; TeX output 1999.11.03:0816S7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr983T*7 &e841:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYN cmbx123.DualObjectsArt!qtheendofthe rstsectioninCorollary3.1.15wesawthatthedualofang cmmi12HV-moSduleScanbeconstructed.WVedidnotshorwthecorrespondingresultforcomodules.InkfactsucrhaconstructionforcomoSdulesneedssome nitenessconditions.Withthisrestriction'thenotionofadualobjectcanbSeinrtroducedinanarbitrarymonoidalcategoryV.CCDe nition3.3.1.vaùLet(!", cmsy10C5; )bSeamonoidalcategoryM;2aWC;Tbeanobject.DAnobject,sM@2 K cmsy8 72OCߨtogetherwithamorphismev:M@2\ tM34!Iiscalleda,@ cmti12leftodualforM+ifthereexistsamorphismdbN*:URIF4!M M@2 됹inCݹsucrhthatsʍx(M26|{Ycmr8dbr 1p ]!M M@2 M21 evp6 !@M@)UR=12cmmi8Mm߹(M@22 V:1 dbp Ҳ( ! M@2 M M@22 V:ev" 1p Ua!!TM@2)UR=1M"!q% cmsy6 D:Amonoidalcategoryiscalledleft35rigidifeacrhobjectM62URCݹhasaleftdual.Symmetricallywrede ne:canobject2oM62URCbtogetherwithamorphismev2:Mh3 'O2SM4!Iis,calledarightodualforMmqifthereexistsamorphismdb7:zI4!2VMi ׅMinCsucrhthat7ȍʍxU(M261 dbp h!9M 2jM M26evb 1p@ ]!M@)UR=1MmW(2M26dbr 1p ]!2"dM M 2jM21 evp6 !@2 M@)UR=1UTM D:NAmonoidalcategoryiscalledright35rigidifeacrhobjectM62URCݹhasaleftdual.ThemorphismsevkanddbarecalledtheevaluationrespSectivrelythedual35bffasis.CCRemark3.3.2.j6If(M@2;ev /)isaleftdualforMǹthenobrviously(M;ev)isarighrtdualforM@2 됹andconrverselyV.8OneusesthesamemorphismdbN*:URIF4!M M@2.Lemma3.3.3.g5QLffet(M@2;ev /)bealeftdualforM@.{Thenthereisanaturaliso-morphismuf\MorYC (- M;-33)PUR԰n:=Mor%5C*(-35;- M@ );i.35e.fithefunctor-  M6:URC!FCjisleftadjointtothefunctor- M@2 V::URC!FC5.+- cmcsc10Proof.@_WVe# givretheunitandthecounitofthepairofadjointfunctors.WWVede ne(A)%:=1A ۫ ǹdb1:A4!cA M> M@2 eandd (B)%:=1B L ǹev:B M@2 Mf4!B.Theseareobrviouslynaturaltransformations.8WVehavecommutativediagrams\ J(A MQlA M M@2 M5vl32fdY]O line10-QF((A)=I51X.;cmmi6A ^ db | 1X.MSA M@)UR=1A M:|32fdZPά- F((A)=lÍ l1X.A ^ 1X.M evOanda\(BE M@2h*BE M@2 M M@26<32fdSά-QGv(Bd)=;F1X.BX 1"MG s0 dbM3BE M@2)UR=1Bd M"L32fdS0ά-"ZGv (Bd)=s鍒 1X.BX ev { 1"MGRthrustheLemmahasbSeenprovedbyCorollaryA.9.11.Icffxff ̟ff ̎ ̄cff#Theconrverseholdsaswell.GIf- m