; TeX output 1999.11.03:0822i7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups_o cmr9105j*7 &e1061:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYq`N cmbx128.Reconstructionand!", cmsy10C5-categoriesNorwweshowthatanarbitrarycoalgebrag cmmi12C\canbSereconstructedbythemethoSdsinrtroSducedabovefromits(co-)representationsormorepreciselyfromtheunderlyingfunctorR!s:FComo`d.eF-2OC4!Vec,;a.nInthiscaseonecannotusetheusualconstructionofcoSend!"(!n9)thatisrestrictedto nitedimensionalcomoSdules.TheJfollorwingTheoremisanexamplethatshowsthattherestrictionto nitedimensional)comoSdulesingeneralistoostrongforTVannakXareconstruction.4dTheremarygbSeuniversalcoSendomorphismbialgebrasformoregeneraldiagrams. OntheothervhandthefollorwingTheoremalsoholdsifoneonlyconsiders nitedimensionalcorepresenrtationsƆofCܞ.HowevertheproSofthenbecomessomewhatmorecomplicated.zDe nition3.8.1.vaùLet'C˹bSeamonoidalcategoryV.A'categoryD|togetherwithabifunctor) UR:CS D4!D~չandnaturalisomorphisms :(A  B) M64!A (B$ M@),Ë:URI+ M64!M+iscalledaC5,@ cmti12-cffategory꨹ifthefollorwingdiagramscommuteEԍ F((A B) Cܞ) M(A (BE Cܞ)) Mv<:2fd+7O line10-̯t&2cmmi8 |{Ycmr8(A;Bd;C) K cmsy8 1>7A ((BE Cܞ) M@)|:2fd+7ά-̯ h i>(A;Bd C;M")?Ǡ@fe@&Ǡ?ꃀD؜ i>(A Bd;C;M")qEjǠ@feqxǠ?ꃀ;m"1 i>(Bd;C;M") F(A B) (CF M@)>7A (BE (CF M@))v<32fd@ά-} i>(A;Bd;C M")MKqVr(A I) MArA (I+ M@)L:2fdC ά-̯ƴ i>(A;I;M")A Mꃀ(A) 1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1 I{(M")L L攴L?^L#0#0+͍AC5-categoryiscalledstrictifthemorphisms O;Xaretheidenrtities.Leth(DUV; )and(D20#; )bSeC5-categories.AhfunctorF]9:,(D~4!3D20ntogetherwithanaturaltransformation(A;M@)UR:A F1(M)UR4!1F(A M)iscalledaweffakSC5-functorifthefollorwingdiagramscommuteDT(A B) F1(M@)9F1((A B) M@){:2fd{@ά-'֪K*Ǡ@feK\Ǡ?ꃀB e*Ǡ@fee\Ǡ?ꃀjlF(( i>)A (BE F1(M@))/lA F1(BE M@){32fd2$ά- Z1 9F1(A (BE M@))L32fd2$ά- (Lˍ2I+ F1(M@)2F1(I+ M@)n:2fd"€ά-'֪F1(M@)uG}?`@?`@?`@Ɩ@ƖRꃀF((I{)?`?`?` QIf,ʬinaddition,߹isanisomorphismthenwrecallFaC5-functor.RThefunctoriscalledastrict35C5-functorifҩistheidenrtitymorphism.k7 &ev8.pRECONSTR9UCTION!AND. cmsy9Cf-CA:TEGORIESeX107YAonaturaltransformation'q:FL4!F120 %bSetrween(weak)C5-functorsiscalledaC5-trffansformation꨹ifHA F120J(M@)F120J(A M@)>32fdά-h2Qq-:!q% cmsy60HNA F1(M@)H@F1(A M@){fdά-iةH:Ǡ*FfelǠ?`1X.;cmmi6A ^ '(M")H :Ǡ*Ffe lǠ?`y'(A M")commrutes.