; TeX output 2002.03.07:11007 VznDtGGcmr17Lectures7ton;?荑`DtqGcmr17QUANTUMBGRuNOUPSAND&::NONCOMMUT_ATIVEBGEOMETRYH؍SXQ ff cmr12Prof./Dr.B.Pareigis1?荒 XQ cmr12SummerSemester2002*7 Vr7 VNff cmbx12TfableffofContents"Chapter1. CommrutativeandNoncommutativeAlgebraicGeometryFG1 InrtroSduction_s1 1. ThePrinciplesofCommrutativeAlgebraicGeometry~C2 2. QuanrtumSpacesandNoncommutativeGeometry712 3. QuanrtumMonoidsandtheirActionsonQuantumSpaces\o17Chapter2. HopfAlgebras,Algebraic,FVormal,andQuanrtumGroups>31 InrtroSductionY31 1. HopfAlgebras;|32 2. MonoidsandGroupsinaCategoryM38 3. AneAlgebraicGroupsD42 4. FVormalGroups748 5. QuanrtumGroups+`50 6. QuanrtumAutomorphismGroupsf56 7. DualitryofHopfAlgebrasS62Chapter3. RepresenrtationTheoryV,ReconstructionandTannakXaDualitry)W67 InrtroSductionY67 1. RepresenrtationsofHopfAlgebrasؠ68 2. MonoidalCategoriesp74 3. DualObjects@82 4. Finitereconstruction_90 5. ThecoalgebracoSend<96 6. ThebialgebracoSend=T98 7. ThequanrtummonoidofaquantumspaceL100 8. Reconstructionand !", cmsy10C5-categoriesm103Chapter4. TheIn nitesimalTheoryk113 1. InrtegralsandFVourierTransformsկ113 2. DerivXationsDȔ126 3. TheLieAlgebraofPrimitivreElements#131 4. DerivXationsandLieAlgebrasofAneAlgebraicGroups\ 134Bibliographry^;139],o cmr937 VRCHAPTER1nCommutativeffandNoncommutativeffAlgebraicGeometry"-N cmbx12Intro`ductionThroughoutMMwrewill xabase eld) msbm10K.mThereadermayconsideritasrealnumbSersorcomplexnrumbSersoranryotherofhismostfavorite elds.A.fundamenrtal/BandpSowerfultoSolforgeometryistoassociatewitheacrhspaceg cmmi12Xethet-algebraoffunctionsOUV(X)fromXtothebase eld(ofcoSecienrts). oThedreamofgeometryisthatthisconstructionisbijectivre,i.e.'Rthattwodi erentspacesareEmappSedtotrwoEdi erentfunctionalgebrasandthateachalgebraisthefunctionalgebraofsomespace.ActuallyRthespacesandthealgebraswillformacategoryV.pThereareadmissiblemaps. FVorugalgebrasitisquiteclearwhatthesemapswillbSe.FVorspacesthisislessjobrvious,partlyduetothefactthatwedidnotsayclearlywhatspacesexactlyare.)Thenthe.@ cmti12drffeam{ofgeffometrywrouldbSethatthesetwocategories,ęthecategoryof(certain)spacesandthecategoryof(certain)algebras,aredualtoeacrhother.AlgebraicgeometryV,noncommrutativegeometryV,andtheoreticalphysicshaveasabasis&thisfundamenrtalidea, thedualityoftwocategories, thecategoryofspaces(statespaces/inphrysics)andthecategoryoffunctionalgebras(algebrasofobservXables)inphrysics.WVewillpresentthisdualityinthe1.chapter.CertainlythetypSeofspacesaswrellasthetypSeofalgebraswillhavetobSespeci ed.TheoreticalIphrysicsusesthecategoriesofloScallycompactHausdor spacesandof;commrutativeCܞ2!K cmsy8-algebras.jA6famoustheoremofGelfand-Naimarksaysthatthesecategoriesaredualsofeacrhother.(Ane)algebraicgeometryusesadualitrybSetweenthecategoriesofanealgebraicscrhemesandof(reduced) nitelygeneratedcommutativealgebras.TVo9sgetthewholeframewrorkofalgebraicgeometryoneneedstogotomoregen-eral`zspacesbrypatchinganespacestogether.VOnthealgebrasidethisamountstoconsideringlshearvesofcommutativealgebras.,WVeshallnotpursuethismoregeneralapproacrhtoalgebraicgeometryV,sincegeneralizationstononcommutativegeometryarestillinthestateofdevrelopmentandincomplete.Noncommrutativegeometryuseseither(imaginary)noncommrutativespacesandnotQnecessarilycommrutativeQalgebrasor(imaginary)noncommrutativeQspacesandnotnecessarilycommrutativeCܞ2-algebras.WVecwilltakreanapproachtothedualitybSetweengeometryandalgebrathatheavilyuses-$functorialtoSols,=especially-$represenrtablefunctors.TTheane(algebraic)spaces]1 7 25+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVwre)usewillbSegivenintheformofsetsofcommonzerosofcertainpSolynomials,PvwherethezeroscanbSetakreninarbitrary(commutative)K-algebrasB.Soananespacewillconsistofmanrydi erentsetsofzeros,depSendingonthechoiceofthecoSecientalgebraB.WVe rstgivreashortintroSductiontocommutativealgebraicgeometryinthissetupanddevrelopadualitybSetweenthecategoryofane(algebraic)spacesandthecate-goryof( nitelygenerated)commrutativealgebras.Thenwrewilltransferittothenoncommutativesituation.>ThefunctorialapproachtoalgebraicgeometryisnottoSooftenusedbutitlendsitselfparticularlywrelltothestudyyofthenoncommrutativeysituation.MEveninthatsituationoneobtainsspace-likeobjects.Thecrhapterwillclosewitha rststeptoconstructautomorphism\groups"ofnoncommrutativespaces.*SincetheconstructionofinrversespresentsspSecialproblemswrewillonlyconstructendomorphism\monoids"inthischapterandpSostponethestudyofinrvertibleendomorphismsorautomorphismstothenextcrhapter.Arttheendofthechapteryoushouldڍ +〹knorwhowtoconstructananeschemefromacommutativealgebra, +〹knorwhowtoconstructthefunctionalgebraofananescheme, +〹knorwwhatanoncommutativespaceisandknowexamplesofsuch, +〹understand||andbSeabletoconstructendomorphismquanrtummonoidsof+certainnoncommrutativespaces, +〹understand,Uxwhry@endomorphismquantummonoidsarenotmadeoutofen-+domorphismsofanoncommrutativespace.&K5Pn1.FnThePrinciplesofCommutativeAlgebraicGeometryWVeHwillbSeginwithsimplestformof(commrutative)Hgeometricspacesandseeadualitry$bSetweentheseverysimple\spaces"andcertaincommutativealgebras.TThisexampleľwillshorwhowtheconceptofafunctionalgebracanbSeusedtoful llthedream ofgeometryinthissituation.ItwillalsoshorwthefunctorialmethoSdsthatwill[bSeappliedthroughoutthistext.vItisaparticularlysimpleexampleofadualitryas$menrtionedintheintroSduction.RTThisexamplewillnotbeusedlateron,CsowrewillonlyskretchtheproSofsforsomeofthestatements.ڍExample1.1.1.cConsiderJasetofpSoinrtswithoutanyadditionalgeometricstruc-ture.zSo/thegeometricspaceisjustaset.WVeinrtroSducethenotionofitsalgebraoffunctions.LetR&XCbSeaset. o[ThenK22cmmi8X :=8Map(XJg;K)isaK-algebrawithcomponenrtwiseaddition[andmrultiplication:V+W Ogn9)UR= K2he(fG)+ OK2h(gn9).Considerthehomomorphismo:URK2X =Q K2Y  !K2XY,Dde nedbryW(fg gn9)(x;y)UR:=fG(x)gn9(y).XIn,ordertoobtainauniquehomomorphism-de nedonthetensorproSductwrehavetoshowthatW20z:URK2X }*_K2Y  !K2XYPisabilinearmap:&W20%V(f+fG208;gn9)(x;y)UR=(fŒ+}fG208)(x)gn9(y)UR=(fG(x)+f208(x))gn9(y)UR=fG(x)gn9(y)+fG208(x)g(y)UR=(W20%V(f;gn9)+W20(fG208;gn9))(x;y)givres?theadditivityinthelefthandargument. TheadditivityintherighthandargumenrtandthebilinearityischeckedsimilarlyV.kOnecancheckthatRƹisalwaysinjectivre.8IfX+orYare nitethenAŹisbijective.BWAsvaspSecialexamplewreobtainamultiplicationrUR:K2X NK2X2up r4!"K2XX2*ppmsbm8K-:Aacmr6p!5!0WK2Xwhere!h:Xv!qXJXrinSet(qisthediagonalmap(x):=(x;x).޵FVurthermorewre{get#aunitË:URK2fg2QK-:;cmmi6p^!'@ K2X Awhere:XF``!fgistheuniquemapinrtotheoneelementset.OneXvreri eseasilythat(K2X;n9;r)XisaK-algebra.TwropropSertiesareessentialhere,theassoSciativitryandtheunitofKandthefactthat(XJg;;)isa\comonoid"inthecategorySet :J͍wX+XoX+XXl32fd&H`O line10-'|{Ycmr81Z&XZiX+XM{fd?ά-̍ȕHKʟǠ*Ffe~Ǡ?` H=ʟǠ*FfepǠ?p#|1LԠt XԠX+X:2fdnά-ζ vzҟǠ@fez8Ǡ??AҬ>UH'<w ׁ w w w S̟>S̟> #K2Y#5lK2Xܞ32fd"€ά-VK-:fVcommrute.ThrusZ$iK -:URSet[u_!ʳK-c AlgisaconrtravXariantfunctor.ByLthede nitionoftheset-theoretic(cartesian)proSductwreknowthatK2X ͠=#u cmex10Q UWXrK.0This`idenrtitydoSesnotonlyholdonthesetlevel,բitholdsalsoforthealgebraRJstructuresonK2X 1resp.8Q7XK.WVenorwconstructaninversefunctor@ SpSec6:URK-c Alg f!*Set :FVorJ}eacrhpSointxq2X<thereJ}isamaximalidealmx dofQԟX[Kde nedbymx AX:=qff@p2Map+Q(XJg;K)jfG(x)=0g.IfYIXJ̹isa nitesetthentheseareexactlyallmaximalidealsofQHVXeK.QTVoshorwthisweobservethefollowing.QThesurjectivehomomorphismpxk:Q UWXrKi!Khaskrernelmx ?rhencemxisamaximalideal.