EMisleftadjointto- m EM@2 ϲthentheunitgivresamorphismdbw:= (I):I4!oPMw 6.M@2 |andthecounit givesamorphismev:=UR (I):M@2 M64!I+satisfyingtherequiredpropSerties.8ThruswehaveU7 &e*3. %DUAL!OBJECTSפ85YCorollary3.3.4.sWIf,- h KM6:URC!FCais,leftadjointto- KM@2 V::URC!FCathenM@2 isa35leftdualforM@.NCorollary3.3.5.sW(M@2;ev /)`isaleftdualforMDifandonlyiftherffeisanaturalisomorphismh ιMorH˟C(M@  -;-33)PUR԰n:=Mor%5C*(-35;M -);=i.ze.`άmZg1 evFTN@2 M M@2d32fd$>pά 1 f 1H`*Ffe64`? 1 gI{ 1 f 1O@^gN@2~ 퍍uevpO 1>_ZHgZRnZ\vZf}ZpZzZZZ'dZ'd~ Su1 db'd'd%'dƥv'd%l'dեb'd%X'd䥀N'd%D'd>>= %cffxff ̟ff ̎ ̄cffTProblem3.3.2.nR1.)In:thecategoryof( msbm10N-gradedvrectorspacesdetermineallob-jectsM+thatharvealeftdual.2.mInthecategoryofcrhaincomplexesK-Comp,J*determineallobjectsM`fthatharvealeftdual.3.7In=thecategoryofcoScrhaincomplexesK-Co`comp7TdetermineallobjectsM~thatharvealeftdual.4.LetZ(M@2;ev /)bSealeftdualforM@.Shorwthatdb{:I`4!0M7] yM@2 ZisuniquelydeterminedbryM@,M2,andev.8(Uniquenessofthedualbasis.)5.Leth(M@2;ev /)bSealeftdualforM@.Shorwthatev+o:,JM@2  Mm.4!IZwisuniquelydeterminedbryM@,M2,anddb.W(7 &e*3. %DUAL!OBJECTSפ87YCorollary3.3.9.sWLffet<M;N|havetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesMPE0N N@2@N M@2:z̞32fd2ά-漍1 fǟ-:YIY\M M@27{fd& @ά-udb綟X.MHpǠ*FfeǠ?edbjX.NHjǠ*FfeǠ?`f 1NProof.@_ThefollorwingdiagramcommutesY:P$M$N N@2 M,Tfd@`ά-bdb̄ 1^ٍYN^ٍٚN N@2 N;ԟrfd!`ά-ٜdb̄ 1@9Ԍfem$Ԍ?'Åf@rԌfe⤟Ԍ?'$1 1 f@NrǠfe⤟Ǡ?ݍ$1 ev@1HHHqƕHqƕj܍This"implies(f ޹1M" D)db.MF=((1N " evgNS)(1N 1N"?v fG)(db 0N'r 1M ) 1M" D)db.MF=(1N b) Hev NX 1M" D)(1N H1N" } f 1M")(db 0N'r 1M 1M")db.Mp=UR(1N b) ev NX 1M")(1N b) 1N"X 4f| 1M" D)(1N N 1N" dbBM)db.N=(1N N (ev 1Ns 1M" D)(1N" f| 1M" D)(1N" db 0M ))db.N|Ĺ=UR(1N fG2)db.N'r.oDŽcffxff ̟ff ̎ ̄cff Corollary3.3.10.zLffetIM;N-havetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesN0phN@2 NrI:32fd$ߠά-MשTev #X.NYN@2 MY.M@2 MԄ{fdؐά-q<͝Lfǟ-:7 1H"Ǡ*Ffe TǠ?`91 fHOǠ*FfeԟǠ?-5Tev#X.MoProof.@_This:statemenrtfollowsimmediatelyfromthesymmetryofthede nitionofaleftdual.ccffxff ̟ff ̎ ̄cff Example3.3.11.uQLetk1M$2ٟR lMR ^ĹbSeanRJ-R-bimodule. |Thenk1Hom亟R$M(M:;R:)isǃanRJ-R-bimoSduleǃbry(rfGs)(x)=rfG(sx). qFVurthermoreǃwrehavethemorphismev:URHom۟R"n(M:;RJ:) R ;M64!Rde nedbryev(f Rm)UR=fG(m).