; TeX output 2000.06.26:15317 YRXQ cmr12CHAPTER4Nff cmbx12TheffIn nitesimalTheory_o cmr9130*7 &eC2.pDERIVA:TIONSNf131YN cmbx122.Deriv@ationsDe nition4.2.1.vaùLet7g cmmi12AbSea) msbm10K-algebraand2cmmi8A MAbe7anA-A-bimodule(withidenrticalK-actiononbSothsides).8AlinearmapD:URA !", cmsy10n!1M+iscalleda,@ cmti12derivationifv]DS(ab)UR=aD(b)+D(a)b:ThesetofderivXationsDer*ppmsbm8Kgg(A;AMA)isaK-moSduleandafunctorinA ȌMA.ByinductiononeseesthatD>6satis esލgDS(a|{Ycmr81:::anP)UR= kn#u cmex10X ㇍Si=1a1:::ai!K cmsy81AVDS(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.- cmcsc10Proof.@_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'ЍO line10--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)7 &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.;cmmi6A: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:'Ѡ7 &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)^