\IfmiQXmKisamaximalidealandai=( 1;:::ʜ; nP)2mthenforanry i]6=i0weget(0;:::ʜ;0;1id;0;:::;0)i=L(0;:::ʜ;0; 1 i p ;0;:::;0)( 1;:::; nP)UR2m.hencethei-thfactor0A:::"K:::0.ofQ UWXrKisinm.Sotheelemenrtsai2mmusthaveatleastonecommoncompSonent jX=qN0sincem6=K.-fButmorethanonesucrhacompSonentisimpSossiblesincewewrouldvgetzerodivisorsintheresidueclassalgebra..ThusmB=mx wherevx2Xgisthejӹ-thelemenrtsoftheset.OnewcaneasilyshorwmorenamelythattheidealsmxarepreciselyallprimeidealsofMap(XJg;K).With)eacrhcommutativealgebraAwecanassoSciatethesetSpec(A)ofallprimeideals:ofA.)Thatde nesafunctorSpSec:K-AlgDz^! ;Set .Appliedtoalgebrasoftheform~K2X Mx=/QFXKwitha nitesetXthisfunctorrecorvers~XasXP!r԰:Y=SpSec,z6(K2X).Thrusthedreamofgeometryissatis edinthisparticularexample.CThe*abSorveexampleshowsthatwemayhopSetogainsomeinformationonthespace(set)XkbryknowingitsalgebraoffunctionsK2X qandapplyingthefunctorSpSectoit.FVor nitesetsandcertainalgebrasthefunctorsK2-}EandSpSecactuallyde neacategorydualitryV.8Wearegoingtoexpandthisdualitrytolargercategories.WVe)shallcarrysomegeometricstructureinrtothesetsXandwillstudythecon-nectionbSetrweenthesegeometricspacesandtheiralgebrasoffunctions. UMFVorthispurpSosewrewilldescribesetsofpoinrtsbytheircoSordinates.ExamplesarethecircleortheparabSola.MoregenerallythegeometricspaceswrearegoingtoconsideraresoEcalledanescrhemesdescribSedbypSolynomialequations.=WVewillseethatsuch77 ?41.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y;65VgeometricspacesarecompletelydescribSedbrytheiralgebrasoffunctions.HeretheYVonedaLemmawillplaryacentralr^ ole.WVetwill,Yhorwever,taketadi erenrtapproachtofunctionsalgebrasandgeometricspaces,0than"onedoSesinalgebraicgeometryV.We"usethefunctorialapproacrh,whichlends Mitselftoaneasieraccesstotheprinciplesofnoncommrutative MgeometryV.Wewill/dde negeometricspacesascertainfunctorsfromthecategoryofcommrutativealgebras.Xtothecategoryofsets.Thesesetswillharve.Xastronggeometricalmeaning.TheN%functorswillassoSciatewitheacrhalgebraAthesetofpoinrtsofa\geometricvXarietry",wherethepSointshavecoSordinatesinthealgebraA.UDe nition1.1.2.ֹThe`functorAUR=A21V:K-c Alg f!*Setf(the`underlyingfunctor)thatassoSciateswitheacrhcommutativeK-algebraAitsspace(set)ofpSoints(elements)A꨹iscalledtheane35line.Lemma1.1.3.The35functor\aneline"isarffepresentable35functor.*O/- cmcsc10Proof.@_ByέLemma2.3.5therepresenrtingobjectisK[x].7Observethatitisuniqueuptoisomorphism.DT& msam10De nition1.1.4.!ThefunctorA22 ¹:IK-c AlgG!#Set*thatassoSciateswitheacrhcommrutativealgebraAthespace(set)ofpSoinrts(elements)oftheplaneA22iscalledtheane35plane.ULemma1.1.5.The35functor\aneplane"isarffepresentable35functor.Proof.@_SimilarDtoLemma2.3.9therepresenrtingobjectisK[x1;x2].ThisDalgebraisuniqueuptoisomorphism.Letp1(x1;:::ʜ;xnP);:::;pmĹ(x1;:::;xnP)2K[x1;:::;xnP]bSeafamilyofpolynomials.WVeMwranttoconsiderthe(geometric)vXarietyofzerosofthesepSolynomials.atObservethatsKmarynotcontainsucientlymanyzerosforthesepSolynomials.$Thuswearego-ing9toadmitzerosinextension eldsofKormoregenerallyinarbitrarycommrutativeK-algebras.In2ythefollorwingrathersimplebuildupofcommutativealgebraicgeometryV,nthereadershouldcarefullyvrerifyinwhichstatementsandproSofsthecommutativityisreally\needed.MostofthefollorwingwillbSeverballygeneralizedtonotnecessarilycommrutativealgebras.UDe nition1.1.6.ٹGivren asetofpSolynomialsfp1;:::ʜ;pmg}K[x1;:::ʜ;xnP].ThefunctorXthatassoSciateswitheacrhcommutativealgebraAthesetXӹ(A)ofzerosoftheNpSolynomials(pidڹ)inA2n uiscalledananealgebrffaicvarietyNorananescheme(in`A2nP)withde ningpSolynomialsp1;:::ʜ;pmĹ.<TheelemenrtsinXӹ(A)arecalledtheA-pffoints꨹ofXӹ.UTheorem91.1.7.n&The/aneschemeXFinA2n ,withde ningpffolynomialsp1;:::ʜ;pmis35arffepresentable35functorwithrffepresenting35algebra} OUV(Xӹ)UR:=K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ);G7 65+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVcffalled35theanealgebraofthefunctorX.Proof.@_FirstwreshowthattheaneschemeXt,:YK-c Alg2m!%Setwiththede ning8pSolynomialsp1;:::ʜ;pm isafunctor.ΐLetf:̾AI! Bb>be8ahomomorphismof commrutativealgebras. JTheinducedmapfG2n Gʹ:W{A2n V! pB2n vde nedbyapplica-tion`]off\onthecompSonenrtsrestrictstoXӹ(A)jA2n as`]X(fG):X(A)!aX(B).Thismapiswrell-de nedforlet(a1;:::ʜ;anP)r2Xӹ(A)bSeazeroforallpolynomi-als{p1;:::ʜ;pm ?thenpidڹ(fG(a1);:::ʜ;f(anP))=fG(pidڹ(a1;:::ʜ;anP))=fG(0)=0{forallihencefG2nO(a1;:::ʜ;anP)=(fG(a1);:::ʜ;f(anP))2B2n isazeroforallpSolynomials.GThrusXӹ(fG)UR:X(A)n!1X(B)iswrell-de ned.8FVunctorialityofX{isclearnorw.NorwweshowthatXo͹isrepresentablebyOUV(Xӹ)UR=K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ).$Ob-servreCthat(p1;:::ʜ;pmĹ)denotesthe(two-sided)idealinK[x1;:::ʜ;xnP]generatedbythepSolynomialslUp1;:::ʜ;pmĹ.WVeknorwthateachn-tupSel(a1;:::ʜ;anP)UR2A2n uniquelylUdeter-mines7analgebrahomomorphismf :ؙK[x1;:::ʜ;xnP]$!ʿA7ȹbryfG(x1)ؙ=a1;:::ʜ;fG(xnP)=anP.=(ThepSolynomialringK[x1;:::ʜ;xn]inK-c Algpisfreeorverthesetfx1;:::ʜ;xng,orK[x1;:::ʜ;xnP]togetherwiththeemrbSedding s:fx1;:::ʜ;xnPg#!.sK[x1;:::ʜ;xnP]isacounivrersalsolutionoftheproblemgivenbytheunderlyingfunctorA&:K-c Alg!nSet̆andthesetfx1;:::ʜ;xnPgUR2Set .),ThishomomorphismofalgebrasmapspSoly-nomialsp(x1;:::ʜ;xnP)inrtofG(p)f=p(a1;:::ʜ;anP).WHence(a1;:::ʜ;anP)isacommonzeroofthepSolynomialsp1;:::ʜ;pm ifandonlyiffG(pidڹ)j=pi(a1;:::ʜ;anP)=0,i.e.]~p1;:::ʜ;pmarehJinthekrerneloffG. lThishappSensifandonlyiffIvXanishesontheideal(p1;:::ʜ;pmĹ)orinotherwrordcanbSefactorizedthroughtheresidueclassmaptcJi:URK[x1;:::ʜ;xnP]n!1K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ)ThisinducesabijectionMor/SK-0#fcmti8cAlgJ6u(K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ);A)UR3fQ7!(fG(x1);:::ʜ;f(xnP))UR2Xӹ(A):Norwitiseasytoseethatthisbijectionisanaturalisomorphism(inA).4IfnopSolynomialsaregivrenfortheaborveconstruction,GthenthefunctorunderthisconstructionZ;istheanespaceA2n ofdimensionn.BygivingpSolynomialsthefunctorXbSecomes]asubfunctorofA2nP,zbecauseitde nessubsetsXӹ(A)URA2nP(A)=A2n. Bothfunctorsarerepresenrtablefunctors.TheembSeddingisinducedbythehomomorphismofalgebras:URK[x1;:::ʜ;xnP]n!1K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ).]"Problem1.1.1.(1)Determinetheanealgebraofthefunctor\unitcircle"+Sן21 ]inA22.(2)+Determinetheanealgebraofthefunctor\unitsphere"Sן2n1"KinA2nP.(3)+DetermineEOtheanealgebraofthefunctor\torus"TRand ndan\emrbSed-+ding"ofTuinrtoA23.=ActuallyډTVcanbSeconsideredasproductx21 Mx21(B)=x21(B)Mx21(B).+TVakrebTthe rstcopyofx21(B)asthecirclewithradius2,}thenwehavex21RA1(B)3+S21RA2r۹(B)UR=f(u;vn9;x;y)ju22j+v22=UR4;x22j+y22=UR1g.8TheemrbSeddingistbqu(u;vn9;x;y)UR7!(u;v;0)+1=2x(u;v;0)+(0;0;y)Wڠ7 ?41.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y;67V+〹hencethealgebramapis..F2cK[x;yn9;z]URn!1K[u;v]=(u 2j+v 24) K[x;y]=(x 2j+y 21)+withGɍʍnxUR7!u+1=2xu;nyË7!URv+1=2xvn9;nz57!URyn9:(4)+LetU*denotetheplanecurvrexyË=UR1.ThenUisnotisomorphictotheane+line.*F(Hinrt:"AnisomorphismK[x;x21 \|]URn!1K[yn9]sendsxtoapSolynomialp(y)+whicrh-!mustbSeinvertible.KConsiderthehighestcoSecientofp(yn9)andshow+thatp(yn9)UR2K.8ButthatmeansthatthemapcannotbSebijectivre.)+Uisalsocalledtheunit35functor.8Canyrouexplain,why?(5)+LetXidenotetheplanecurvrey~=Ex22.pQThenXisisomorphictotheane+line.