyde nedcoSecienrtsFōf@f[zF ΍@Xi2URBE B op ~#:ڋInthecommrutativesituationwrehaveuniquecoSecientsō`@fWF[zF ΍@Xi2URK[Xidڹ]:4.8WVegivrethefollowingexamplesforpart3:[=ō@Xi4[zv ΍@Xj/=URijJ;ō@X1X2[z# ΍@8@X17l=UR1 X2;ō@X1X2[z# ΍@8@X27l=URX1j 1;ōB@X1X2X3B[z2\P ΍p@X2'j=URX1j X3;ōR@X1X3X2R[z2\P ΍p@X27l=URX1X3j 1:[Thisisobtainedbrydirectcalculationorbytheprffoduct35ruleōf@fGgd[zF ΍@XiJ=UR(1 gn9)ō<@f33[zF ΍@XiT+(f 1)ō@g33[zF ΍@Xif:07 &e134h4.pTHE!INFINITESIMALTHEOR:YYTheproSductrulefollorwsfromʍ8C&d(fGgn9)UR=d(f)g+fd(gn9)=X((1 g)ō<@f33[zF ΍@XiT+(f 1)ō@g33[zF ΍@Xif)d(Xidڹ):QLetAUR=KhXidi=I.8IffQ2I+then`zQ1 pd(fG)+=dA(: z f)=0hencevקXōi@f`0[zF ΍@XiϓdA(\-z %F ӍXi %F)UR=0:݄These.arethede ningrelationsfortheA-A-bimoSdule A withthegeneratorsdA(\-z %F ӍXi %F).%FVorHmotivXationofthequanrtumgroupcaseweconsiderananealgebraicgroupGwithrepresenrtingcommutativeHopfalgebraA.RecallthatHomD(A;RJ)isanalge-braizwiththeconrvolutionizmultiplicationforeveryR82K-cAlg$andthatG(RJ)=K-cAlgo(A;RJ)BHom(A;RJ)isasubgroupofthegroupofunitsofthealgebraHomy(A;RJ).%De nitionandRemark4.2.4.ßA linear mapT-:AO!1A iscalledleftPtrffansla-tion35invariant,ifthefollorwingdiagramfunctorialinRn2URK-cAlg"commutes:@J[fTG(RJ)Hom$1(A;R) NHom!(A;RJ)̘ܞ32fd8?ά-[fTG(RJ)Hom$1(A;R) NHom!(A;RJ)̘ܟ:2fd8?ά-Ǡ@feǠ?ꃀai1 HomD(TV;R )$xǠ@fe$Ǡ?ꃀ)^0then35Lie8(HV)35isarffestricted35Liealgebrffaorap-Liealgebra.Proof.@_Letka;b2HYbSeprimitivreelements. ;Then([a;b])=(abba)=(aT 1+1 a)(b 1+1 b)(b 1+1 b)(a 1+1 a)UR=(abTba) 1+1 (abba)hence(Lie}(HV)URH2L Vis(aLiealgebra.,`IfthecrharacteristicofKispUR>0(thenwehave(ag 1+1 a)2p=URa2p. 1+1 a2p].-ThrusLie*(HV)isarestrictedLiesubalgebraofH2LZ΍withthestructuremaps[a;b]UR=abba꨹anda2[p] ՗=URa2p].;cffxff ̟ff ̎ ̄cffCorollary4.3.3.sWLffetHPbeaHopfalgebra.Thenthesetoflefttranslationin-variantderivationsDO:qHP2!LH isaLiealgebrffaunder[DS;D20!ǹ]=DD20uSD20!D.IfZ΍crhar3|=URp35thenthesederivationsarffearestrictedLiealgebrawithDS2[p] )%=URDS2p.mProof.@_Themap m:HV2o !}HV22`p  !#7End8DN(HV)isahomomorphismofalgebrasbryF4.2.6. MHence (dd20fd20d)+=(dd20d20d)+=(d)(d209)(d20)(d).IfdisaprimitivreelementinHV2o ~thenby4.2.7and4.3.1theimageDn.:= (d)inEnd (HV)misalefttranslationinrvXariantmderivationandalllefttranslationinrvariantderivXationsuareofthisform.Since[d;d209]UR=dd20 d20duisagainprimitivrewegetthat[DS;D20!