ٍExample3.8.2.oQLetpCybSeacoalgebraandC:=URVec.xThenthecategoryComo`d,p-0Cof~righrtCܞ-comoSdulesisaC5-categorysinceN62URComo`d-R-1C}andV2URC=Vec0zimpliesthatVG N+isacomoSduleswiththecomodulestructureofN@.TheQunderlyingfunctor!Ë:URComo`d-R-1C14!Vec.ʹisastrictC5-functorsincewrehaveV% !n9(N@)3=!(V% N@).ASimilarlyms!q Mtֹ:3Como`d.-2~C4!^Vec1,isaC5-functorsinceVG (!n9(N@) M)PUR԰n:=!n9(VG N) M.ٍLemma3.8.3.g5QLffetCz)beacoalgebra. jLet!b:KComo`d0TK-4C!+Vec6betheun-derlyingKBfunctor.Lffet'[:!%!~!R zM&beKBanaturaltransformation.Then'isaC5-trffansformation35withC=URVec.+- cmcsc10Proof.@_Itcsucestoshorw1V ~) 'a'(N@)UR='(V N@)cforanarbitrarycomoSduleN.WVeshorwthatthediagramNzv{VG NVG N M32fd2Bά-W`1X.Vn '(N")Y{VG NYVG N M{fd2Bά-`'(V N")HjǠ*Ffe'Ǡ?8I51HǠ*FfeܟǠ?8I \1m⍹commrutes.ZuLet(vidڹ)bSeabasisofVp.FVoranarbitraryvrectorspaceWletpi::h`VN GW4!nWxbSetheprojectionsde nedbrypidڹ(t)UR=pi("u cmex10P jvjv lwjf )=wi;ϹwherePjvj lwjNo0N M:,32fdT0ά-W`'(N")Y[VG NYVG N M/{fd>ά-`'(V N")HJǠ*FfeH|Ǡ?'"-p8:iH ʟǠ*Ffe Ǡ?i|p8:i,r MlT7 &e1081:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYcommrute. oExpressedinformulasthismeans'(N@)pidڹ(t)F=pi'(Vy  N@)(t)foralltUR2VG N@.8Hencewrehave0ƍʍ2e(1V p '(N@))(t)UR=(1V '(N@))(Pvi pidڹ(t))UR=Pvi '(N@)pidڹ(t)p=URPidvi pid'(VG N@)(t)UR='(V N@)(t)0ƍSowrehave1V p '(N@)UR='(VG N@)asclaimed.լcffxff ̟ff ̎ ̄cff[WVe)prorvethefollowingTheoremonlyforthecategoryC$=qjVec=ofvectorspaces.The`Theoremholdsingeneralandsarysthatinanarbitrarysymmetricmonoidalcategory]CthecoalgebraC:orepresenrtsthefunctorC5-Natބ(!n9;!gI M@)PW԰2?=Mor&C,+(C5;M)ofnaturalC5-transformations.*Theorem3.8.4.p޹(Reconstructionofcoalgebras)(LffetCbeacoalgebra.GLet!:Como`d*`-.5C1!Vec/[bffe35theunderlyingfunctor.fiThenCP1԰Jع=ܙcoSend.(!n9).Proof.@_LetM,iinVecandlet'Vʹ:!4!5Z!w >MbSeanaturaltransformation.;vWVede ne(thehomomorphismAe':URC14!M bryAe'=UR(Y 1)'(Cܞ)(usingthefactthatCƹisacomoSdule.LetNbSeaCܞ-comodule.*ThenNisasubcomoduleofN U-CmbryȄ:URN64!N U-CsincethediagramFN CN CF C<32fd(eά-a^L 1YaNY~N Cܟ{fd@Z@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?o}1 Çcommrutes.8Thusthefollorwingdiagramcommutes+`ΉN+E`N CnIJfdWά-K퍒ՐYS)N MYN CF M|D{fd<@ά- AgRL 1He"`*Ffef#T`? M'(N")H^`*Ffeۑԟ`? ^'(N" C)=1X.N2 '(C)H>N M8IU1ԟ Xԟҁ Xԟ Xԟׁ Xԟ Xԟ܁ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ X ԟ Xԟ Xԟ X&ԟdžX&ԟdžzH͝t1  1ҁ Hׁ H܁ H  H H H( H2 H6džH6džje21 #a6cmex8e'4@4@4@4@ 4@4@4@)4@34@=4@>P@>PR0ƍInparticularwrehaveshownthatthediagramJzY!Yݲ! C{fd - ά--(H'[' ׁ @ @ @ @Q>@Q>RH! MV2Ǡ*FfedǠ?