(6)+LetK[=CbSethe eldofcomplexnrumbSers.DShowthattheunitfunctorU :+K-c Alg f!*Set蓹inProblem(3)isnaturallyisomorphictotheunitcircleSן21r۹.+(Hinrt:?ThereWisanalgebraisomorphismbSetweentherepresentingalgebras+K[e;e21 \|]andK[c;s]=(c22j+s221).)(7)+〟26_+LetKbSeanalgebraicallyclosed eld. Letpbeanirreduciblesquare+pSolynomialLinK[x;yn9].LetZ@+betheconicsectionde nedbrypwiththeane+algebradRK[x;yn9]=(p).ShorwthatZX^isnaturallyisomorphiceithertoX&%orto+Ufromparts(4)resp.8(5).KRemark1.1.8.CAnemalgebrasofanescrhemesare nitelygeneratedcommu-tativreԒalgebrasandanysuchalgebraisananealgebraofsomeanescheme,sinceAPUR԰n9=K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ)(HilbSertbasistheorem).The8pSolynomialsp1;:::ʜ;pm =arenotuniquelydeterminedbrytheanealgebraofananescrheme."NoteventheidealgeneratedbythepSolynomialsinthepolynomialring[K[x1;:::ʜ;xnP]isuniquelydetermined.dAlsothenrumbSer[ofvXariablesx1;:::ʜ;xn isnotuniquelydetermined.TheŴK-pSoinrts( 1;:::ʜ; nP)UR2Xӹ(K)ŴofananeschemeX(withcoSecientsinthebase eldK)arecalledrffationalpoints.pqTheydonotsucetocompletelydescribSetheanescrheme.LetforexampleKP=RѹthesetofrationalnrumbSers. [IfXandYgareanescrhemes6withanealgebrasOUV(Xӹ)UR:=K[x;yn9]=(x22+;y22i׹+1)6andOUV(Y)UR:=K[x]=(x22+1)then[2bSothscrhemeshavenorationalpSoints. TheschemeY,Uhowever,has[2exactlytrwo5complexpSoinrts(withcoecienrtsinthe eldCofcomplexnumbSers)andthescrheme:XF hasin nitelymanycomplexpSoints,henceXӹ(C)Z6P԰= Y(C).ThisdoSesnotresultfromtheemrbSeddingsintodi erentspacesA22 Wresp.>8A21.InfactwealsohaveOUV(Y)UR=K[x]=(x22+1)P԰n9=K[x;yn9]=(x22+1;yn9),-socYcanbSeconsideredasananescrhemeinA22.Since&SeacrhaneschemeX&isisomorphictothefunctorMor\PK-cAlg3?'(OUV(Xӹ);-)wewillhenceforthidenrtifythesetwofunctors,thusremovingannoyingisomorphisms.p7 85+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVDe nitionv1.1.9.߹LetK-A pdenotethecategoryofallcommrutative nitelygenerated(oranecf.ԣ1.1.8)K-algebras.Ananealgebrffaicvarietyisarepresenrtablefunctor K-A 3U(A;-)/:K-A !@Set .The anealgebraicvXarietiestogetherwiththenaturalPtransformationsformthecffategory|aofanealgebrffaicvarietiesPK-VarorverK.]ThefunctorthatassoSciateswitheacrhanealgebraAitsanealgebraicvXarietyrepresenrtedbyAisdenotedbySpSecx:URK-A 2!K-Var ,SpSec&(A)=K-A 3U(A;-).BytheYVonedaLemmathefunctor7؍نSpSecV:URK-A 2!K-VarisananrtiequivXalence(orduality)ofcategorieswithinversefunctor/O:URK-Varo1!!NK-A 3U:AnanealgebraicvXarietryiscompletelydescribSedbyitsanealgebraOUV(Xӹ).וThusthedreamofgeometryisrealized.Arbitrary^(notnecessarily nitelygenerated)commrutative^algebrasalsode nerepresenrtablefunctors(de nedonthecategoryofallcommutativealgebras).Thuswealso|harve\in nitedimensional"vXarietieswhichwewillcallgeffometricVspacesoranevarieties.8WVedenotetheircategorybryG\effom(K)andgetacommutativediagramP؍\ K-c AlgոG\effom(K)532fd,'ά- T덍PP԰iع=צHnoHOK-A HVK-Var̟{fd6@ά-}SpSecH:Ǡ*FfelǠ?H:Ǡ*FfelǠ?Z썑WVecalltherepresenrtablefunctorsXg:K-c Alg 9%!vSet ageffometric^spaces߹oranevarieties,andbtherepresenrtablefunctorsX㗹:!K-A Un!jSetiGaneschemesoranealgebrffaic35varieties.8ThisisanotherrealizationofthedreamofgeometryV.The:HgeometricspacescanbSeviewredassetsofzerosinarbitrarycommutativeK-algebras[B#ofarbitrarilymanrypSolynomialswitharbitrarilymanyvXariables.@ThefunctionalgebraofX{willbSecalledtheane35algebrffaofXinbSothcases.덑Example31.1.10."AUsomewhatVlesstrivialexampleisthestatespaceofacircularpSenduluma(oflength1). 9 ThelocationisinL+=f(a;b)2A22ja224+0b22 묹=1g,themomenrtumuisinM6=URfp2Aguwhichisastraightline.SothewholegeometricspaceforthepSendulumis(LM@)(A)X=f(a;b;p)ja;b;pX2A;a22k+b22=1g.>ThisgeometricspaceisrepresenrtedbyK[x;yn9;z]=(x22j+y221)sinceT(LM@)(A)UR=f(a;b;p)ja;b;pUR2A;a 2Bƹ+b 2V=1gP԰n9=K-c Alg f(K[x;yn9;z]=(x 2Bƹ+y 21);A):7؍The4#trwoantiequivXalencesofcategoriesabSovegiverisetothequestionforthefunc-tioncalgebra.tIfarepresenrtablefunctorX%=URK-c Alg f(A;-)isviewedasgeometricsetsofu8zerosofcertainpSolynomials,i.e.asspaceswithcoordinatesinarbitrarycommruta-tivre%algebrasB,4(plusfunctorialbSehavior),4thenitisnotclearwhytherepresentingalgebraȭAshouldbSeanrythinglikeanalgebraoffunctionsonthesegeometricsets.-It U7 ?41.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y;69VisnotevrenclearwherethesefunctionsshouldassumetheirvXalues.OOnlyifwecanshorw`thatAcanbSeviewedasareasonablealgebraoffunctions,|LweshouldtalkabSoutaLBrealizationofthedreamofgeometryV.]ButthiswillbSedoneinthefollorwingtheo-rem.WVehgwillconsiderfunctionsasmaps(coSordinatefunctions)fromthegeometricsetDiXӹ(B)tothesetofcoSordinatesB,ZmapsthatarenaturalinB.F#SucrhcoSordinatefunctionsarejustnaturaltransformationsfromX{totheunderlyingfunctorA.`⍑Theorem221.1.11.!LffetXbeageometricspacewiththeanealgebraAUR=OUV(Xӹ).Then}APUR԰n9=Nat#@(X;A)asK-algebrffas,where}AUR:K-35c fAlg "F!wS*et}9istheunderlyingfunc-torEoraneline.DTheisomorphismAPUR԰n9=Nat#@(X;A)inducffesanaturaltransformationAXӹ(B)UR!B;(naturffal35inB).+Proof.@_FirstYwrede neanisomorphismbSetweenthesetsAandNatT(X;A).BecauseofX=JMorGK-cAlg6(A;-)J=:K-c Alg f(A;-)andAJ=MorGK-cAlg6(K[x];-)=:K-c Alg f(K[x];-)theYVonedaLemmagivresusǍ!Natb7(X;A)UR=Nat(K-c Alg f(A;-);K-cAlg f(K[x];-))PUR԰n9=K-cAlg f(K[x];A)UR=A(A)P԰n9=Aonthesetlevrel. Let>:A!h Nat.(X;A)denotethegivenisomorphism. isde ned\bry(a)(B)(p)(x)0:=p(a). By\theYVonedaLemmaitsinverseisgivenby21 \|( h:=UR ((A)(1)(x).Nat%@(X;A)carriesanalgebrastructuregivrenbythealgebrastructureofthecoSef- cienrts.FVor)acoSecientalgebraB,8aB-pSoinrtps:A!sBin)Xӹ(B)=K-Alg f(A;B),and ; j2%Nate(X;A)wrehave (B)(p)%2A(B)=B. -Hence( !+ʒ O)(B)(p):=( (B)+ O(B))(p)i= (B)(p)+ O(B)(p)4Land(  O)(B)(p)i:=( (B) O(B))(p)i= (B)(p) O(B)(p)makreNat+Q(X;A)analgebra.Let-abSeanarbitraryelemenrtinA.7oBytheisomorphismgivenabSovethisele-menrtMinducesanalgebrahomomorphismga Z:VK[x]pL!AMmappingxontoa.Thisalgebraӽhomomorphisminducesthenaturaltransformation(a):XW q*!A. Onthe|B-levrelitisjustthecompSositionwithgaϹ,i.e.^(a)(B)(p)#=(K[x]h HgaJ ?|!)Ah q"pJ ?|!B). L~Since2sucrhahomomorphismiscompletelydescribSedbytheimageofxweget](a)(B)(p)(x)=p(a).TVocomparethealgebrastructuresofAandNat(X;A)leta;a202 A. WVeharve(a)(B)(p)(x)=p(a)and(a20:)(B)(p)(x)=p(a209),hence(a0j+a209)(B)(p)(x)=p(a0j+a20)=p(a)0j+p(a20)=(a)(B)(p)(x)0j+(a20)(B)(p)(x)=((a)(B)(p)*+(a209)(B)(p))(x)UR=((a)(B)*+(a209)(B))(p)(x)UR=((a)*+(a209))(B)(p)(x).Analogouslywreget(aa209)(B)(p)(x)V=p(aa20)=p(a)p(a20)=((a)(a20))(B)(p)(x),andWthrus(auU+a209)P=(a)uU+(a20)Wand(aa20)P=(a)uU(a20).HenceWadditionandmrultiplicationinNat(X;A)arede nedbytheadditionandthemultiplicationofthevXaluesp(a)+p(a209)resp.8p(a)p(a20).WVe{BdescribSetheaction n9(B)Kt:A Xӹ(B)Ktd!uBHof{BAonXӹ(B).LetpKt:Ad!uBbSePaB-poinrtinK-c Alg f(A;B)|=Xӹ(B).AFVorPeacrha|2APtheimage(a)|:X>XE!.Aismanaturaltransformationhencewrehavemaps n9(B)m:AXӹ(B)mS!B*ssucrhmthat n9(B)(a;p)<=p(a).wMFinally!eacrhhomomorphismofalgebrasf;:@>RHD6Xӹ(B)GǠ*Ffez̟Ǡ?3f1X(BI)commrutes. I WVewillshowthisresultlateronfornoncommutativealgebras. I TheunivrersalpropSertyimpliesthatthefunctionalgebraAofangeometricspaceXmisuniqueuptoisomorphism.Let-XbSeangeometricspacewithfunctionalgebraAǹ=OUV(Xӹ).oIf-p:AR!Kis arationalpSoinrtofXӹ,Ri.e.ahomomorphismofalgebras,thenImp(p)?=K henceKer(p)isamaximalidealofAofcoSdimension1._Conrverselylet1%n eufm10mbeamaximalideal&ofAofcoSdimension1thenthisde nesarationalpoinrtpG:A!A=mP԰.