ǹ]==DSD20D20Dzis'alefttranslationinrvXariant'derivationsothatthesetoflefttranslationinrvXariantderivXationsDerxgHgKٹ(HF:;HV)isaLiealgebraresp.darestrictedLiealgebra.]cffxff ̟ff ̎ ̄cffDe nition4.3.4.vaùLetSHbSeaHopfalgebra.Anelemenrtc2HisScalledcffocom-mutativeifW(c)=(c),i.e.ifPic(1) ;0c(2)"=PiQc(2) c(1) \|.LetCܞ(HV):=fc2HVjc꨹iscoScommrutativeaTg.LetG(HV)denotethesetofgrouplikreelementsofHV.Lemma4.3.5.g5QLffetw?HdbeaHopfalgebra.'ThenthesetofcocommutativeelementsCܞ(HV)0isasubffalgebra0ofHandthegrffouplikeelementsG(H)formalineffarlyinde-pffendentsubsetofCܞ(HV).UFurthermoreG(HV)isamultiplicativesubgroupofthegroupof35unitsU@(Cܞ(HV)).C7 &egjC3.pTHE!LIEALGEBRAOFPRIMITIVEELEMENTSVt137YProof.@_It3isclearthatCܞ(HV)isalinearsubspaceofH. Ifa;bk2Cܞ(H)3then(ab) =(a)(b)=(W)(a)()(b) =W((a)(b))=(ab)aand(1) =1Ǿ 1 =W(1).8ThrusCܞ(HV)isasubalgebraofH.ThehgrouplikreelementsobviouslyarecoScommutativeandformamultiplicativegroup,henceasubgroupofU@(Cܞ(HV)). TheyarelinearlyindepSendenrtbyLemma2.1.14.Ցcffxff ̟ff ̎ ̄cffsProp`osition4.3.6.OLffetHbeaHopfalgebrawithSן22-=URid H9:.VThenthereisaleftmoffdule35structuren:Cܞ(HV) Lie(H)UR3c aUR7!caUR2LieZ(H)s2withc}aUR:=rHD(rH _ }1)(1 W)(1 S/ 1)( 1)(c a)UR=Pc(1) \|aS׹(c(2))suchthat]⍒c[a;b]UR=X[c(1)$a;c(2)b]:In35pffarticularG(HV)actsbyLieautomorphismsonLie8(H).sProof.@_ThegivrenactionisactuallytheactionH= HG !H withha=Ph(1) \|aS׹(h(2)),theso-calledadjoint35action.WVej rstshorwthatthegivenmaphasimageinLieo(HV).FVorc/2Cܞ(H)janda/2LieU(HV)wrehave(c0a)UR=(Pc(1) \|aS׹(c(2)))UR=P(c(1) \|)(a0 1+1 a)(S(c(2) \|))UR=P(c(1) \|)(ah 1)(S׹(c(2)))+Pv(c(2))(1 a)(S׹(c(1)))UR=Pc(1)aS׹(c(4))h c(2)S׹(c(3))+`Pc(3) \|S׹(c(2)) c(4) \|aS(c(1))#=ca 1+1 ca sinceciscoScommrutative,qS22=#id TǟHandaisprimitivre.WVe6shorwnowthatLie<:(HV)isaCܞ(H)-moSdule.(cd)ސa=Pc(1) \|d(1)aS׹(c(2)d(2))=Pc(1) \|d(1)aS׹(d(2))S(c(2))UR=c(da).8FVurthermorewrehave1aUR=1aS׹(1)=a.TVoshorwthegivenformulaleta;bPG2LieU(HV)andcPG2Cܞ(H).eThencup[a;b]PG=Pc(1) \|(abba)S׹(c(2))UR=Pc(1) \|aS(c(2))c(3)bS(c(4))Pc(1)bS(c(2))c(3)aS(c(4))UR=P(c(1) \|a)(c(2) Ipb)P (c(1)b)(c(2)a)UR=P[c(1)Ipa;c(2)b]dagainsincecUR2Cܞ(HV)discoScommrutative.