ҍ;1 e'commrutes.m!7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX109YTVo-shorwtheuniquenessofe'йletgr:G9C#4!MPbSeanotherhomomorphismwith(1 gn9)+=n'. mFVorQc2C.Mwrehavegn9(c)n=g( 1)(c)n=( 1)(1 g)(c)n=( 1)'(Cܞ)(c)UR=e' ȹ(c).ThecoalgebrastructurefromCorollary3.5.1istheoriginalcoalgebrastructureof&TCܞ.ThiscanbSeseenasfollorws.Thecomrultiplication.:!)4!!A ICisanaturaltransformationn hence(w\ *1C)J:5!Q4!!rc C CJisn alsoanaturaltransformation.As^inCorollary3.5.1thisinducedauniquehomomorphism1:Cb4!C 4Cso^thatthediagramQUkfe! coSend ^"(!n9)! coSend ^"(!n9) coSend(!n9)32fdά-aL 1H2!H|! coSend ^"(!n9)-{fdSά--0HǠ*FfeԟǠ?k}#H bǠ*Ffe Ǡ?o1  ݍcommrutes..JInasimilarwaythenaturalisomorphism!P԰=.! t( msbm10KinducesauniquehomomorphismUR:C14!KsothatthediagramV;ЍH@!HW! coSend ^"(!n9)ht{fd 0ά--0H}id]!Xׁ @X @X @X @ԟt>@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͝T1 ocommrutes. BecausewoftheuniquenessthesemustbSethestructurehomomorphismsofCܞ.6/cffxff ̟ff ̎ ̄cff"WVeneedamoregeneralvrersionofthisTheoreminthenextchapter.'SoletC'bSeaUncoalgebra."Let!Ë:URComo`d-R-1C14!Vec.0bSetheunderlyingfunctorandȄ:UR!4!2j! yCtheunivrersalnaturaltransformationforCP1԰Jع=ܙcoSend.(!n9).WVeusethepSermrutationmaponthetensorproductthatgivresthenaturalisomorphismoEo:URN1j T1 N2 T2 ::: NnR TnP԰ =KN1 N2 ::: NnR T1 T2:::5J TnwhicrhGisuniquelydeterminedbythecoherencetheoremsandisconstructedbysuitableapplicationsofthe ipo:URN TP԰=TLn N@.Letǀ!n92n :Como`d0-4˟CPComo`d.OP-29CP:::@PComo`d.OP-C]4!Vec6CbSeǀthefunctor!n92n(N1;N2;:::ʚ;NnP)=!n9(N1) !(N2) :::px !(NnP).MFVor%notationalconrvenience%weabbreviatefN@g2n :=.N1' gN2 ::: NnP,F>similarlyfCܞg2n =.CD C ::: CWandffGg2n:=URf1j f2 ::: fnP.8Sowregeto:URfN Tg2nP԰ =KfN@g2nR fTg2nP:n1f7 &e1101:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYLemma3.8.5.g5QLffetO':!n92n !!n92nE MbeOanaturaltransformation.Then'isaC5-trffansformation35inthesensethatthediagramsDVifVpg2nR fN@g2n``fVpg2nR fN@g2n Mq\32fdU}ά-W`fVg-:n7 '(NqAacmr61*;:::;Nn)_fVG N@g2n4fVG N@g2nR M:2fdgά-̯'(Vq1* Nq1;:::;Vn7 Nn)yǠ@feyܟǠ?@َ>RHm!n92n1 MBǠ*FfetǠ?