=K.IfKisalgebraiclyclosedandmanarbitrarymaximalidealofA,mthenA=misa nitelygeneratedK-algebraanda eldextensionofK,henceitcoincideswithK.iThrusthecoSdimension|ofmis1.ThesetofmaximalidealsofAiscalledthemaximal͍spffectrumSpSecmĹ(A).Thisistheapproacrhofalgebraicgeometrytorecoverthegeometricspaceof(rational)pSoinrtsfromthefunctionalgebraA.ǐWVewillnotfollowthisapproachsinceitdoSesnoteasilyextendtononcommrutativegeometryV.Problems1.1.2.M91.LetAXbSeangeometricspacewithanealgebraA.ShorwthatthealgebraAisunivrersalwithrespSecttothepropertryV,޻thatforeachcommutativealgebrawDandeacrhnaturaltransformationa:DXӹ(-)0!HO-#ntherewisaunique ٠7 ?41.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y6p811VhomomorphismofalgebrasfQ:URDk!A,sucrhthatthetriangleJam3AXӹ(B)B:^,32fd$0ά-W`@ I{(Bd)H`K(Bd)༟ׁ @༟ @༟ @༟ @'<>@'<>RH$D6Xӹ(B)u*Ǡ*Ffe˨\Ǡ?3@f1X(BI)2.8LetX{bSeananescrhemewithanealgebraW@AUR=K[x1;:::ʜ;xnP]=(p1;:::ʜ;pmĹ):De ne[\coSordinatefunctions"qi %:KXӹ(B)5!R#B'awhicrh[describethecoordinatesofB-pSoinrtsandidentifythesecoSordinatefunctionswithelementsofA.ύNorwwewillstudymorphismsbSetweengeometricspaces.+Theorem !1.1.13.>GLffetOX$A2r 5andY!A2s bffeanealgebraicvarietiesandletUR:X%}!Y/bffe35anaturaltransformation.fiThentherearepolynomialsW@]p1(x1;:::ʜ;xrb);:::;psn<(x1;:::;xrb)UR2K[x1;:::;xrb];such35thatpۍPL(A)(a1;:::ʜ;arb)UR=(p1(a1;:::ʜ;arb);:::;psn<(a1;:::;arb)); forT-allAlo2K-35A andT-all(a1;:::ʜ;arb)lo2Xӹ(A),ki.e.T-themorphismsbffetweenT-anealgebrffaic35varietiesareofpolynomialtype.éProof.@_LetqOUV(Xӹ)K=K[x1;:::ʜ;xrb]=IandOUV(Y)=K[y1;:::ʜ;ysn<]=Jr. =FVorA2K-AlgEandj(a1;:::ʜ;arb)2Xӹ(A)letf+:K[x1;:::ʜ;xrb]=IԔ!2AwithfG(xidڹ)=aiϖbSethehomomorphismobtainedfromXӹ(A)P_9԰x =K-Alg f(K[x1;:::ʜ;xrb]=I;A). ThenaturaltransformationoKisgivrenbycompSositionwithahomomorphismgO:7K[y1;:::ʜ;ysn<]=J!nK[x1;:::ʜ;xrb]=I+hencewregetW@(A)UR:K-c Alg f(K[x1;:::ʜ;xrb]=I;A)UR3fQ7!fGgË2K-cAlg f(K[y1;:::ʜ;ysn<]=J:;A):SincegXisdescribSedbrygn9(yidڹ)UR=pi(x1;:::ʜ;xrb)2K[x1;:::ʜ;xrb]wreget"W@ʍl(A)(a1;:::ʜ;asn<)UR=(fGgn9(y1);:::ʜ;fgn9(ysn<))l=UR(fG(p1(x1;:::ʜ;xrb));:::;fG(psn<(x1;:::;xrb)))l=UR(p1(a1;:::ʜ;arb);:::;psn<(a1;:::;arb)):!pٍI`éAnanalogousstatemenrtholdsforgeometricspaces.+Example@1.1.14.ԹTheisomorphismbSetrweentheaneline(1.1.2)andthepara-bSolaeisgivrenbytheisomorphismfn:&K[x;yn9]=(yl~x22)@u!gaK[z],fG(x)=z,fG(yn9)=z22thathastheinrversefunctionfG21 {(z)z=x.=OntheanescrhemesA,theaneline,andP,theparabSola,theinducedmapisf:>A(A)3a7!(a;a22)2P(A)resp.fG21͹:URP(A)3(a;b)7!a2A(A). 7 125+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YV=[2.NQuantumSpacesandNoncommutativeGeometryNorw8wecometononcommutativegeometricspacesandtheirfunctionalgebras.ManryǙofthebasicprinciplesofcommutativealgebraicgeometryasintroSducedin1.1carry,orvertononcommutativegeometryV.nOurmainaim,bMhowever,is,tostudythesymmetries0p(automorphisms)ofnoncommrutative0pspaceswhicrhleadtothenotionofaquanrtumgroup.Sincelktheconstructionofnoncommrutativelkgeometricspaceshasdeepapplicationsintheoreticalphrysicswewillalsocallthesespacesquantumspaces."8㍑De nition 1.2.1.Let)AbSea(notnecessarilycommrutative))K-algebra.UcThenthe"functorX*4:=haK-Alg f(A;-):K-Algu!#PSet)*represenrted"byAiscalled(ane)noncffommutative~(geometric)space<ǹorquantumspfface./=TheLffet(Dbeasetand&|:DQRXӹ(-35)!eA(-)(bffeanaturaltransfor-mation.fiThen35therffeexistsauniquemapfQ:URDK!dAsuchthatthediagram;vAXӹ(B)HB̶32fdCά-W` I{(Bd)@"D6Xӹ(B)ǠfeǠ?b{f1@b\(Bd)Ǫ@|>RHdD6Xӹ(B)JjǠ*Ffe}Ǡ?`f1commrutes.zDe nition1.2.5.ThenoncommrutativespaceAߍ2j0]GqG$withthefunctionalgebra /OUV(A 2j0ڍq \|)UR:=Khx;yn9i=(xyq 1 ʵyx)Ԣwithx:qË2URKnf0giscalledthe(deformeffd)quantumplane.Thenoncommrutativespace"!Aߍ0j2]GqG$withthefunctionalgebra|SOUV(A 0j2ڍq \|)UR:=Khs;n9i=( 2Tw;n9 2.=;+qn9s)iscalledthedual35(deformeffd)quantumplane.8WVeharveeq|A 2j0ڍq \|(A)UR=q USqʍ*xlyqȍ$U 38$U (Ux;yË2A;xyË=qn9 1 ʵyn9x2Kq andhGNA 0j2ڍq \|(A)UR=G UTGʍVs;") eGȍ.g 38.g 2gs;Ë2A;s 2ɹ=0;n9 2=0;sË=qn9G :橠7 K;2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRYA 15VDe nitionY1.2.6. LetXȅbSeanoncommrutativespacewithfunctionalgebraAand letXc ;MbSetherestrictionofthefunctorX :C:K-AlgN!#SetVtothecategoryofcommrutativeYtalgebras:xXc@a:K-c Algxq! Set .DThenwrecallXcthecffommutative partofthenoncommrutativespaceXӹ.Lemma1.2.7.kThewcffommutativepartXc oofanoncommutativespaceX9isanane35variety.Proof.@_The underlyingfunctorAй:K-c Alg Y !K-Algrahas aleftadjoinrtfunctorK-AlgS3A7!A=[A;A]2K-c Algҹwhere6I[A;A]denotesthetrwo-sided6IidealofAgenerated&brytheelementsabba.8 In&factforeachhomomorphismofalgebrasfQ:URA!NBlwith;facommrutative;falgebraBthereisafactorizationthroughA=[A;A]sincef2vXanishesontheelemenrtsabba.HenceEifAUR=OUV(Xӹ)EisthefunctionalgebraofX^thenA=[A;A]istherepresenrtingalgebraforXc.y.]4Remark1.2.8.FVoranrycommutativealgebra(ofcoSecients)BthespacesXand_Xc harvethesameB-pSoints:!Xӹ(B)7=Xc.y(B). The_trwospacesdi eronlyfornoncommrutativegalgebrasofcoSecienrts.Inparticularforcommutative eldsBamaswualgebrasofcoSecienrtsthequantumplaneAߍ2j0]GqThasonlyB-pSointsonthetwoaxessincethefunctionalgebraKhx;yn9i=(xyq21 ʵyx;xyyx)PUR԰n9=Kܞ[x;y]=(xy)de nesonlyB-pSoinrts(b1;b2)whereatleastoneofthecoSecienrtsiszero.YProblemx1.2.4.nLetFS3 JbSethesymmetricgroupandA:=K[S3]Fbethegroupalgebra onS3.DescribSethepoinrtsofXӹ(B)=K-Alg f(A;B) asasubspaceofA22(B).WhatisthecommrutativepartXc.y(B)ofX{andwhatistheanealgebraofXc?TVo$understandhorwHopfalgebras tintothecontextofnoncommutativespaceswrehavetobSetterunderstandthetensorproductinK-Alg f.De nition6W1.2.9.LetAUR=OUV(Xӹ)andA20#=OUV(Y)bSethefunctionalgebrasofthenoncommrutativetWspacesX6*resp.Y.TwrotWB-pSointsp?:AY9!B]inXӹ(B)andp20 :A20!nBinY(B)arecalledcffommuting35points꨹ifwrehaveforallaUR2A꨹andalla20#2URA20d$Ap(a)p 09(a 0)UR=p 09(a 0)p(a);i.e.8iftheimagesofthetrwohomomorphismspandp20commrute.Remarknp1.2.10.iTVo/shorwthatthepSointspandp20hcommute,itissucienttocrheckuthattheimagesofthealgebrageneratorsp(x1);:::ʜ;p(xmĹ)commrutewiththeimagesofthealgebrageneratorsp209(y1);:::ʜ;p20(ynP)underthemrultiplication. y*Thismeansthatwrehave}bidb 0ڍj\=URb 0ڍjf bivfortheB-pSoinrts(b1;:::ʜ;bmĹ)UR2Xӹ(B)and(b20RA1;:::ʜ;b20RAnP)UR2Y(B).De nition1.2.11.Thefunctord$KP(X%?URY)(B):=f(p;p 09)2Xӹ(B)Y(B)jp;p 0commrute5e3giscalledtheorthoffgonal35product꨹ofthenoncommrutativespacesX{andY.7 165+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVRemark|1.2.12.TVogetherawithX#candY^&theorthogonalproSductX%?URYisagainafunctor, sincehomomorphismsf˚:B8,!B20oarecompatiblewiththemrultiplicationandthruspreservecommutingpSoints.8HenceX%?URY>isasubfunctorofXl{Y. Lemma1.2.13.IfaDX#andY]arffenoncommutativespaces,lthenXlp?Y]isanon-cffommutative35spacewithfunctionalgebraOUV(X%?URY)=O(Xӹ) O(Y).IfjX,andYgWhave nitelygenerffatedjfunctionalgebrffasthenthefunctionalgebraofX%?URY/is35also nitelygenerffated.<ۍProof.@_Let A:=OUV(Xӹ)andA20<:=OUV(Y).Let(p;p209)2(Xy?Y)(B)bSeapairof1commrutingpSoints. Thenthereisauniquehomomorphismofalgebrash*:A A20!nBsucrhthatthefollowingdiagramcommutesFӍ/ HA/ ƏA A20A{fdά- OH pׁ @ @ @Ü @✟>@✟>RHѵ"B:ʟǠ*FfeǠ? ̤h/ /  A20 ,{fdk ά{20H|p20ܟׁ ܟ ܟ ܟ S\>S\> 卹De neh(a a209)W:=p(a)p20(a20)andcrheckthenecessarypropSerties.ObservrethatforanqbSepoinrtssuchthat((pxH;py );pzʮ)>isapairofcommutingpSointsin((X?Y)UR?Z )(B).5Inparticular(pxH;py )isalsoapairofcommrutingpSoints.5ThuswehaveforallaUR2A:=OUV(Xӹ),a20#2A20:=OUV(Y),anda200q2URA200:=OUV(Z )pxH(a)py (a 09)pzʮ(a 00r)UR=(px;py )(a a 09)pzʮ(a 00r)UR=pz(a 00r)(pxH;py )(a a 09)UR=pz(a 00r)pxH(a)py (a 09)and1pxH(a)py (a 09)UR=py(a 09)pxH(a):卹IfwrechoSosea˹=1thenwegetpy (a209)pzʮ(a200r)=pz(a200r)py (a209).FVorarbitrarya;a20;a200 "wrethenget{XmUR:M?Mn!1M꨹and|e:E i"!xHMbSemorphismsinQSǹsucrhthatthediagramsNr*EMUR?M*5MN,32fd9lά-e捒mskڲMUR?M?MskEMUR?MڒL{fd`ά-mUR?1HE:Ǡ*FfexlǠ? 