; TeX output 2000.06.26:15347 YRXQ cmr12CHAPTER4Nff cmbx12TheffIn nitesimalTheory_o cmr9135*7 &e136h4.pTHE!INFINITESIMALTHEOR:YY]N cmbx123.n[TheLieAlgebraofPrimitiveElementsLemma4.3.1.g5Q@ cmti12Lffetrg cmmi12H` beaHopfalgebraandHV22cmmi8o beitsSweedlerdual. $Ifd!!", cmsy102Derݟ+ppmsbm8K|(HF:;"l* msbm10K"lй)<Hom(H;K)PisaderivationthendisaprimitiveelementofHV2o.Furthermorffe35everyprimitiveelementdUR2HV2o Kis35aderivationinDerK(HF:;"lK"lй).- cmcsc10Proof.@_Letd߹:H5!2KbSeaderivXationandleta;b2HV.mThen(b*d)(a)=d(ab)ɺ="(a)d(b)+d(a)"(b)ɺ=(d(b)"+"(b)d)(a)[hence(bɺ*d)=d(b)"+"(b)d.Consequenrtly{wehaveHVdL+=(H9*d)K" e+KdsothatdimHVdL+2<1.Thisshorws'odp2HV2o.5FVurthermorewrehavehd;aU bip=hd;abi=d(ab)=d(a)"(b)U+"(a)d(b)ͼ=hd ";a bi+h" d;a biͼ=h1H ;cmmi6o d+d 1Ho F;a bi1fhence(d)ͼ=d 1Ho+1Ho d꨹sothatdisaprimitivreelementinHV2o.Conrversely*fletdu2HV2o |bSe*fprimitivre.thend(ab)u=h(d);aZ biu=d(a)"(b)Z+"(a)d(b).|cffxff ̟ff ̎ ̄cffProp`ositionandDe nition4.3.2.ajLffetHbeaHopfalgebra.7:Thesetofprimi-tiveelementsofHx=willbffedenotedbyLie<(HV)andisaLiealgebra..PIfcrhari(K)UR=p>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|{Ycmr8[p] ՗=URa2p].;cffxff ̟ff ̎ ̄cffCorollary4.3.3.sWLffetHPbeaHopfalgebra.Thenthesetoflefttranslationin-variantderivationsDO:qHP2!LH isaLiealgebrffaunder[DS;D2"K cmsy80!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.if$u cmex10Pic(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)).7 &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 ̎ ̄cffsProblem4.3.1.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.7 &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-:#q% cmsy60[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@O line10-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