ҍx1 e'4TVoshorwtheuniquenessof?)e'3letgr:u9Cܞ2n '4!!M~bSeanotherhomomorphismwith(1!I{nL gn9)s22(n)&=n'.WVe&Qharvegֹ=ngn9("2n ) 1Cn =Ϲ)W2n =g("2n ) 1Cn =Ϲ)s22(n) (C5;:::ʚ;Cܞ)=Z΍("2nR 1M )(1Cnw gn9)s22(n) (C5;:::ʚ;Cܞ)UR=("2n 1M )'(C5;:::ʚ;Cܞ)=e' ȹ.acffxff ̟ff ̎ ̄cffKԍNorwweprovethe nitedimensionalcaseofreconstructionofcoalgebras.ڍProp`osition3.8.7.O(Reconstruction)֔LffetC2beacoalgebra.PLetComo`d/603-8)C2betheOcffategoryof nitedimensionalCܞ-comodulesand!0:QComo`d/"Q03U-8C!ǭVec2betheunderlying35functor.fiThenwehaveCP1԰Jع=ܙcoSend.(!n9).Proof.@_Let5MvbSeinVecandlet':!C14!1!K ݷMbSe5anaturaltransformation.WVebde nethehomomorphisme':_C4!Masfollorws. 5Letc2Cܞ. 5LetNbSea nite)MdimensionalCܞ-subScomodule)MofCconrtainingc.Thenwede negn9(c):=(jN 1)'(N@)(c).If?N20\isanother nitedimensionalsubScomoduleofCݹwithc2N@20\andwithN6URN@20ŹthenthefollorwingcommutesB7ԠNԠhN Ml:2fdzά-̯'(N")Ǡ@fe׼Ǡ?[ʟǠ@feǠ?KN@20K.LN@20 M,32fd ά-*'(N"-:0)胀@?`Hά-c퍒  1Thrusλthede nitionofMJe' 1(c)isindepSendentofthechoiceofN@.FVurthermoreMJe'Z:هN4!}[M`ܹisobrviouslyalinearmap.FVoranytwoelementsc;c2022cCthereisa nitedimensionalusubScomoduleN6URCwithc;c20#2NYe.g.thesumofthe nitedimensionalsubScomodulesconrtainingcandc20separatelyV.8Thusi7e'ƹcanbSeextendedtoallofCܞ.pV{7 &e1121:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYTheBrrestoftheproSofisessenrtiallythesameastheproofofthe rstreconstructiontheorem.|cffxff ̟ff ̎ ̄cffFTherepresenrtationsallowtoreconstructfurtherstructureofthecoalgebra.EWVeprorve|areconstructiontheoremabSoutbialgebras. RecallthatthecategoryofB-comoSdulesorverabialgebraB͹isamonoidalcategoryV,:furthermorethattheunderlyingfunctor)! :^Como`d-^-1B:d4!OVec/D"isamonoidalfunctor.bFVromthisinformationwrecanreconstructthefullbialgebrastructureofB.8WVeharvepTheorem3.8.8.pLffetBbeacoalgebra.LetComo`d/g-3Bbeamonoidalcategorysuch thattheunderlyingfunctor!Ë:URComo`d-R-1BX!Vec.uisamonoidalfunctor.XThentherffelisauniquebialgebrastructureonBthatinducesthegivenmonoidalstructureon35thecfforepresentations.Proof.@_Firstwwreprovetheuniquenessofthemultiplicationr\$:B5 z/B*4!lBandKcoftheunith0:K4! {B.[TheKcnaturaltransformationm):!4!{!Z BibSecomesamonoidaljnaturaltransformationwithrUR:B@ BX4!_7BandjË:K4!1BWVejshorwthatr꨹andXareuniquelydeterminedbry!ands2.Letr20<:B 9BV 4!*BWӹandn920u:BV 4!*KbSemorphismsthatmakre/amonoidalnaturaltransformation.8ThediagramsVH@!n9(X) !(Yp)Hq!n9(X) !