; TeX output 2000.06.26:15377 YRXQ cmr12CHAPTER4Nff cmbx12TheffIn nitesimalTheory_o cmr9139*7 &e140h4.pTHE!INFINITESIMALTHEOR:YY(qN cmbx124.9Deriv@ationsandLieAlgebrasofAneAlgebraicGroupsLemmaandDe nition7.4.1.G@ cmti12Lffet7!!", cmsy10G+:]* msbm10K-35cAlg"B$d!4cSetK()beagroupvaluedfunc-tor.fiThe35kernelLie #(G.)(g cmmi12RJ)oftheseffquenceyFι0t1Lie #(G.)(RJ)Oj<fd!YO line10-M4G.(RJ(s2))fdά-MMMtfd)P- ("K cmsy8Gv|{Ycmr8(2cmmi8p)t32fd)PάW`YmGv(jv)&G.(RJ)e0DBfdά-MMis35cffalledtheLiealgebraofGcandisagroupvaluedfunctorinRJ. - cmcsc10Proof.@_FVorevreryalgebrahomomorphismf:kR_! S@thefollowingdiagramofgroupscommrutesGT͍ѭF0ѭt1Lie #(G.)(RJ)Oj<fd!Yά-Tѭѭ4G.(RJ(s2))fdά-TTTṫfd)P- (Gv(p)t:2fd)Pά^`YmGv(jv)ѭ&G.(RJ)ѭe0DBfdά-Tέ7n`@fejLn`?έjڟn`@fe n`?*@PGv(f(L))έ3n`@fe3̟n`?*@8LGv(f)Fι0tLie #(G.)(S׹)Oj<fd!ά-MÚG.(S׹(s2))\fd@ά-MMM|fd*Ѝ- (Gv(p)|32fd*Ѝ#0άW`YmGv(jv)'\G.(S׹)e0CLfdj0ά-MB\ %cffxff ̟ff ̎ ̄cff Prop`osition4.4.2.OLffet?G":lK-35cAlg"Q$%!4$SetKMbeagroupvaluedfunctorwithmul-tiplicffation35.fiThentherearefunctorialoperations~x]G.(RJ)Lie #(G)(RJ)UR3(gn9;x)7!gx2Lie #(G)(RJ)UVjvSRLie #(G.)(RJ)UR3(a;x)7!ax2Lie(G.)(RJ)such35that(ʍOg(x+yn9)UR=gx+gyn9;h(gx)UR=(hgn9)x;%1a(x+yn9)UR=ax+ay;K(ab)xUR=a(bx);og(ax)UR=a(gx):1 Proof.@_FirstobservrethatthecompSosition+onLie #(G.)(RJ)isinducedbythemrultiplicationofG.(RJ(s2))soitisnotnecessarilycommutative.WVede negIAxUR:=G.(jӹ)(gn9)xG(jӹ)(gn9)21 \|.7ThenG(p)(gIAx)UR=G.(p)G(jӹ)(gn9)G(p)(x)G.(p)G(jӹ)(gn9)21ι=URg1g21 =UR1hencegx2Lie #(G.)(RJ).Norwqleta2RJ.+~f1ډ=f0 ~1+f1 ډ2gWHV2o HV2o=gWHV2o  K(s2). Then" fj isahomomorphism1ofalgebrasi fG(ab)9=f(a)f(b)1andf(1)9=11i f0(ab)9=f0(a)f0(b)and6f1(ab)UR=f0(a)f1(b);+f1(a)f0(b)6andf0(1)UR=16andf1(1)UR=06i Ho F(f0)UR=f0 ;f0andHo F(f1)UR=f0 f1+f1 f0and"Ho F(f0)UR=1and"Ho F(f1)=0i (Ho  Ϲid sK(L)u)(f0 1+f1Ѭ s2)UR=f0 f0 1+f0 f1 ڹ+f1 f0 Ȅ=UR(f0 1+f1 s2) K(L)v(f0 1+f1 s2)W7 &e142h4.pTHE!INFINITESIMALTHEOR:YYandu("Ho hzid 3K(L) )(f0(~ hz1+f1 s2)/=1hz 1ui (Ho hzid 3K(L) )(fG)/=fy K(L)|fItand("Ho id uLK(L)N)(fG)UR=1i fQ2URGK(L)e(HV2o3 K(s2)).Hence;wwrehaveabijectivemap!M:K-cAlgo(HF:;K(s2))3f&ݹ=f0+f1R7!f0 ᬹ1+f1 Rm2GK(L)e(HV2o K(s2)).