1UR?mH7:Ǡ*FfejlǠ?9mrandF53ksd{E i?URMP԰n9=MP԰n9=M?E3knEMUR?ML{fd!`ά-iQ^idf?H5Ǡ*FfehǠ?`w0I{?idH':Ǡ*Ffe'lǠ?`,r*MUR?M* M>32fdPά-l~rH3,1X.M٬ҁ H٬ׁ H٬܁ H٬ H٬ H٬ H٬ H٬ H fdžH fdžjL+commrute.8Then(M;m;e)iscalledaquantum35monoid.Prop`ositionT1.3.2.Lffet~MbeanoncommutativespacewithfunctionalgebraHV.Then35H isabialgebrffaifandonlyifMisaquantummonoid.W卍Proof.@_SincethefunctorsM ?M,M?Es׹andEt ?MarerepresenrtedbyH tH)resp..H KP԰=H)resp..K HP&԰ =MHtheH`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.X(2)+Let#AbSeaK-algebra.SAK-algebraM@(A)togetherwithahomomorphism+ofalgebras:[A(!8CM@(A) Aiscalledanalgebrffa(coactinguniversally+on=A(orsimplyauniversalalgebrffafor=A)ifforevreryK-algebraBCand+evreryhomomorphismofK-algebrasf:A&!Be :_Athereexistsaunique+homomorphism#)ofalgebrasg#:M@(A) !B/sucrh#)thatthefollowingdiagram+commrutesD H@AHM@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.AobBytheunivrersalpropSertiestheuniversalalgebraM@(A)forAandtheuniversalquanrtumspaceM(Xӹ)forX{areuniqueuptoisomorphism.'Prop`osition1.3.8.(1)S'LffetՉAbeaK-algebrawithuniversalalgebraM@(A)+and/:'A!M@(A)e4 A.YThenM@(A)isabialgebrffaandAisanM(A)-+cffomodule35algebrabys2.(2)+If{Bisabialgebrffaandiff:A!ҕB9/ )A{de nesthestructureofaB-+cffomodule algebraonAthenthereisauniquehomomorphismgË:URM@(A)!B+of35bialgebrffassuchthatthefollowingdiagramcommutesNYH2PAHKM@(A) Aw{fdPЍά-H`f$ׁ @$ @$ @$ @>@>RH(BE AҟǠ*FfeQǠ?Wg 1AN%ThecorrespSondingstatemenrtforquantumspacesandquantummonoidsisthefollorwing.f7 245+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVProp`osition1.3.9.(1)S'Lffet}XzPbeaquantumspacewithuniversalquantum+spffacezbM(Xӹ)and!:M(X)?A?!A.;ThenzbM(X)isaquantummonoid+and35XisanM(Xӹ)-spfface35by.(2)+IfY}isanotherquantummonoidandiffQ:URYQ?X%}!XBde nesthestructurffe+of3haY-spfface3honX;thentherffeisauniquemorphismofquantummonoidsgË:URY+.I!=?M(Xӹ)35suchthatthefollowingdiagrffamcommutesPW$ M(Xӹ)UR?XX:Ԟ32fdά-W>H`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.X qProof.@_WVegivretheproSofforthealgebraversionofthepropSosition.v4ConsiderthefollorwingcommutativediagramF؍lgM@(A) A[fM@(A) M(A) A32fd9Nά-c!1M(A)pR hAgM@(A) A :2fdfά-mxટǠ@feܟǠ?J~Q˪Ǡ@feܟǠ?%\ 1X.A!Dwhere themorphismofalgebrasisde nedbrytheuniversalpropSertyofM@(A)withrespSecttothealgebramorphism(1M"(A) %Is2).U6FVurthermorethereisauniquemorphismofalgebrasUR:M@(A)n!1K꨹sucrhthatSdčHAHM@(A) Al{fdPЍά--,H31X.Auׁ @u @u @u @׼>@׼>RH"%APUR԰n9=K AJǠ*Ffe|Ǡ?3 1X.Adcommrutes.Thecoalgebraaxiomsarisefromthefollorwingcommutativediagramsu؍UA'_M@(A) AbCDA2fdPά-m;Z1Π@feZeΠ?Q_BΠ@feBԟΠ?%GT 1X.A?_M@(A) AE^M@(A) M(A) AxT:2fd@ά-}C21M(A)pR YBǠ@feY8tǠ?%< 1X.A\"Ǡ@fe\TǠ?Sap1M(A)pR @\bǠ@fe@Ǡ? 1M(A)pR 1X.ACBǠ@feDtǠ?H1M(A)pR  1X.A'M@(A) M(A) AzM@(A) M(A) M(A) AL32fdg0ά-vC1M(A)pR 1M(A) Z7 4p3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES+~t25Vando`g|AM@(A) As”A2fdpά-mzk2Π@fekdΠ?QpΠ@feKΠ?%. 1X.A3SrǠM@fe3Ǡ?89$1M(A)pR 1X.APM@(A) A+M@(A) M(A) Al:2fd9Nά-}C1M(A)pR x 1M(A)pR 1X.AܔP P?_P攴P P?^PPʍPԍ?]P&D P&D q@M@(A) APUR԰n9=M(A) K AǠ@feKǠ?x 1M(A)pR  1X.AtTando`rAM@(A) A~tA2fdfά-mpǠM@feq$Ǡ?ӍbB1X.A|2Π@fe|dΠ?QΠ@fe$Π?% 1X.AyDoM@(A) A.M@(A) M(A) A$:2fd-@ά-}Cl 1M(A)pR |2Ǡ@fe|dǠ?kS 1X.AǠ@fe$Ǡ?x  1M(A)pR 1X.ArA,M@(A) APUR԰n9=K M(A) A:~t32fd5氍ά-mtTInnfactthesediagramsimplybrytheuniquenessoftheinducedhomomorphismsofalgebras*b( 1M"(A)~L)=(1M"(A)TW  ),:Q(1M"(A) )=1M"(A)and*b (1M"(A)~L)=1M"(A)~L.8FinallyAisanM@(A)-comoSdulealgebrabrythede nitionofand.NorwXassumethatastructureofaB-comoSdulealgebraonAisgivenbyabialgebraBandf:VApq!YBFO IA.;Thenthereisauniquehomomorphismofalgebrasg:M@(A)!nBsucrhthatthediagramN{RH@AHM@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.A!Ѝcommrutes.8ThenthefollowingdiagramMѭTAѭM@(A) A`bDfd ά-h-nRѭM@(A) AѭAM@(A) M(A) ATTTԟ̇fd9Nά-ȣ*y 1X.Aԟ:2fd9Nά-c1M(A)pR έ*@oafdؤݐʬQnؤ;tQxؤQؤQlQlsέbn`@fen`? gI{ 1X.Aέ4n`@fehn`? gI{ g 1X.A@ZxBE A@ZJQBE B AM4fdXPά-(X.BX 1X.A432fdXPά- =1X.BX f implies((gg R.gn9) 1A)kT="(g g 1A)( 1A)kT="(gg g 1A)(1M"(A)z s2)kT=(gP K(g 1A)s2)S=i(1B 0 (g 1A)s2)(g 1A)S=i(1B 0 fG)f(h=(B 0 1A)f(h=(B 0 7 265+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YV1A)(g 1A)Ȅ=UR(BN>g 1A)]ڹhence(g gn9)UR=BN>g.8FVurthermorethediagramp1w[A,M@(A) ADA2fdOά-mpԠ48BE A@f3ԟF`H3ԟF`H3ԟF`H3ԟF`H3ԟF`H3ԟF`HDTHDTjΠ@feΠ?WMżgI{ 1X.AbǠM@fe1Ǡ?Ӎ  1X.A"/APUR԰n9=K AӍ(1X.AtF`@tF`@tF`@tF`@tF`@tF`@tF`@|`@|`RǠ@feǠ?kSR_X.BX 1X.AWimpliesBN>gË=UR.8ThrusgXisahomomorphismofbialgebras.|SinceunivrersalalgebrasforalgebrasAtendtobSecomeverybigtheydonotexistingeneral.pButatheoremofTVamrbara'ssaysthattheyexistfor nitedimensionalalgebras(orvera eldK).?De nitionZ(1.3.10.ҼIfXXI+isaquanrtumspacewith nitedimensionalfunctionalgebrathenwrecallX{a nite35quantumspfface.ThenfollorwingtheoremisthequantumspaceversionandequivXalenttoatheoremofTVamrbara.Theorem1.3.11.Lffet'NX!bea nitequantumspace.BThenthereexistsa(uni-versal)quantumspffaceM(Xӹ)withmorphismofquantumspffaces:M(X)?Xfk!X.ThealgebravrersionofthistheoremisTheoremi1.3.12.ْ(TVamrbara)LffetAbea nitedimensionalK-algebra. NThentherffeexistsa(universal)K-algebraM@(A)withhomomorphismofalgebras:Afk!M@(A) A.Proof.@_WVeTaregoingtoconstructtheK-algebraM@(A)quiteexplicitly.oFirstwreobservrevthatA2V=URHom۟K"(A;K)isacoalgebra(cf./[AdvXancedAlgebra^Q,]Problem2.25)withthestructuralmorphismUR:A2V .!5(An A)2PV԰.==A2Vr A2.5Denotethedualbasisbry"P*In U_Ii=1!#aib K %a2i=2A% A2.VONow"letTƹ(A% A2)"bSethetensoralgebraofthevectorspaceA A2.8Considerelemenrtsofthetensoralgebra(ԫʍMxy =S2URA A2;Mx y ()UR2A A A2j A2PV԰.==A A2 A A2;M1 =S2URA A2;M(1)UR2K:ThefollorwingelementsW(1)#xy x y ()and(2)01 (1)W7 4p3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES+~t27Vgenerateatrwo-sidedidealIFURTƹ(A A2).8Norwwede neYM@(A)UR:=Tƹ(A A )=Iand\thecoSoperation\:NA3a0!H)P*"n U_"i=114,(a@ ùa2i ) ai|(2NTƹ(A A2)=I A.Thisisawrell-de nedlinearmap.TVoshorwthatthismapisahomomorphismofalgebraswe rstdescribSethemul-tiplication(IofAbryaidaj =qPk@3 2kRAijJak#.ThenthecomultiplicationofA2 Misgivenbyn(%a2k N)=Pliji 2kRAijpgJa2i .ȹ Ea2jsince((%a2k);al+  EamĹ)=(%a2k;al!amĹ)=Plr 2rylKm (%a2k;arb)=E 2kylKm1=lPRijb4 2kRAijJ(%a2i ;al!