`Norw^letg;2͚G(HV). GThenga=gn9aS׹(g)=gag21(Թsince^S׹(g)=g21(Թfor^anrygroup,likreelement.FVurthermoregEא[a;b]Ř=[gאa;gb],henceg׹de nesaLiealgebraautomorphismofLie(HV).$鯄cffxff ̟ff ̎ ̄cffsProblem4.3.2.nRShorwthattheadjointpactionH GHB3URh a7!Ph(1) \|aS׹(h(2))2HmakresHanHV-moSdulealgebra.De nitionandRemark4.3.7.ßTheJalgebraK(s2)K=K[]=(2236)Jiscalledtheal-gebraofdual35numbffers.8ObservrethatK(s2)UR=KK]ڹasaK-moSdule.WVeconsider]ڹasa"smallquanrtity\whosesquarevXanishes.Themapsp :K(s2)!!K⺹withp()=0andj0: K!!K()arealgebrahomomor-phismsatisfyingpj%=URid .LetK(;s220Ak):=K[;s220Ak]=(s22236;s220 u^25).ThenK(;s220Ak)=KCK:uKs220Ks220.ThemapK(s2)37!20L2K(;20Ak)Sisaninjectivrealgebrahomomorphism.FVurthermoreforevrery h2URKwehaveanalgebrahomomorphism' J:URK(s2)3Ȅ7! 2K(s2).ThesemdalgebrahomomorphismsinducealgebrahomomorphismsH" K(s2)URn!1H K(s2)resp.8H K()URn!1H K(;20Ak)forevreryHopfalgebraHV.X7 &e138h4.pTHE!INFINITESIMALTHEOR:YYProp`osition4.3.8.OThe35mapz4oNne L- /:URLieZ(HV)UR!H K(s2)URH K(;s2 0Ak)with35e2La hɹ:=UR1+a Ȅ=1+s2a35iscffalled35theexpSonenrtialmapandsatis es!񪝍e2L(a+b)D=URe2La we2Lb2;=Te2L a`=UR' (e2La w); e2L-:0[a;b]p=URe2La we2L̟-:0b K(e2La)21 \|(e2L̟-:0b)21:#kFurthermorffe^allelementse2La[2Hu |K(s2)aregrouplikeintheK(s2)-HopfalgebraH K(s2).DProof.@_1.8e2L(a+b)D=UR(1+s2(a+b))=(1+s2a)(1+b)UR=e2La we2Lb2.2.8e2L a`=UR1+s2 a=' (1+s2a)=' (e2La w).ꍑ3.$Sincet(1+s2a)(1a)UR=1twrehave(e2La w)UR=1s2a.$Sotwegete2L-:0[a;b]p=UR1+s2[a;b]=1N+s2(aa)+20Ak(bb)+20Ak(ababba+ab)UR=(1N+a)(1+20Akb)(1a)(120Akb)UR=e2La we2L̟-:0b K(e2La)21 \|(e2L̟-:0b)21.4.3K(L)e(e2La w)@=(1W+a s2)@=1W K(L)ֹ1+(a 1+1 a) yr=@1 K(L)1+s2aQ K(L)1+1 K(L)s2a+a K(L)a=(1Q+a) K(L)(1+a)=e2La e K(L)e2La Jand"K(L)e(e2La w)UR="K(L)(1+s2a)UR=1+"(a)UR=1.ucffxff ̟ff ̎ ̄cff\Corollary4.3.9.sW(LieU(HV);e2L-ݹ) xisthekernelofthegrffouphomomorphismpUR:GK(L)e(H K(s2))!G(HV).TvProof.@_pUR=1_ pUR:H _K(s2)n!1H _K=HisahomomorphismofK-algebras.WVePshorwthatitpreservesgrouplikeelements.kObservethatgrouplikeelementsinH  K(s2)v3arede nedbrytheHopfalgebrastructureoverK(s2).ہLetg2BGK(L)e(H K(s2)).8Then(H 1)(gn9)UR=g K(L)gXand("H 1)(gn9)=12K(s2).SincepUR:K(s2)n!1Kisanalgebrahomomorphismthefollorwingdiagramcommutes@3TqU(H K) (H K)AH H K:ۑԞ32fd33@ά-R΍⍍88=ӘZV۹(H K(s2)) K(L)(H K(s2))Ә }_H H K(s2)T:2fdά-@n⍍=ÒǠ@feğǠ?