; TeX output 1999.11.03:0815E7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups"N cmbx12Intro`ductionOne,ofthemostinrterestingpropSertiesofquantumgroupsistheirrepresentationtheoryV.It.hasdeepapplicationsintheoreticalphrysics.Themathematicalsidehastodistinguish[bSetrweentherepresentationtheoryofquantumgroupsandtherepresen-tationtheoryofHopfalgebras.7InbSothcasestheparticularstructureallorwstoformtensorproSductsofrepresenrtationssuchthatthecategoryofrepresentationsbSecomesamonoidal(ortensor)categoryV.Theproblemwrewanttostudyinthischapteris,thowmuchstructureofthequanrtum׈grouporHopfalgebracanbSefoundinthecategoryofrepresentations.2WVewillAshorwthataquantummonoidcanbSeuniquelyreconstructed(uptoisomorphism)from1itsrepresenrtations.|TheadditionalstructuregivenbytheantipSode1isitimitelyconnectedўwithacertaindualitryofrepresentations. WVewillalsogeneralizethisproScessofreconstruction.OnBtheotherhandwrewillshowthattheproScessofreconstructioncanalsobeusedto?DobtaintheTVamrbaraconstructionoftheuniversalquantummonoidofanoncom-mrutative=geometricalspace(fromcrhapter1.).2ThuswewillgetanotherpSerspectiveforthistheorem.Arttheendofthechapteryoushoulds2#!", cmsy10 #understandrepresenrtationsofHopfalgebrasandofquantumgroups,# #knorw>thede nitionand rstfundamentalpropSertiesofmonoidalortensor #categories,# #bSeotfamiliarwiththemonoidalstructureonthecategoryofrepresenrtationsof #Hopfalgebrasandofquanrtumgroups,# #understandwhrythecategoryofrepresentationscontainsthefullinformation #abSout6thequanrtumgroupresp.theHopfalgebra(TheoremofTVannakXa-Krein),# #knorwtheproScessofreconstructionandexamplesofbialgebrasreconstructed #fromcertaindiagramsof nitedimensionalvrectorspaces,# #understandSbSettertheTVamrbaraconstructionofauniversalalgebrafora nite #dimensionalalgebra./^*o cmr969F*7 &e701:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYj4T1.TRepresentationsofHopfAlgebrasLetg cmmi12AbSeanalgebraorveracommrutativering' msbm10K.cLetA-Mo`d#sbethecategoryofA-moSdules.8AnA-moduleisalsocalleda+@ cmti12rffepresentationofA.ObservretthattheactionAI M64!MAXsatisfyingtthemoSduleaxiomsandanalgebrahomomorphismBA4!=End*(M@)areequivXalenrtdescriptionsofanA-moSdulestructureontheK-moSduleM@.ThewfunctorUl:v[A-Mo`d#=4!6_%K-Mo`d#YwithU1(2cmmi8AM@)=M[andU1(fG)=fviscalledtheforffgetful35functorortheunderlyingfunctor.IfB isabialgebrathenarffepresentation֧ofBisalsode nedtobSeaB-module.OItwill=turnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofrepresenrtationsinacanonicalwayV.LetC}KbSeacoalgebraorveracommrutativeringK. 