-SincethegroupmrultiplicationinK-cAlgo(HF:;K())Homy(HF:;K(s2))istheconrvolutionandthegroupmrultiplicationinGK(L)e(HV2oe NOK())URHV2o E K(s2)} istheordinaryalgebramrultiplication,wherethemultiplicationofHV2oagain{istheconrvolution,Lit{isclearthat!1isagrouphomomorphism.FVurthermorethĕrighrthandsquareoftheabSovediagramcommutes..ThuswegetanisomorphismeZ:Lie_(HV2o)t!έLie #(G.)(K)#onthekrernels.RThismapisde nedbye(d)Z=1'+d2K-cAlgo(HF:;K(s2)).TVo)*shorwthatthisisomorphismiscompatiblewiththeactionsofKresp.fG(HV2o)let 2{K,a2HV,andd2Lieܹ(HV2o).|8WVeharvee( d)(a){="(a)+ d(a)={u ("(a)+d(a)s2)UR=(u ?(1+d))(a)UR=(u ?e(d))(a)=( e(d))(a)hencee( d)UR= e(d).FVurthermoreletg2G(HV2o)=K-cAlgo(HF:;K), a2HV,andd2Lie(HV2o).Thenwreharvee(g~d)(a) g =e(gn9dg21 ʵ)(a) g =(1+gn9dg21 ʵs2)(a) g ="(a)+gn9dg21 ʵ(a) ;=$u cmex10Pgn9(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$idO,"HQHQJI=I22U<:2fd{ά-č3?<D9|?`H9|?`H9|?`H9|?`H9|?`H9|?`HIHIj?<D?`@?`@?`@\@\RM:Ǡ@feǠ?ꃀ m cffxff ̟ff ̎ ̄cff2;7 &e1Һ4.pDERIVA:TIONS!ANDLIEALGEBRASOFAFFINEALGEBRAICGR9OUPS l143YProp`osition4.4.6.OLffetQHybeacommutativeHopfalgebraandH M5beanHV-moffdule.۳Thenwehave HP U԰ m=H) I=I22 a*andd;:H)~!=H I=I22 a*isgivenbyd(a);=f`Pa(1)$ `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.$(HF:;M@)bry(u";D"lй))y7!(1;0)e(u";D"lй))y=(1;1eD"lй))y2G1(B)withinrversemap(1;D1))y7!(S ;0)Ö(1;D1)=(u";SvmÖD1)2G"lй(B).HenceFwrehaveisomorphismsDer#K(HF:;M@)P԰=Derݟ"0(HF:;M@)PUR԰n9=Hom(y(I=I22;M@)PUR԰n9=Hom(yH0ȹ(H I=I22;M@).Thehunivrersal"-derivXationforvectorspacesis: z id ʢ"#4:+AEX!q'I=I22.Thehuniversalf`"-derivXation07forHV-moSdulesisD"lй(a)u=1O `z3 p(id ʢ")(a)<p2uA I=I22. The07univrersal1-derivXation}uforHV-moSdulesis1D" Ewith}u(1D"lй)(a)=Pa(1)c `z>n( p(id ʢ")(a(2) \|)H-2`A I=I22.wcffxff ̟ff ̎ ̄cffK;7  +ppmsbm8* msbm10$u cmex10"K cmsy8!!", cmsy10 ;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8- cmcsc10@ cmti12o cmr9N cmbx12Nff cmbx12XQ cmr12O line10Y