(DualBasisLemma:)AtmoSduleM62URMR \iscalled nitelySgenerffatedandprojectiveifthereareelemenrtsm1;:::ʚ;mn2URM+undm21;:::ʚ;m2n2URHom۟R"n(M:;RJ:)sucrhthatcڍM8mUR2M6: kn"u cmex10X ㇍Si=1midm i(m)UR=m:*ꍑActually;thisisaconsequenceofthedualbasislemma. ,>Butthisde nitionisequivXalenrttotheusualde nition.X:o7 &e881:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYLetM62URR HMR.+MasarighrtRJ-moSduleis nitelygeneratedandprojectivei Mhasaleftdual.8TheleftdualisisomorphictoHomd1R#WĹ(M:;RJ:).IfjMR is nitelygeneratedprojectivrethenweusedb:URRn4!{M R ;Hom!:R(͹(M:;RJ:)withdbN(1)=P*In U_Ii=1#mi`) R m2idڹ. BInfactwrehave(1O R ev)(db. R1)(m)=(1 R$ev 1)(Pmi R 9m2i Rm)ٝ=PHmidm2i(m)ٝ=m.>WVeharvefurthermore(ev / R1)(1E Rdb 0)(fG)(m)UR=(ev / R1)(P* n U_ i=1f\ R ~miy Rm2idڹ)(m)UR=Pf(midڹ)m2i(m)UR=f(Pmidm2i(m))=URfG(m)forallm2M+hence(ev / R1)(1 R ;dbk)(fG)=f.ConrverselyifMhasaleftdualM@2 thenev:M@2 7C R )Mw4!R"de nesahomo-morphismUR:M@2 V:4!Hom3>R:25(M:;RJ:)inR "MRbry(m2)(m)UR=ev(m2] R m). WVede neP* n U_ i=1miw{ m2i,:=URdbc(1)UR2MS M@2,then:m=(1 ev )(db. 1)(m)=(1 ev )(Pmiw{ $m2i% m)=P7jmid(m2i)(m)9sothatm1;:::ʚ;mn52M<and(m21);:::ʚ;(m2nP)2HomyRm(M:;RJ:)GlformadualbasisforM@,i.e. O-MPis nitelygeneratedandprojec-tivreasanRJ-moSdule.8ThusM@2 됹andHomd1R#WĹ(M:;RJ:)areisomorphicbythemap.AnalogouslyoHom0R#ù(:M;:RJ)oisarighrtdualforMi Mis nitelygeneratedandprojectivreasaleftRJ-moSdule.-׍Problem3.3.3.nRFindanexampleofanobjectMpinamonoidalcategoryCdthathasaleftdualbutnorighrtdual.De nition3.3.12.}!ùGivrenW objectsM;NinC5. ~ Anobject[M;N@]iscalledaleftS/innerHom#ofMdڹandNifthereisanaturalisomorphismMorYCDz(-j M;N@)Pj԰=Mor5C(-;[M;N@]),i.e.8ifitrepresenrtsthefunctorMor Cd(-P M;N@).IfthereisanisomorphismMorNCw(P xRM;N@)PW|԰pd=OMor):LC. (PS;[M;N@])naturalinthethreevXariableM;N;PnthenthecategoryCݹiscalledmonoidal35andleftcloseffd.If%thereisanisomorphismMor[ϟCɎ(M K#h (PS;HomyK!2i(M;N@))P԰N=Hom*:JK0,(P/ iM;N@)n|canbSere- #strictedtoanisomorphismtbHom{\NHu(PS;HomyK!2i(M;N@))PUR԰n:=Hom(yH0ȹ(PLn M;N@); #bSecause