7LetCܞ-FComo`d2bethecategoryofCܞ-comoSdules.8AC-comoSduleisalsocalledacfforepresentation꨹ofC.TheYfunctorUc:URCܞ-FComo`d2|4!CwK-Mo`d#;withU1(2CM@)=M=andU1(fG)=fXiscalledtheforffgetful35functorortheunderlyingfunctor.IfYB_isabialgebrathenacfforepresentation`oofYBisalsode nedtobSeaB-comodule.ItpawillturnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofcorepresenrtationsinacanonicalwayV.UsuallyznrepresenrtationsofaringareconsideredtobSemodulesorverznthegivrenring.fTheroleofcomoSdulescertainlyarisesintheconrtextofcoalgebras.Butitisnot{quiteclearwhatthegoSod{de nitionofarepresenrtationofaquantumgrouporitsrepresenrtingHopfalgebrais.FVor(thispurpSoseconsiderrepresenrtationsMikofanordinarygroupG.|Assumefore1thesimplicitryoftheargumentthatGis nite.|RepresentationsofGarevectorspaces?ZtogetherwithagroupactionGM4!M@. 6EquivXalenrtly?ZtheyarevectorspacesztogetherwithagrouphomomorphismG'4!۹Aut-(M@)ormoSdulesorverzthegroupalgebra:K[G]j M64!M@.!InthesituationofquanrtumgroupsweconsidertherepresenrtingHopfalgebraHasalgebraoffunctionsonthequantumgroupG.Then`thealgebraoffunctionsonGistheHopfalgebraK2G,thedualofthegroupalgebradK[G].AneasyexerciseshorwsthatthemoSdulestructureK[G]4 MK4!?MtranslatestothestructureofacomoSduleM64!K2G u@MandconrverselyV.ܚ(ObservethatGis nite.)Sowreshouldde nerepresentationsofaquantumgroupascomoSdulesorvertherepresenrtingHopfalgebra.!De nition3.1.1.vaùLetvGbSeaquanrtumgroupwithrepresentingHopfalgebraHV.ArffepresentationofGisacomoSduleorvertherepresentingHopfalgebraHV.FVromthisde nitionwreobtainimmediatelythatwemayformtensorproSductsofrepresenrtationsofquantumgroupssincetherepresentingalgebraisabialgebra.WVexcomenorwtothecanonicalconstructionoftensorproSductsof(co-)represen-tations.G 7 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+71YLemma3.1.2.g5QLffetHBbeabialgebra.LetM;Nz2WB-;Mo`d%`betwoB-modules.ThenMy 8N͠isaB-moffdulebytheactionb(m n)UR=!u cmex10Pb|{Ycmr8(1) \|m b(2)n..IffQ:URM6!QM@2K cmsy80andeg!I:N! qN@20tarffehomomorphismsofB-modulesinB-;Mo`d$Ithenf& 'g!I:M N!M@20 N@20BRis35ahomomorphismofB-moffdules.,- cmcsc10Proof.@_WVeharvehomomorphismsofK-algebras qʹ:^;BA4!q End*}(M@)and :BA4!End (N@):de ningtheB-moSdulestructureonM@andN.fThruswegetahomomorphismof~algebrascan( r  O)UR:BX4!_7Bh BX4!End*k(M@) Endڢ(N)UR4!1End)(M N).ThrusM' NN]is yaB-moSdule.&Thestructureisb(m n)UR=canF( p O)(Pb(1) \|) b(2))(m n)UR=can(P (b(1) \|) O(b(2)))(m n)UR=P (b(1))(m) O(b(2))(n)UR=Pb(1)m b(2)n.FVurthermorewrehave1(m n)UR=1m 1mUR=m n.If[f;garehomomorphismsofB-moSdules,Hthenwrehave(f Дgn9)(b(m n))4 =(f gn9)(Pb(1) \|m b(2)n)UR=PfG(b(1)m) gn9(b(2)n)UR=Pb(1)fG(m) b(2)gn9(n)UR=b(fG(m) gn9(n))UR=b(f g)(m n).8Thrusf gXisahomomorphismofB-moSdules.4 cffxff ̟ff ̎ ̄cffCorollary3.1.3.sWLffetBbeabialgebra.LThen UR:B-;Mo`d";B-;Mo`d#!5MB-;Mo`dwith35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.Proof.