ȹ)(%a2j ;amĹ)l=(P ij 2kRAijpga2i[ La2j;al amĹ).eNorw%)write1l=PP O2kak#.Thenwreget(%a2i ۹)R = O2i since(%a2i)=(%a2i;1)=Pjb O2jY(%a2i;ajf )= O2i). |Sowrehaves2(a)(b)UR=(P* n U_ i=1(a Ga2i 2) aidڹ)(P* n U_ jv=13(b Ga2j 3Ϲ) ajf )UR=PijJ(a b Ga2ic Ga2j) aidaj\=EP ijvk! 2kRAijJ(a1 b Wca2i Wca2j ) akx=URPk#(a b (%a2k N)) ak=URPk#(ab Wca2k s) ak=URs2(ab).FVurthermorelwrehaves2(1)UR=Pidٹ(1p a2i 9K) ai,=URPiZda2i(1) ai=UR1 PTixa2iHй(1)ai=UR1 1.Hence]ڹisahomomorphismofalgebras.NorwrwehavetoshowthatthereisauniquegCforeachfG.yFirstofallfQ:URAn!1Bw tqAinducesDuniquelydeterminedlinearmapsfi,:URAn!1B@JwithDfG(a)=Pidfidڹ(a) ai sincetheai\formabasis._Sincef?isahomomorphismofalgebraswregetfromPSkfk#(a) ak j=G,fG(ab)=f(a)f(b)=Pٟij<(fidڹ(a)2 ai)(fjf (b) aj)G,=Pٟijof algebrasg:"Tƹ(Aɜ A2)1 eUR=1;>1 =S=UR1 2V=0;>x22j eUR=(x e)(x e);>x22j =S=UR(x )(x e)+(x e)(x );>x22j 2V=UR(x 2)(x e)+(x )(x )+(x e)(x 2);? ʍ-C0UR=x23j e=(x22 e)(x e);-C0UR=x23j =S=(x22 )(x e)+(x22 e)(x );-C0UR=x23j 2V=(x22 2)(x e)+(x22 )(x )+(x22j e)(x 2)ާWVeusetheabbreviationfu;vn9gUR:=u22v+uvu+vu22andharve!9ʍa23V=UR0;fa;bgUR=0;fa;cg+fb;agUR=0:Thecondition(1 s2)Ȅ=UR( 1)]ڹimplies69ʍx(a)UR=a 1+b a+c a22;xι(b)UR=b b+c (ba+ab);xι(c)UR=b c+c b22j+c (ca+ac);x(a)UR=0;x(b)UR=1;x(c)UR=0:5/6.S5Aohasthebasis1;x;yn9;xy.S5TheodualbasisofA2sisdenotedbrye;s;;S.S5Thediagonalis#>(ʍuJ(e)UR=e e;uJ(s)UR=? e+e ;uJ(n9)UR= e+e ;uJ(S)UR=6 e+e +? +qn9 s:%ZThrusethecoSendomorphismbialgebrahasthealgebrageneratorsa C fwitheaW2f1;x;yn9;xygܹandn2 mfe;s;;Sg. |Thegeneratorsoftherelations(ofI)aregivrenbrytheequations1.1and1.2. ATheyimplythat1 8 e퍹istheunitelement,nFthat1 Ź=UR1 Ë=1 =0andthat($ʍab eUR=(a e)(b e);ab Ź=UR(a s)(b e)+(a 1)(b );ab Ë=UR(a n9)(b e)+(a 1)(b );ab =UR(a S)(b e)+(a 1)(b )+(a s)(b n9)+q(a )(b s):FVurthermoreforabwrehavetotakeintoaccounttherelationsinA.WVede ne9ʍZY{aUR:=x e;3"bUR:=x s;cUR:=x n9;dUR:=x S;[eUR:=y e;fQ:=URy s;ݕgË:=URy n9;&hUR:=x S;7 305+?1.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YVandv]gets2(x)UR=a 1+b x+c y+W+d xy䖹andv](yn9)UR=e 1+f x+g+W y+h xyn9.HenceBisgeneratedbrya;:::ʜ;hasanalgebra.8Therelationsare<;ʍY'a22V=URe22=0;Y'ab+baUR=ac+caUR=ef+fGe=eg+gn9e=0;Y'ad+da+bc+qn9cbUR=eh+he+fGg+qgfQ=UR0;Y'aeUR=qn9ea;Y'af+beUR=qn9(fGa+eb);Y'ag+ceUR=qn9(ga+ec);Y'ahqn9ha+deqed+bgq22.=gb+qcfqfGcUR=0:Thediagonalis';ʍE(a)UR=a 1+b a+c e+d ae;E(b)UR=b b+c f+d (af+be);E(c)UR=b c+c g+d (ag+ce);E(d)UR=b d+c h+d (ah+de+bg+qn921 ʵcfG)etc.ՠ7 VRCHAPTER2sHopfffAlgebras,Algebraic,Fformal,andQuantumGroups"Intro`ductionIn1the rstcrhapterwehaveencounteredquantummonoidsandstudiedtheirroleasmonoidsopSeratingonquanrtumspaces.The\elements"ofquantummonoidsopSeratingonquanrtumspacesshouldbeunderstoodasendomorphismsofthequanrtumspaces.9Indtheconstructionofthemrultiplicationforuniversalquantummonoidsofquanrtum;spaceswehaveseenthatthismultiplicationisessentiallythe\compSosition"ofendomorphisms.WVe3are,~horwever,primarily3interestedinautomorphismsandweknowthatauto-morphismsshouldformagroupundercompSosition.EThiscrhapterisdevotedto ndinggroupstructuresonquanrtummonoids,i.e.8tode neandstudyquantumgroups.Thisiseasyinthecommrutativesituation,?Hi.e. iftherepresenrtingalgebraofaquanrtummonoidiscommutative. -Thenwecande neamorphismthatsendselemenrtsrofthequantumgrouptotheirinverses.Thiswillleadustothenotionofanealgebraicgroups.Inthenoncommrutativesituation,5however,itwillturnoutthatsucrhaninversionmorphismX(ofquanrtumspaces)doSesnotexist.OItwillhavetobSereplacedbyamorecomplicatedconstruction.ThrusquantumgroupswillnotbSegroupsinthesenseofcategoryFtheoryV.StillwrewillbSeabletoperformoneofthemostimportanrtandmostbasicconstructionsingrouptheoryV,theformationofthegroupofinrvertibleelementsofcamonoid. pInthecaseofaquanrtummonoidactinguniversallyonaquantumspace& thiswillleadtothegoSod& de nitionofaquanrtumautomorphismgroupofthequanrtumspace.InordertoharvetheappropriatetoSolsforinrtroducingquanrtumgroupswe rstinrtroSduce%HopfalgebraswhichwillbSetherepresentingalgebrasofquantumgroups.FVurthermorewreneedthenotionofamonoidandofagroupinacategory.Wewillsee,horwever,thatޘquantumgroupsareingeneralnotgroupsinthecategoryofquantumspaces.WVe rststudythesimplecasesofanealgebraicgroupsandofformalgroups.TheywillharveHopfalgebrasasrepresenrtingobjectsandwillindeedbSegroupsinreasonableAcategories.Thenwrecometoquantumgroups,W(andconstructquantumautomorphismgroupsofquanrtumspaces.Arttheendofthechapteryoushould +〹knorwwhataHopfalgebrais,/^31 ߠ7 32242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSV +〹knorwwhatagroupinacategoryis, +〹knorwpthede nitionandexamplesofanealgebraicgroupsandformalgroups, +〹knorwɔthede nitionandexamplesofquantumgroupsandbSeabletoconstruct+quanrtumautomorphismgroupsforsmallquantumspaces, +〹understand$UwhryaHopfalgebraisareasonablerepresentingalgebrafora+quanrtumgroup.Ӎ1.׈HopfAlgebrasThe=di erencebSetrween=amonoidandagroupliesintheexistenceofanadditionalmapSX::G3gs7!gn921o2GforagroupGthatallorwsforminginverses.Thismapsatis estheequationS׹(gn9)gË=UR1orinadiagrammaticform@5{^G:f1g:2fd&*Pά-č" ?G❔:2fd&*Pά-Z1@Ǡ@fesǠ??<ٵGGH5GGd32fdME ά-ׁ̃Sr}idǠ@fe4?`6JmÎultWVeCwranttocarrythispropSertyovertobialgebrasB0Iinsteadofmonoids.iAn\inversemap"shallbSeamorphismS:hB!BwithasimilarpropertryV.YThiswillbecalledaHopfalgebra.LDe nition2.1.1.~Ableft5HopfalgebrffaHisabialgebraHtogetherwithaleftantipffodeS:_HM f!OHV,jJi.e.јaK-moSdulehomomorphismSgsucrhthatthefollowingdiagramcommrutes::j)"QH)"$K̟:2fd)@ά-čׄ" 1HU<:2fd.bά-(70Ǡ@feǠ??<P@3H HH H32fdJsά-ׁ'Sr} idSjǠ@fe?`6?<9rhSymmetrically{wrede nearightHopfalgebrffa{HV.A{Hopfalgebra{isaleftandrighrtHopfalgebra.8ThemapSiscalleda(left,righrt,two-sided)antipffode.LUsing{theSwreedlernotation(2.20)thecommutativediagramabSovecanalsobSeexpressedbrytheequation(XS׹(a(1) \|)a(2)ι=URn9"(a)p[for6alla 2HV. Observre6thatwedonotrequirethatS߹: H^ !Hoisanalgebrahomomorphism.}&Problem2.1.8.(1)Let8,H%bSeabialgebraandS2DHomR(HF:;HV).!nThenSis+ananrtipSodeforH(andHisaHopfalgebra)i S}isatrwosidedinversefor+idgSinthealgebra(Homy(HF:;HV);;n9")gS(see2.21).InparticularS*isuniquely+determined.(2)+LetHGbSeaHopfalgebra.ThenSȹisananrtihomomorphismofalgebrasand+coalgebras{i.e.,|SxR\inrvertstheorderofthemultiplicationandthecomultipli-+cation".!M7 1.pHOPF!ALGEBRAS33V(3)+Let(FHandKbSeHopfalgebrasandletf3:4H!KKbSeahomomorphismof+bialgebras.8ThenfGSH n=URSK;f,i.e.8f2iscompatiblewiththeanrtipSode.JDe nitionh"2.1.2.g+BecauseofProblem(22)evreryhomomorphismofbialgebrasbSetrweenq[Hopfalgebrasiscompatiblewiththeanrtipodes.Sowrede neahomomor-phismofHopfalgebrffasItoK&bSeahomomorphismofbialgebras.ZZThecategoryofHopfalgebraswillbSedenotedbryK-Hopfg.Prop`osition2.1.3.-Lffet]HrbeabialgebrawithanalgebrageneratingsetX.\LetS:7!H$w!HV2op}bffe%analgebrahomomorphismsuchthatPWS׹(x(1) \|)x(2)=7!n9"(x)forallxUR2X.fiThen35S isaleftantipffode35ofHV."OProof.@_Assume;a;b2HܑsucrhthatPS׹(a(1) \|)a(2)mV=n9"(a)andPS׹(b(1) \|)b(2)mV=n9"(b).8Then%ʍ-&0P;S׹((ab)(1) \|)(ab)(2)=URPS׹(a(1) \|b(1))a(2)b(2)ι=URPS׹(b(1))S(a(1))a(2)b(2)=URPS׹(b(1) \|)n9"(a)b(2)ι=UR"(a)"(b)="(ab):$4SinceeevreryelementofHSGisa nitesumof niteproSductsofelementsinX,|forwhichtheequalitryholds,thisequalityextendstoallofHbyinduction.XY1"OExample2.1.4.7(1)XFLetVbSeavrectorspaceandTƹ(Vp)thetensoralgebraoverVp."WVemharveseeninProblem2.2thatTƹ(Vp)isabialgebraandthatVݹgeneratesT(Vp)as analgebra.$VDe neS):URV M!`Tƹ(Vp)2op &bryS׹(vn9):=vBforallvË2Vp.$VBytheunivrersalpropSertryofthetensoralgebrathismapextendstoanalgebrahomomorphismSӈ:Tƹ(Vp)URn!1T(V)2op.SinceHe(vn9)UR=vt _;1+1 vwrehavePS׹(v(1) \|)v(2)ι=URr(S _;1)(vn9)=vo+&6vo=60=n9"(v)!forallv26Vp,̀henceTƹ(V)isaHopfalgebrabrytheprecedingpropSosition.(2)ZLetVbSeavrectorspaceandS׹(Vp)thesymmetricalgebraoverV(thatiscommrutative).WVeDhaveseeninProblem2.3thatS׹(Vp)isabialgebraandthatVgeneratesKS׹(Vp)asanalgebra.[De neS:i!V!?S(Vp)bryS(vn9)i!:=vdforKallvZ2i!Vp.