m))G=(f)(Ph(1) \|p> h(2)m)G=PwfG(h(1)p)(h(2)m)= #P,5(h(1) \|(fG(p)))(h(2)m)=PVh(1) \|(f(p)(S׹(h(2))h(3)m))=h(f(p)(m))=h((f) #(p m))p]andconrverselyp](h(fG(p)))(m)UR=Ph(1) \|(f(p)(S׹(h(2))m))UR=Ph(1) \|((f) #(pֱ S׹(h(2) \|)m))i=Pn(fG)(h(1)pֱ h(2)S׹(h(3))m)i=(fG)(hpֱ m)i=f(hp)(m). #ThrusHV-Mo`d$0isleftclosed.2#If M42PHV-Mo`d$Eisa nitedimensionalvrectorspacethenthedualvector #spaceM@2 N:=M+HomƴK$(M;K)againisanHV-moSdulebry(hfG)(m)M+:=f(S׹(h)m): #FVurthermoreM@2 됹isaleftdualforM+withthemorphismsfx^db:URK317!X ㇍ imi m i,2M M@ g #and}uev:URM@  M63f m7!fG(m)2KA #where=miչandm2iareadualbasisofthevrectorspaceM@. 2Clearlywehave #(1 ev [)(db. 1)UR=1M 5andE(ev / 1)(1 db )UR=1M"։sinceM>)isa nitedimensional #vrectorfspace.WVehavetoshowthatHdbandHev'areHV-moSdulehomomorphisms. #WVeharve&tʍ+#5(hdb.(1))(m)UR=(h(Pmi m2idڹ))(m)=(Ph(1) \|mi h(2)m2idڹ)(m)UR=+#5P7(h(1) \|midڹ)((h(2)m2i)(m))UR=P(h(1) \|mi)(m2i(S׹(h(2))m))UR=+#5P9h(1) \|S׹(h(2))mUR="(h)m="(h)(Pmi m2idڹ)(m)="(h)db.(1)(m)=+#5db81e("(h)1)(m)UR=dbc(h1)(m);)t #hencehdb.(1)UR=dbc(h1).8FVurthermorewrehavetʍ(hev /(f m)UR=hfG(m)=Ph(1) \|f(S׹(h(2))h(3)m)UR=P(h(1) \|f)(h(2)m)UR=(ꉟP74evC+e(h(1) \|f h(2)m)UR=ev(h(f m)):4. #Let^HL>bSeaHopfalgebra. KThenthecategoryofleftHV-comodules(seeExample #3.2.433.)isamonoidalcategoryV.LetM2џHV-Como`d3kbSea nitedimensional #vrectorspace. $LetmibSeabasisforMϕandletthecomultiplicationofthe #comoSdule9bes2(midڹ)UR=Phij z /8mjf .$Thenwrehave(hikl)UR=Phij z /8hjvk .$M@2 V::= #Hom8K?U(M;K)tbSecomesaleftHV-comodules2(m2jf )UR:=PS׹(hijJ){ m2idڹ.ZOnetvreri es #thatM@2 됹isaleftdualforM@.0Lemma3.3.14.mQLffet35M62URCjbeanobjectwithleftdual(M@2;ev /).fiThen1.fiM M@2 4is35analgebrffawithmultiplicationtL rUR:=1M . ev@ 1M".:M M@  M M@  V:!M M@ Za7 &e901:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYand35unit3ߍuUR:=db:IF!M M@ ;2.fiM@2 Mtis35acffoalgebra35withcffomultiplicationDKyйUR:=1M" db 1M B:M@  M6!QM@  M M@  Mand35cffounit5"UR:=ev@:M@  M6!QI:B5Proof.@_1.nThe#-assoSciativitryisgivenby(ro 1)riD=(1M _ oev 1M"X o1M 1M" D)(1M " ev4 1M")'=1M ev4 ev- 1M"k=(1M 1M"w 1M ev4 1M" D)(1M ev / 1M" D)UR=(1 r)r.The&axiomfortheleftunitisr(u 1)UR=(1M ev 1M")(db. 1M . 1M" D)UR=1M 1M" D.2.8isdualtothestatemenrtforalgebras.