@_ThefollorwingareobviousfromtheordinarypropSertiesofthetensorproSduct/orverK. 1M  ~1N n=UR1M" N۹and(f gn9)(fG20W g20;B"K 1)UR=(1 1 )( 1 1)(1  1)UR=>_c( 1 1)(1 1 W)(1  1)UR=( 1 1)(1 Ma>;Bd Bt)thatusesthefactthat(1is gn9)(fr 1)2N=(f is1)(1 gn9)holdsandthatBd Ma>;B$¹=( 1)(1 W)andMa>;Bd B"ƹ=UR(1 )( 1).Square(5)and(6)commrutebythepropSertiesofthetensorproduct.H7 &e721:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYSquare(7)commrutessinceBisabialgebra.%ߍ*Y*Y\BE M@ $g 6969BE N0%`09M N099 T,ҲfdDvpO line10-)wS٠L *Y*Y=BE B $g 6969M N0%`0909,ҲfdAά-)퍒A1 r 10909cBE M N&},Ҳfd9Ѝά-)4r 1 11Rl @"j2c*Ffe"dc?Q 'OL 1Rl @زc*Ffe c?Pd 1  11Rl @&Rc*FfeYc?P   1 11R@1fe?t/ah 1 1py4BE M pytBE B M@ jܠ|/2BE N|zJBE B Nvvv='sH2fd4AЍά-o쁍D1 L 1 pypyBE B B jܠ||%BE M NvvvtsH2fd5npά-n>51-:Aacmr62* X.;cmmi6MB";BI q% cmsy6 By 1wǀ~"j2F`*Ffe"dF`?L]'O1 r 1wǀ~زF`*Ffe 䟪F`?ud1 X.BI MB";By 1 1wǀ~&RF`*FfeYF`?L] 1 r 1 1 1D jBE B DvjVBE B B R 9M N9xM BE N`$$;fd7Opά-bD1 1 L DDBE B B R 99%BE M N`$$'fd6@ά-j퍒91 1 1 r 1=@"j2*Ffe"d?鯍'Or 1 1=@ز*Ffe ?鯍dr 1 1 1 1=@&R*FfeY?鯍 r 1 1 1 1yyBE B Ǡ x xxM BE NiBE M NE#Ğ32fd0aά-ׁP1 L yy=BE B Ǡ x xgiBE M ND32fd2~ά-m-1 1 r 1WBE B M N-㤞32fd&fpά-/-2l1 r 1 1R@U(1)R@(2)OU(3)O(4)@U(5)@(6)Ln"(7)ThelarwofthecounitisL|99\BE M@ `yyBE NΠYM NY9 T{fdDvpά- AS٠L 99=BE B `yyM NΠYY{fdAά-)-A1 r 1YYcBE M N&}{fd9Ѝά-24r 1 1H8Idؤ13$ҁ H=$ׁ HG$܁ HQ$ H[$ He$ Ho$ Hy$ H}$džH}$džjPM NM NyԞ32fdHά-YУ"1oPM N#T32fdHά-YF1'زǠ$g`fe Ǡ?-d" 1 " 1' G2Ǡ$g`fe zdǠ?-," " 1 1HǠ*FfeǠ?͝d" 1 1Qύwherethelastsquarecommrutessince"isahomomorphismofalgebras.Norwletf2andgXbSehomomorphismsofB-comodules.8ThenthediagramN󍍍99\BE M@ `yyBE NΠYM NY9 T{fdDvpά- AS٠L 99=BE B `yyM NΠYY{fdAά-)-A1 r 1YYcBE M N&}{fd9Ѝά-24r 1 1' "j2耄@fe"d耬?`'Of g' ز耄@fe 耬?`d1 f 1 g'  G2耄@fe zd耬?`,1 1 f g' 耄@fe耬?`d1 f g++@BE M@20 Ǡ k k6BE N@20K ՘M@20 N@20K;Ԟ32fd@<0ά-ׁTL yy=BE B Ǡ k kM@20 N@20KKĞ32fd@xPά-mU1 r 1KKaBE M@20 N@20&}32fd7Pά-/3*r 1 1VӍcommrutes.8Thusf gXisahomomorphismofB-comoSdules.scffxff ̟ff ̎ ̄cffϪCorollary3.1.5.sWLffetBbeabialgebra.Then ܹ:B-;Como`d2k=B-;Como`d3!B-;Como`d3apwith35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.Prop`osition3.1.6.OLffet3B*9beabialgebra./Thenthetensorproduct UR:B-;Mo`d!zB-;Mo`d#!5MB-;Mo`d$osatis es35thefollowingprffoperties:(獍1. #Thes9assoffciativityisomorphism m:(M1 M2) M3!-M1 (M2 M3)s9with # ((mL n) p)=mL (n p)isanaturffaltransformationfromthefunctorI097 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+73Y # q( Id Ү)tothefunctor q(Id ; )inthevariablesM1,ۚM2,andM3 yin #B-;Mo`d M@)u=A SMkvandUfw(gn9)=g˹wheream:=fG(a)mfora2AandmUR2M@.8ThefunctorUf aǹisalsocalledforffgetfulorunderlying35functor.The2lactionofAonaB-moSduleMsPcanalsobeseenasthehomomorphismAUR4!1B4!n߹End&{(M@).WVedenotetheunderlyingfunctorspreviouslydiscussedbryHNUA 36:URA-Mo`d"44!4K-Mo`d#Cresp.