[SextendsYtoanalgebrahomomorphismS):URS׹(Vp)n!1S(Vp).)SinceY(vn9)=v@ J1+1 vwrewhaveP"S׹(v(1) \|)v(2);=Er(S  ӹ1)(vn9)=vy +v=0=n9"(v)wforallv2EVp,8henceS׹(Vp)isaHopfalgebrabrytheprecedingpropSosition.Exampleo2.1.5.E(GroupAlgebras)ܦFVoreacrhalgebraAwecanformthegrffoupofunitsgU@(A)* :=fa2Aj9a212AgwiththemrultiplicationofAascompSositionofthegroup.) ThenUisacorvXariantfunctorUŹ:nK-Algj!!]G.r.Thisfunctorleadstothefollorwinguniversalproblem.Let4GbSeagroup.;AnalgebraKGtogetherwithagrouphomomorphismyQ:G!"U@(KG)TKiscalleda(the)grffoupNalgebraofG,nifTKforevreryalgebraAandforeverygrouphomomorphismfQ:URGn!1U@(A)thereexistsauniquehomomorphismofalgebras"'7 34242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVgË:URKGn!1A꨹sucrhthatthefollowingdiagramcommutesMHGHU@(KG)\{fd@ά- oH`Ç%flׁ @ƾl @оl @ھl @>@>RHiU@(A): Ǡ*Ffe=LǠ? guThegroupalgebraKGis(ifitexists)uniqueuptoisomorphism.ItisgeneratedaslanalgebrabrytheimageofG.vThemap2:GLy!iU@(KG)KGlڹisinjectiveandtheimageofGinKGisabasis.TheDAgroupalgebracanbSeconstructedasthefreevrectorspaceKGwithbasisGandxthealgebrastructureofKGisgivrenbyKG KG3g+" h7!gn9h2KGxandtheunitË:URK3 h7! e2KG.ThegroupalgebraKGisaHopfalgebra.ThecomrultiplicationisgivenbythediagramFtj⍒0Gj⍒=SKGd{fd% ά-TDH`f tׁ @ t @ t @ t @Q>@Q>RH"њKG KGW"Ǡ*FfeTǠ?`<߂withfG(gn9):=g 6g&whicrhde nesagrouphomomorphismf:GA!U@(KG6 KG).Thecounitisgivrenbyj⍒x&Gj⍒ꄻKG4̟{fd% ά-қH`7fRܟׁ @Rܟ @Rܟ @Rܟ @\>@\>RH""'K󞊟Ǡ*FfeѼǠ?]D<"gBwherefG(gn9)\=1forallg 2G.|OneshorwseasilybyusingtheuniversalpropSertyV,thatCiscoassoSciativreandhascounit".C~De neanalgebrahomomorphismSn:KG!n߹(KG)2op ŹbryG2 j⍒Gj⍒[KGl{fd% ά-%LH`f|ׁ @| @| @| @">@">RH (KG)2op(*Ǡ*Ffe[\Ǡ?` S㍹withfG(gn9)rF:=g21\ƹwhicrhisagrouphomomorphismfE:rFG!U@((KG)2op)./ThenoneshorwswithPropSosition1.3thatKGisaHopfalgebra.!TheӌexampleKGofaHopfalgebragivresrisetothede nitionofparticularel-emenrts4inarbitraryHopfalgebras,thatsharecertainpropSertieswithelementsofagroup.8WVewilluseandstudytheseelemenrtslateronininchapter4.# 7 1.pHOPF!ALGEBRAS35VDe nitiong2.1.6.LetբHbSeaHopfalgebra.1AnelemenrtgË2URHF:;g6=0բiscalledagrffouplike35element꨹ifd?(gn9)UR=g g:5Observrethat"(gn9)4=1foreachgrouplikeelementginaHopfalgebraHV.2InfactwreohavegË=URr(" 1)(gn9)="(g)gË6=0ohence"(gn9)=1.4xIfthebaseringisnota eldthenoneaddsthispropSertrytothede nitionofagrouplikeelement.Problem2.1.9.(1)Let[KbSea eld.ShorwthatanelementxUR2KG[satis es+(x)UR=x x꨹and"(x)UR=1ifandonlyifxUR=gË2G.(2)+ShorwthatthegrouplikeelementsofaHopfalgebraformagroupunder+mrultiplicationoftheHopfalgebra.63Example&2.1.7.)ع(UniversalEnvelopingAlgebras)@T>RH#ӎA2LG:͂Ǡ*FfeǠ? 4g<TheGunivrersalenvelopingalgebraU@(g)is(ifitexists)uniqueuptoisomorphism.Itisgeneratedasanalgebrabrytheimageofg.TheunivrersalenvelopingalgebracanbSeconstructedasU@(g)=Tƹ(g)=(x[ y4ya Kx[x;yn9])UXwhereTƹ(g) =KKgg g:::isUXthetensoralgebra.xThemap :g!nU@(g)2L 2isinjectivre.$,q7 36242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVTheunivrersalenvelopingalgebraU@(g)isaHopfalgebra.8ThecomultiplicationisgivrenbythediagramER荍H egHےU@(g){fd$ά-WH`-fItׁ @It @It @It @Տ>@Տ>RHRU@(g) U(g)"Ǡ*FfeTǠ?`zwith*fG(x)v}:=x 1+1 xwhicrhde nesaLiealgebrahomomorphismf|:v}g!(U@(g) U(g))2LGع.8ThecounitisgivrenbyJHٝgHa5U@(g)T{fd$ά-%LH`fׁ @ @ @ @^,>@^,>RH""KcZǠ*Ffe񖌟Ǡ?]DI "KwithMvfG(x)=0forallx2g.aLOneshorwseasilybyusingtheuniversalpropSertyV,f*that{}iscoassoSciativreandhascounit". `De neanalgebrahomomorphismSm:U@(g)!n߹(U@(g))2op ŹbryEHT5gHU@(g)T{fd$ά-ɟH`vfDׁ @ĒD @ΒD @ؒD @ğ>@ğ>RHz(U@(g))2opǠ*Ffe$Ǡ?`äSˍwithfG(x)UR:=xwhicrhisaLiealgebrahomomorphismfQ:gn!1(U@(g)2op)2LGع.ThenoneshorwswithPropSosition1.3thatU@(g)isaHopfalgebra.(Observre,[that79themeaningofUxinthisexampleandthepreviousexample(groupofNunits,funivrersalenvelopingalgebra)istotallydi erent,finthe rstcaseUcanbSeappliedtoanalgebraandgivresagroup,inthesecondcaseUعcanbSeappliedtoaLiealgebraandgivresanalgebra.)čThe precedingexampleofaHopfalgebragivresrisetothede nitionofparticularelemenrts~inarbitraryHopfalgebras,thatsharecertainpropSertieswithelementsofaLiealgebra.8WVewilluseandstudytheseelemenrtslateroninchapter4.De nitiont>2.1.8.TLet HbbSeaHopfalgebra. S Anelemenrtx:h2His calledaprimitive35element꨹ifI$`(x)UR=x 1+1 x:VLet_$g2HLzbSeagrouplikreelement.VAnelementx2HLzis_$calledaskewGprimitiveorgn9-primitive35element꨹if(x)UR=x 1+g x:VProblem2.1.10.ShorwpthatthesetofprimitiveelementsPƹ(HV)UR=fx2Hj(x)=x 1+1 xg꨹ofaHopfalgebraHisaLiesubalgebraofHV2L5..%:7 1.pHOPF!ALGEBRAS37VProp`ositionx2.1.9.Lffet->HbeaHopfalgebrawithantipodeS.dlThefollowingareeffquivalent:(1)35Sן22-=id.(2)35PS׹(a(2) \|)a(1)ι=URn9"(a)35foralla2HV.(3)35Pa(2) \|S׹(a(1))UR=n9"(a)35forallaUR2HV.ލProof.@_LetSן22-=id.8ThenʍF/PTS׹(a(2) \|)a(1)㷹=URSן22r۹(PS׹(a(2) \|)a(1))=S׹(PS(a(1) \|)S22r۹(a(2)))=URS׹(PS(a(1) \|)a(2))=S׹(n9"(a))="(a)bryusingProblem(21).Conrverselyassumethat(2)holds.8Thenʍ_oKS]Sן22r۹(a)a=URPS׹(a(1) \|S22r۹(a(2))UR=S׹(PS(a(2))a(1)a=URS׹(n9"(a))="(a):nThrusSן22){andid7areinversesofSiwintheconvolutionalgebraHom0)(HF:;HV),:8Cisnotnecessarilyfaithfully+ at[Scrhauenburg])(2)+CallZ-acoalgebraC6admissibleifitadmitsanalgebrastructuremakingita+Hopfalgebra.!TheconjecturestatesthatC}isadmissibleifandonlyifevrery+ nitesubsetofCFliesina nite-dimensionaladmissiblesubScoalgebra.=(Remarks./((a)CѬBothimplicationsseemhard./' (b)CѬThereݬisacorrespSondingconjecturewhere\Hopfalgebra"isreplacedCѬbry\bialgebra".0uD(c)CѬThereisadualconjectureforloScally nitealgebras.)=(Noresultsknorwn.)(3)+A qHopf algebraofcrharacteristic0hasnonon-zerocentralnilpSotentelements.=(Firstcounrterexamplegivenby[Schmidt-Samoa].ΙIfH'isunimoSdularand+not!asemisimple,Ie.g.aDrinfel'ddoubleofanotsemisimple nitedimensional+HopfValgebra,!thentheinrtegralsatis esUR6=0,22V="()=0VsinceDS(HV)+isnotsemisimple,andaUm="(a)="(a)=a긹sinceDS(HV)isunimodular+[Sommerh auser].)&G7 38242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSV(4)+(Nicrhols).LetxbSeanelementinaHopfalgebraHwithantipSodeS׹.Assume+thatforanryainHwehave󍍍5uXbidxS׹(ci)UR="(a)x+〹whereaUR=Pbi cidڹ.8Conjecture:xisinthecenrterofHV.=(axUR=Pa(1) \|x"(a(2))UR=Pa(1) \|xS׹(a(2))a(3))UR=P"(a(1) \|)xa(2)ι=xa:)=In'theremainingsixconjecturesHr}isa nite-dimensionalHopfalgebra+orveranalgebraicallyclosed eld.(5)+If$Hzissemisimpleoneitherside(i.e.1_eitherHorthedualHV2 ~issemisimple+asanalgebra)thesquareoftheanrtipSodeistheidentityV.=(YVesifcrhar(K)=0[Larson-Radford],yesifchar(K)islarge[Sommer-+h auser])(6)+TheAsizeofthematricesoSccurringinanryfullmatrixconstituentofH.hdivides+thedimensionofHV.=(YVesisHopfalgebraisde nedorverZ[Larson];_ingeneralnotknorwn;+wrorkby[Montgomery-WitherspSoon],[Zhru],[Gelaki])(7)+If\HissemisimpleonbSothsidesthecrharacteristicdoesnotdividethedi-+mension.=(Larson-Radford)(8)+IfthedimensionofHisprimethenHiscommrutativeandcoScommrutative.=(YVesincrharacteristic0[Zhu:81994])=Remark.8Kac,Larson,andSwreedlerhavepartialresultson5{8.=(WVasalsoprorvedby[Kac])=In thetrwo nalconjecturesassumethatthecrharacteristicdoSesnotdivide+thedimensionofHV.(9)+ThedimensionoftheradicalisthesameonbSothsides.=(Counrterexampleuby[Nichols];چcounterexampleinFVrobSenius-Lusztigker-+nelofUq(slC(2))[Scrhneider])(10)+Thereareonlya nitenrumbSer(uptoisomorphism)ofHopfalgebrasofa+givrendimension.=(YVesforsemisimple,cosemisimpleHopfalgebras:8Stefan1997)=(Counrterexamples:8[Andruskiewitsch,Schneider],[Beattie,others]1997)&mgg2.xMonoidsandGroupsinaCategoryBeforewreuseHopfalgebrastodescribSequantumgroupsandsomeofthebSetterknorwn#groups,ȁsuchasanealgebraicgroupsandformalgroups,ȁweintroSducetheconcept/gofageneralgroup(andofamonoid)inanarbitrarycategoryV.UsuallythisconceptJisde nedwithrespSecttoacategoricalproductinthegivrencategoryV.ButinʊsomecategoriesthereareingeneralnoproSducts..,Still,onecande netheconceptofagroupinavrerysimplefashion.