ʜDŽcffxff ̟ff ̎ ̄cffB5Lemma3.3.15.mQ1.PLffetAbeanalgebrainCandleftM62URCbffealeftrigidobjectwith)ileftdual(M@2;ev /).c%TherffeisabijectionbetweenthesetofmorphismsfQ:URA MU!M making(MaleftA-moffduleandthesetofalgebramorphismsVze*f !:URA!M| ;M@2.2. (LffetpCbeacoalgebrainCΥandleftM62URCbffealeftrigidobjectwithleftdual(M@2;ev /).TherffeɋisabijectionbetweenthesetofmorphismsfQ:URM6!QM C)makingM oaright*Cܞ-cffomodule35andthesetofcffoalgebra35morphismsV\e*f.:URM@2 M6!QC.B5Proof.@_1." By$Lemma3.3.14theobjectM_ M@2 isanalgebra.GivrenfQ:URA M4!Msucrh;thatMbSecomesanA-module. ByLemma3.3.3wreassociateVe*f1ܹ:=(f2 ꊹ1)(1 db):A4!A M+n M@2 4!M M@2.R]TheH|compatibilitryofVqe*fwiththemrultiplicationisgivenbythecommutativediagramvJA AvpAnLsH2fd[`ά-p rYsWA A M M@2Y A M M@2wfd6ά-92r 1 1$A M M@2$|M M@2ƹfdM@ά-`后f 1^ٍxA M M@2 M M@2^ٍ A M M@2rfd ά-l==md1 1 ev { 1&jM M@2 M M@2^1M M@232fdȒά-r b1 ev { 1sZ} Ǡ|z@feZ<Ǡ?+㎍F#a6cmex8erEsfJj% ㎍erfO򊟱R fe%R ? <1 f 1@Ԍfe%Ԍ?̈́Q<1 db | 1 1O-݊R fe.R ? f 1stS Ǡ|z@fet<Ǡ?+㎍z„ery8fsOэn1 1 dbk M`Hu M`H M`HUHUjsOb蜟M`@l蜟M`@v蜟M`@蜟M`@蜟M`@\O@\ORs8r1 ㎍erf@\\z\p\f\b蜟ob蜟o 胀1㎍i; ergEfld 1 1@bnf 1 1|rk ƕk ƕ@ͻɍ|1|U>|UY)X\16I@@I@JI@TI@^I@bMܟo@bMܟoR@b9f 1>|HH|HR|HZ|ƕHZ|ƕjTheunitaxiomisgivrenby;I}%IM M@2ȴRfd1ʀά-Ԡ db{AWA M M@2d32fd$ά-ׁ=1 dbHtM M@232fdά- x&f 1 Ǡ fe54Ǡ?+vu 5ŸǠ fehǠ?Mu 1 M1 1ٴHٴHٴHٴH +uH +uj[t7 &e*3. %DUAL!OBJECTSפ91YConrverselyletg::NA4!)M @M@2 ޹bSeanalgebrahomomorphismandconsidervegԹ:=(1 evJ)(g" 1)nR:A M64!7M M@2 M64!M@.dThenXM:LF`9LF`4LF`/LF`*LF`&欟`&欟` s^MgI{ g 1C}`XMM`XW`XaM`Xk`XuM`X`XM`X`XM`X`XM`XXzPMigI{ 1CǠMGWǠaGkǠuGǠGǠGǠGۿۿ:ꬽD1 evۿ`X ?`X`X?`X'`X1?`X;`XE?`XO`XY?`Xc`Xm?`XoXoz0=h1 evoF`AtF`AyF`A~F`AF`AF`AF`AvL`AvL`Us^MlgI{ 1ZM`~ZM`tZM`sܟsܟ eand3 oA MM M@2 Mu32fd ά-o͍MgI{ 17M_$32fd,Ѝά-r 1 ev {MǠ@fe$Ǡ?ꬽm}Ju 1񍍒,db 1h?`Hh?`Hh?`Hh?`Hh?`Hh?`HydHydjid1٤ۿ`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`XXzycommrute.2.(M@2;ev /)isaleftdualforMĖinthecategoryC6ifandonlyif(M@2;db.)istherighrtTdualforMinthedualcategoryC52op R.wXSoifwedualizetheresultofpart1.wXweharvetocrhangesides,hence2.cffxff ̟ff ̎ ̄cff;7  ,@ cmti12+- cmcsc10)ppmsbm8( msbm10#a6cmex8"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line10