@,UB :B-Mo`d#I:4!4K-Mo`dX:DProp`osition3.1.10.Lffet)f:BC$!QCbeahomomorphismofbialgebras.FThenUf Tsatis es35thefollowingprffoperties:8#ʍ^Ufw(M N@)UR=Uf(M@) Uf(N@);^Ufw(g h)UR=g h;^Ufw(K)UR=K;^Ufw( )UR= ;33Ufw()=;Ufw()=;^UBN>Ufw(M@)UR=UC(M);^UBN>Ufw(gn9)UR=UC(g):8ؘProof.@_Thisx.isclearsincetheunderlyingK-moSdulesandtheK-linearmapsstaryuncrhanged. nTheނonlythingtocheckisthatUf UgeneratesthecorrectB-moSdulestructureonthetensorproSduct.GFVorUfw(M5 \QN@)y=M \QN0}wrehaveb(m\Q n)y=fG(b)(mF n)UR=Pf(b)(1) \|mF f(b)(2) \|nUR=Pf(b(1) \|)mF f(b(2) \|)nUR=Pb(1)mF b(2)n. dcffxff ̟ff ̎ ̄cffK_Y7 &enYZ1. %REPRESENT:ATIONS!OFHOPFALGEBRASf+75YRemark3.1.11.qN6LetpCM;andD+bSecoalgebrasorverpacommrutativepringK.2LetfQ:C[4!DVϹbSeAahomomorphismofcoalgebras.ThenwrehaveafunctorUfP:1Cܞ-FComo`d-J4!%DS->6Como`d4 withwUfw(2CM@) v=2D M[andUfw(gn9)=gwhereD =(fO P1)C +:M4!nCN r+M64!DŹ M@./AgainthefunctorUf Fiscalledforffgetfulorunderlyingfunctor.WVedenotetheunderlyingfunctorspreviouslydiscussedbrye8:UC t:URCܞ-FComo`d2|4!CwK-Mo`d#Cresp.@,UD :DS->6Como`d14!CbgK-Mo`dX:Prop`osition3.1.12.Lffet)f:BC$!QCbeahomomorphismofbialgebras.FThenUfq:URCܞ-Como`d2%!DDS-Como`d3satis es35thefollowingprffoperties:6Ajʍ^Ufw(M N@)UR=Uf(M@) Uf(N@);^Ufw(g h)UR=g h;^Ufw(K)UR=K;^Ufw( )UR= ;33Ufw()=;Ufw()=;^UCUfw(M@)UR=UBN>(M);^UCUfw(gn9)UR=UBN>(g):7UTProof.@_WVelearvetheproSoftothereader.cffxff ̟ff ̎ ̄cffProp`osition3.1.13.LffetnH\ beaHopfalgebra.LetMandNbffebeHV-modules.ThenDڹHomc(M;N@),BtheDsetK-lineffarmapsfromMtoN@,BbecomesanHV-moduleby(hfG)(m)UR=Ph(1) \|f(S׹(h(2)m).fiThis35structurffemakesevMHom:URHV- Mo`d#9mHV- Mo`d#!5HV- Mo`da35functorcffontravariant35inthe rstvariableandcffovariantinthesecondvariable.Proof.@_TheΆmainparttobSeprorvedΆisthattheactionHo Homm(M;N@)4!Homy(M;N@)nsatis estheassoSciativitrylaw. o2Letf`2JaHom(M;N@),h;k~2JaHV,andmUR2M@.Thenn((hkg)fG)(m)=P(hk)(1) \|fG(S׹((hk)(2))UR=Ph(1) \|k(1)fG(S׹(k(2))S(h(2))m)UR=Ph(1) \|(kgfG)(S׹(h(2))m)UR=(h(kfG))(m).WVelearvetheproSofoftheotherproperties,inparticularthefunctorialproperties,tothereader.cGqcffxff ̟ff ̎ ̄cffCorollary3.1.14.zLffet!ZMb>beanHV-module. 0ThenthedualK-moduleM@2 I=Homy(M;K)35bffecomesanHV-moduleby(hfG)(m)UR=f(S׹(h)m).Proof.@_TheyspaceKisanHV-moSdulevia"UR:HB4!K.)Henceywrehave(hfG)(m)UR=Ph(1) \|fG(S׹(h(2)m)UR=P"(h(1) \|)f(S׹(h(2)m)UR=f(S׹(h)m).~cffxff ̟ff ̎ ̄cffp;7 -%n eufm10,- cmcsc10+@ cmti12*o cmr9(ppmsbm8' msbm10!u cmex10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10