ꃀn'*(1 p) (1 p)1Ǡ@fe1ğǠ?PM6D1 p:TWVeidenrtifyelementsalongtheisomorphisms.cThusweget(H ͟ [1K)(1H p)(gn9)m=(1H Hl p)(H 6 1K(L)e)(gn9)t=((1H p) K(L)(1H p))(g K(L)gn9)t=(1H p)(gn9) (1H K p)(gn9),soe/that1H pUR:GK(L)e(H] K(s2))n!1G(HV). cNorwe/wehave(1H K p)(gn9g20+~f1ډ=f0 ~1+f1 ډ2gWHV2o HV2o=gWHV2o  K(s2). Then" fj isahomomorphism1ofalgebrasi fG(ab)9=f(a)f(b)1andf(1)9=11i f0(ab)9=f0(a)f0(b)and6f1(ab)UR=f0(a)f1(b);+f1(a)f0(b)6andf0(1)UR=16andf1(1)UR=06i Ho F(f0)UR=f0 ;f0andHo F(f1)UR=f0 f1+f1 f0and"Ho F(f0)UR=1and"Ho F(f1)=0i (Ho  Ϲid sK(L)u)(f0 1+f1Ѭ s2)UR=f0 f0 1+f0 f1 ڹ+f1 f0 Ȅ=UR(f0 1+f1 s2) K(L)v(f0 1+f1 s2)7 &e142h4.pTHE!INFINITESIMALTHEOR:YYandu("Ho hzid 3K(L) )(f0(~ hz1+f1 s2)/=1hz 1ui (Ho hzid 3K(L) )(fG)/=fy K(L)|fItand("Ho id uLK(L)N)(fG)UR=1i fQ2URGK(L)e(HV2o3 K(s2)).Hence;wwrehaveabijectivemap!M:K-cAlgo(HF:;K(s2))3f&ݹ=f0+f1R7!f0 ᬹ1+f1 Rm2GK(L)e(HV2o K(s2)).-SincethegroupmrultiplicationinK-cAlgo(HF:;K())Homy(HF:;K(s2))istheconrvolutionandthegroupmrultiplicationinGK(L)e(HV2oe NOK())URHV2o E K(s2)} istheordinaryalgebramrultiplication,wherethemultiplicationofHV2oagain{istheconrvolution,Lit{isclearthat!1isagrouphomomorphism.FVurthermorethĕrighrthandsquareoftheabSovediagramcommutes..ThuswegetanisomorphismeZ:Lie_(HV2o)t!έLie #(G.)(K)#onthekrernels.RThismapisde nedbye(d)Z=1'+d2K-cAlgo(HF:;K(s2)).TVo)*shorwthatthisisomorphismiscompatiblewiththeactionsofKresp.fG(HV2o)let 2{K,a2HV,andd2Lieܹ(HV2o).|8WVeharvee( d)(a){="(a)+ d(a)={u ("(a)+d(a)s2)UR=(u ?(1+d))(a)UR=(u ?e(d))(a)=( e(d))(a)hencee( d)UR= e(d).FVurthermoreletg2G(HV2o)=K-cAlgo(HF:;K), a2HV,andd2Lie(HV2o).Thenwreharvee(g~d)(a) g =e(gn9dg21 ʵ)(a) g =(1+gn9dg21 ʵs2)(a) g ="(a)+gn9dg21 ʵ(a) ;=Pgn9(a(1) \|)"(a(2))gS׹(a(3))+PBKg(a(1) \|)d(a(2))gS׹(a(3))=PPig(a(1))e(d)(a(2))gS׹(a(3))=`(jW{ge(d)jgn921 ʵ)(a)UR=(ge(d))(a)hencee(gd)UR=ge(d).aScffxff ̟ff ̎ ̄cffƏProp`osition4.4.4.OLffet.H beaHopfalgebraandletI ̴:=1KerL(").XThen(vDerݟ"0(HF:;-33)%:Vec F"_!43VecN?isbrffepresentablebyI=I22 andd%:H21"ps{ !NI2 Ȍpw !S(I=I22,inpffarticulare. }Der@Z"E;*(HF:;-33)PUR԰n9=Hom(y(I=I 2;-)3EandK.Lie](HV o)PUR԰n9=Hom(y(I=I 2;K):ƏProof.