(Yp) BE B{fdoˀά-`lL(X) (Y)HbzǠ*FfecǠ?'YH63Ǡ*Ffe6g,Ǡ?R; r-:0I0&!n9(X+ Yp)w>!n9(X+ Yp) BbL32fdЍά-=L(X Y)tandM덍⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jj!n9(K)ᅢ!n9(K) BL32fdH8pά-L()ppmsbm8K)hAcommrute.8InparticularthefollowingdiagramscommuteVHAFd!n9(B) !(B)Ht!n9(B) !(B) BE B@|{fdpά-`L(Bd) (B)HbzǠ*FfecǠ?'YH63Ǡ*Ffe6g,Ǡ?R; r-:0I!n9(BE B)!n9(BE B) B32fdpά-)"L(Bd B)ql7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX113YandS@⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jj!n9(K)ᅢ!n9(K) BL32fdH8pά-L(K)"GHence uwregetP b(1) j#  c(1) b(2) \|c(2)ι=URPb(1) c(1) r209(b(2) c(2) \|) uand1 1UR=1 n920l"Π@feTΠ?ۏ1 "Ǡ@feTǠ?ɞ⍍=sa"ΠM@feaTΠ?ɻfnԍandh50"v2K0"2KA2fd`ά-Y֯Z1ԩ" IK BZ\L(K)NF`HNF`HNF`HNF`HNF`HNF`H^H^j81 @Π@Π@Π@Π@Π@ΠPFPF*ǍF`@F`@F`@F`@F`@F`@F`@Ɩ`@Ɩ`Rҿ㍒<ǠǠǠǠ ǠǠǠ$G$G HB؜*Ǡ@fe\Ǡ?ɞ⍍݁݁=HenceXandrarecoalgebrahomomorphisms.s+7 &ev8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESeX115YTVo}shorwtheassoSciativityofrweidentifyalongthemaps c:O(MO N@) PP԰ =Ml (N Pƹ)^andfurthermoresimplifytherelevXanrtdiagramby xingthatIrepresentsasuitablepSermrutationofthetensorfactors.8Thenthefollowingcommute+BE B B+)BE B B B B BNIJfdV0ά-: U9jI{(L(Bd) (B) (B)++fBE B B/LIJfd3pά-q퍒7   1YBE B BY)BE B B B B B\y\yNfdV0ά-kX\I{(L(Bd B) (B))h9h9N҄fdV0ά-&X\I{(L(Bd) (B B))YYr8BE B/L{fd?ά-)-=    1BE B B\rBE B B BN32fdnά-p7L(Bd B B)~QBl32fdc0ά-m7   1H.U`*Ffe.`? 3;<1H.UǠ*Ffe.Ǡ?8I3;<1H泪`*Ffeܟ`? ɐ1 (r 1)Hj`*Ffe򜟽`? 1 (1 r)H\ꟽ`*Ffe`?>/hr 1Hh`*Ffeܟ`?>/N\1 rH9Ǡ*FfelǠ?o<1 rHʟǠ*FfeǠ?`|reTheuppSerrorwistheidentityhencewegettheassoSciativelaw.FVor39theproSofthatrhasthepropertiesofaunitwremustexplicitlyconsiderthecoherence morphismsandByreasonsofsymmetrywrewillonlyshowonehalfofoftheunitaxiom.]ThisaxiomfollorwsfromthecommutativityofthefollowingdiagramYv#Bv1BE B psH2fdoЍά-n!L(Bd)v"v"kyBE B KV1sH2fdـά-o0ōW-:1v"v"VBE BsH2fdά-o67vvB{ sH2fd/pά-omv 1"#BE K"__BE B K B'ПA2fd4 ά--L(Bd) (K)"".BE K B B0A2fd@ά-mA1 r 1""J~mBE B B*(A2fd`ά-7͍+ 1 1tBE BސA2fdά-m 1 1ԩ"#BE Kԩ"BE K B'П:2fd_ά-̯stL(Bd K)#BVBE B p32fd8ά-VAL(Bd)B{ 32fd/pά-mv 1sޟՠ@feՠ?"}-:1s#^ՠ@feVՠ? 1 1 L(K)sgBՠ@feguPՠ?^MH|p1 1 sՠ@feՠ?^M1 ޟΠ@feΠ? =ޟΠ@feΠ?ۏw1 1 rgBǠM@feguPǠ?XP1 rǠM@feǠ?»rޟǠ@feǠ?uG =s)vL(Bd) 1/ՠ9ՠCՠMՠWՠaՠb'Mb'M*PM:!0 1?`H?`H(?`H2?`H<?`HF?`HG%pHG%pje %cffxff ̟ff ̎ ̄cff;7  . cmsy9,@ cmti12+- cmcsc10)ppmsbm8( msbm10#a6cmex8"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line10n