@_Becausezof"(id ʢu")(a)UR="(a)"u"(a)UR=0zwrehaveImd(id ʢ")URI.LetiUR2I.Then wrehaveiUR=iߍ"(i)UR=(id ʢ")(i) henceIm(id")UR=KerBm(").WVe harveI22 3(id ʢ")(a)(id")(b)i=ab"(a)ba"(b)+"(a)"(b)i=(id ʢ")(ab)"(a)(id")(b)(id ʢ")(b). 7KHence?vwrehaveinI=I22 the?vequation(id ʢ")(ab)M(="(a)(id")(b)@+(id ʢ")(a)"(b)sothatǹ(id")UR:HB\3!IF``!I=I22 /isan"-derivXation.Norw'SletD#:Hv! Mh7bSean"-derivXation.ThenD(1)=D(11)=1D(1)+D(1)1hencecDS(1)(=0.qItfollorwsDS(a)=D(id ʢ")(a).qFVromc"(I)=0wregetDS(I22)"(I)DS(I)+D(I)"(I)UR=0hencethereisauniquefactorization@ sH ұIl:2fdЍά-ށ$idO,"HQHQJI=I22U<:2fd{ά-č3?<D9|?`H9|?`H9|?`H9|?`H9|?`H9|?`HIHIj?<D?`@?`@?`@\@\RM:Ǡ@feǠ?ꃀ m cffxff ̟ff ̎ ̄cff7 &e1Һ4.pDERIVA:TIONS!ANDLIEALGEBRASOFAFFINEALGEBRAICGR9OUPS l143YProp`osition4.4.6.OLffetQHybeacommutativeHopfalgebraandH M5beanHV-moffdule.۳Thenwehave HP U԰ m=H) I=I22 a*andd;:H)~!=H I=I22 a*isgivenbyd(a);=f`Pa(1)$ `z>n( p(id ʢ")(a(2) \|)A.Proof.@_ConsiderIthealgebraB˹:=HŢM-with(a;m)(a209;m20)=(aa20;am20۹+a209m).LetDG_B=K-cAlgo(HF:;-).ThenwrehaveG.(B)Hom"(HF:;B)P԰=Hom+!(HF:;HV)2Homy(HF:;M@).Anselemenrt(';DS)UR2Hom(HF:;B)sisinG.(B)i (';DS)(1)UR=('(1);DS(1))=(1;0),@_hence'(1)=1andDS(1)=0,@_and('(ab);DS(ab))=(';DS)(ab)=(';DS)(a)(';D)(b)J=('(a);D(a))('(b);D(b))J=('(a)'(b);'(a)D(b)- +D(a)'(b),henceH'(ab)='(a)'(b)andDS(ab)='(a)D(b)g+D(a)'(b).SoH(';D)isinG.(B)i 'B2G.(HV)v6andDĹisa'-derivXation.ۊThe-mrultiplicationinHom(HF:;B)isgivenby(';DS)#m('209;DS20!ǹ)>=('#m'209;'DS20E4+Dv'209) bryapplyingthistoanelementa>2HV.Since7(';0)2G.(B)and(u";DS)2G.(B)forevrery"-derivXationDS,[thereisabi-jectionhDer,П"(HF:;M@)P԰=ef(u";D"lй)2G"(B)gP԰=ef(1HD;D1)2G1(B)gP԰=eDer'BK.$(HF:;M@)bry(u";D"lй))y7!(1;0)e(u";D"lй))y=(1;1eD"lй))y2G1(B)withinrversemap(1;D1))y7!(S ;0)Ö(1;D1)=(u";SvmÖD1)2G"lй(B).HenceFwrehaveisomorphismsDer#K(HF:;M@)P԰=Derݟ"0(HF:;M@)PUR԰n9=Hom(y(I=I22;M@)PUR԰n9=Hom(yH0ȹ(H I=I22;M@).Thehunivrersal"-derivXationforvectorspacesis: z id ʢ"#4:+AEX!q'I=I22.Thehuniversalf`"-derivXation07forHV-moSdulesisD"lй(a)u=1O `z3 p(id ʢ")(a)<p2uA I=I22. The07univrersal1-derivXation}uforHV-moSdulesis1D" Ewith}u(1D"lй)(a)=Pa(1)c `z>n( p(id ʢ")(a(2) \|)H-2`A I=I22.wcffxff ̟ff ̎ ̄cffΉ;7  ,- cmcsc10+@ cmti12)ppmsbm8( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12o cmr9XQ cmr12O line10