&IWVewillstartwiththatde nitionandthenshow'U7 q\2.pMONOIDS!ANDGR9OUPSINACA:TEGORYg`39Vthatitcoincideswiththeusualnotionofagroupinacategoryincasethatcategoryhas niteproSducts.5>De nition2.2.1.(۹Let:C߹bSeanarbitrarycategoryV.(LetG݁2CbSe:anobject.(WVeuse)thenotationG(X)t:=Mor}C <(XJg;G))forallXf2tC5,yG(fG):=Mor}C(f;G))forallmorphismsfQ:URXF``!Y8ginC5,andfG(X):=MorOC(XJg;f)forallmorphismsfQ:Gn!1G209.G!͹togetherwithanaturaltransformationmf:G(-)~G(-)fx!gG(-)!iscalledagrffoup(monoid)inthecategoryC5,\ifEfthesetsG(X)togetherwiththemrultiplicationm(X)UR:G(X)G(X)URn!1G(X)aregroups(monoids)forallXF2URC5.Let"(G;m)and(G209;m20)"bSegroupsinC5.XOAmorphismf&:g'G!G20[in"CWiscalledahomomorphism35ofgrffoups꨹inC5,ifthediagramsN̒HQG(X)G(X)H&+G(X)ޮ|{fd$ 0ά-`ƪm(X)G209(X)G20(X)G209(X)32fdά-"&m-:0(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)commruteforallXF2URC5.Let?(G;m)and(G209;m20)?bSemonoidsinC5.A8morphismf:Gl!OG20xin?Ctiscalledahomomorphism35ofmonoidsinC5,ifthediagramsN̒HQG(X)G(X)H&+G(X)ޮ|{fd$ 0ά-`ƪm(X)G209(X)G20(X)G209(X)32fdά-"&m-:0(X)H Ǡ*Ffe<Ǡ?`f(X)f(X)H Ǡ*Ffe<Ǡ?`ef(X)andC8Hfg]DW~uׁ̟ ̟ ̟ ̟ ی>ی> Hnu-:07,ׁ A7, A7, A7, AZl>AZl>U2KG(X)nG209(X)̶32fd_`ά- f(X)IύcommruteforallXF2URC5.5>Problem2.2.11.(1)LetthesetZgbtogetherwiththemrultiplicationm:+Z?ZK!ZbSeamonoid.Shorwthattheunitelementeʭ2Zisuniquely+determined.=Let(Z5;m)bSeagroup.1ShorwthatalsotheinverseiUR:Z1K{!Zisuniquely+determined.=Shorw8^thatunitelementandinversesofgroupsarepreservedbymaps+thatarecompatiblewiththemrultiplication.(2)+Find1anexampleofmonoidsY.andZnϹandamapfQ:URY M!`ZwithfG(y1y2)UR=+fG(y1)f(y2)forally1;y2V2URYp,butfG(eYP)6=eZ8.(aߠ7 40242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSV(3)+LetG(G;m)bSeagroupinCP|andiX :[G(X)!$CG(X)GbSetheinrverseGforall+XF2URC5.8Shorwthatiisanaturaltransformation.=ShorwthattheYVonedaLemmaprovidesamorphismSЇ:G7;!TGƹsuch+thatiX r۹=URMorOC(XJg;S׹)UR=S(X)forallXF2URC5.=FVormrulateandprovepropSertiesofSofthetypSeS]idʞ=UR:::.ԍProp`osition2.2.2.@LffeteCbeacategorywith nite(categorical)products.A2nobjeffctoGinCcarriesthestructuremq:G(-35)'G(-)q|!HG(-)oofagrffoupinCifandonly=iftherffearemorphismsmgw:GGgw!5[G,?tugw:E!rG,and=SN:G!5[Gsuchthatthe35diagrffamsLLZCGGGZjGGl{fdQά-).m1O]GGbGt 32fd*Fά-čmH`\*Ǡ*Ffe`\Ǡ?͞I{1mHѪǠ*FfeܟǠ?]D\mZ|E^GPUR԰n9=GPUR԰n9=GEZLoGG>cl{fd ά-).;1u"GGXgfG̞32fd_ά-č!mHǠ*FfeǠ?͞ABu1H\֊Ǡ*Ffe] Ǡ?]Da,Ǡ?1mi?-H:Ǡ*FfelǠ?(mi?-ڍcommrutesifandonlyifMorѩC?h(-;m(m R1))UR=MorOC(-;m)(Mor5C(-;m) R1)UR=m̹-j(m̹-s1)_R=m̹-j(1m̹-)_R=MorOC(-;m)(1MorK Cʹ(-;m))_R=MorOC(-;m(1m))ifandonlyifm(m1)UR=m(1m)ifandonlyifthediagramLJZ#GGGZmMGG۠ܟ{fdQά-).ۙm1GGdGϦ|32fd*Fά-čZmH^Ǡ*Ffe̟Ǡ?͞~ 1mHǠ*FfeLǠ?]Dm)qN7 q\2.pMONOIDS!ANDGR9OUPSINACA:TEGORYg`41Vcommrutes.8InasimilarwayoneshowstheequivXalenceoftheotherdiagram(s).2Problem2.2.12.6Let!CkVbSeacategorywith niteproducts.ҳShorwthatamorphismfQ:URGn!1G20inCݹisahomomorphismofgroupsifandonlyifJIFZGGZIGI|{fd@ά-:mK̔G20xG20KG2032fdNά-.&m-:0HǠ*Ffe4̟Ǡ?`ffHڟǠ*Ffe Ǡ?`fcommrutes.L؍De nitionz2.2.3.A?cffogroup(comonoid)GinCoisagroup(monoid)inC52op R,i.e.NanwobjectGl2Ob C=ObC52opwɹtogetherwwithanaturaltransformationm(X)l:G(X)۲G(X)Oiw!eG(X)whereG(X)O=MorCmrop(K(XJg;G)=MorC (G;X),sucrhthat(G(X);m(X))isagroup(monoid)foreacrhXF2URC5.Remark-2.2.4.Let`CFbSeacategorywith nite(categorical)coproducts. Anobject0GinCecarriesthestructuremӹ:G(-)ɨG(-)^!_3G(-)0ofacogroupinCeifandonly%*iftherearemorphisms:Gs!]G}qG,3":Gs!I,3and%*Sk:Gs!G%*sucrhthatthediagramsLff7GqGq9GqGqG[d32fdά- [q1bZBGbZ}8GqGO\{fd*ά-̍a\HZǠ*FfeǠ?` 1qHGXڟǠ*FfeG Ǡ?`;0ٚۄ8GqGٚI+qGPUR԰n9=GPUR԰n9=GqI32fdqά- "q1bZQ6GbZVGqG ܟ{fd_Zά-̍ 0Hg Ǡ*FfegA,Ǡ?`k1q"HZǠ*FfeǠ?`+H8Jb1H+ʬQHtQ HQH+Q HrQ*HQ-ܟQ-ܟsMbZ4GbZտGI{fd K0ά-bZbZ&Gbܟ{fd K0ά-"gGqGYGqG32fd6@ά- бw1qSlбwSr}q1H*Ǡ*Ffe\Ǡ?`H*Ǡ*Ffe\ׁ 6` zr柍commrute7dwhererisdualtothemorphismde nedin[AdvXancedAlgebra^Q,]4.2.ThemrultiplicationsarerelatedbyX r۹=URMorOC(;X)UR=(X).Let!CoVbSeacategorywith nitecoproductsandletGandG20ZbecogroupsinC5.ThenElahomomorphismofgroupsf7й:G20 ו!hGisamorphismf:G \!/G20inElCsucrhthatthediagramGZVGZGG{fd@ά-̍ɆnK:G20KG20xG20532fdNά-.y+-:0H6Ǡ*FfeiǠ?`lffHzǠ*FfeǠ?` fLccommrutes.8Ananalogousresultholdsforcomonoids.*7 42242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVRemark42.2.5.Obrviously#hsimilarobservXationsandstatementscanbSemadeforotheralgebraicstructuresinacategoryC5.CSoonecaninrtroSducevectorspacesandcorvectorO spaces,&monoidsandcomonoids,ringsandcoringsinacategoryC5. fThestructures(b)UR=1 b꨹arealgebrahomomorphisms.+Ơ7 aK3.pAFFINE!ALGEBRAICGR9OUPS!O43VSothecategoryK-c AlgQ1has nitecoproSductsandalsoaninitialobjectK.A'more( generalpropSertryofthetensorproductofarbitraryalgebraswrasalreadyconsideredin1.2.13.Observrethatthefollowingdiagramcommutes>RԠnAԠȫ A A:2fd~`ά-͝;8q1ԠԠ`Ak̟:2fd~`쁠ά͝;xq2kS1X.A?`@?`@?`@Ɩ@ƖRkS~d1X.A?`?`?` gA؜*Ǡ@fe\Ǡ?bp݁r^whereristhemrultiplicationofthealgebraandbythediagramthecoSdiagonalofthecoproSduct..De nitionYg2.3.2.`Anane!algebrffaicgroupisagroupinthecategoryofcom-mrutativegeometricspaces.ByrthedualitrybSetweenthecategoriesofcommutativegeometricspacesandcom-mrutativealgebras, ananealgebraicgroupisrepresenrtedbyacogroupinthecate-goryofK-c AlgQ1ofcommrutativealgebras.FVoranarbitraryanealgebraicgroupHwregetbyCorollary2.2.7^UR=1j2V2K-c Alg f(HF:;H HV);V:"UR=e2K-c Alg f(HF:;K);jand4CjS)=(id ʤ) 12K-cAlg f(HF:;HV):+ThesemapsandCorollary2.2.7leadtoProp`osition22.3.3.O^LffetHK62]K-35c fAlg f.HhisarepresentingobjectforafunctorK-35c fAlg "F!wG.r ܷif35andonlyifH isaHopfalgebrffa.!Proof.@_BothestatemenrtsareequivXalenttotheexistenceofmorphismsinK-c Alg^gkUR:HB\3!H H N":HB\3!K S):HB\3!Hsucrhthatthefollowingdiagramscommute?ҍԠۣHԠ,H Hd:2fd?ά-ζ łǠ@feǠ??<0T 胀(coassoSciativitry)>Ǡ@fe>괟Ǡ?ԐC4 1H H AH H Hv$32fd&H`ά-0h1 B獍ԠLHԠ7ȓH Hpl:2fd٠ά-ζ Ҏ"H H"bK HPB԰[=QHPB԰[=H K,32fd<ά-nŬv" 1S,<胀(counit)nǠ@feǠ??<IJǠ@feI|Ǡ?ꬾNy1 "j1JܟܔPJܟ PJܟ?_PJܟ攴PJܟ PJܟ?^PJܟPJܟPJܟ?]P㜟 P㜟 q?tlt胀(coinrverse))"iH)"BK:2fd)@ά-č40 .OHsT:2fd.bά-(7NǠ@fe64Ǡ??<nX7KH H!H Hӳ32fdJsά-ׁESr} idaCidK S3qǠ@fe3?`6?<8W4r,Q7 44242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVI`EThis#PropSositionsarystwothings.FirstofalleachcommutativeHopfalgebraHde nesĜafunctorK-c Alg f(HF:;-)N:K-cAlg.Hb!"SetthatĜfactorsthroughthecategoryof+groupsorsimplyafunctorK-c Alg f(HF:;-)b:K-cAlgv!!G.r.bjSecondly+eacrhrepre-senrtablefunctorK-c Alg f(HF:;-):K-cAlg ! SetthatfactorsthroughthecategoryofgroupsisrepresenrtedbyacommutativeHopfalgebra.1ύCorollaryFg2.3.4.ٚA2n!algebrffaHB2URK-35c fAlgkrepresentsananealgebraicgroupifand35onlyifH isacffommutativeHopfalgebra.ThedQcffategoryofcommutativeHopfalgebrasisdualtothecategoryofanealge-brffaic35groups.InBthefollorwinglemmasweconsiderfunctorsrepresentedbycommutativealge-bras. $Theyede nefunctorsonthecategoryK-c Alg˝aswrellasmoregenerallyonK-Alg f._WVe$ rststudythefunctorsandtherepresenrtingalgebras.ThenwreusethemtoconstructcommrutativeHopfalgebras.Lemma2.3.5.TheKfunctorGa `:\K-35Algz)!#Ab Vde neffdbyGaϹ(A):=A2+x,theunderlying~additivegrffoupofthealgebraA,sisarepresentablefunctorrepresentedbythe35algebrffaK[x]thepolynomialringinonevariablex.EProof.@_Ga ܹis anunderlyingfunctorthatforgetsthemrultiplicative structureofthealgebraandonlypreservrestheadditivegroupofthealgebra.ʃWVehavetodeterminenaturallisomorphisms(naturalinA)GaϹ(A)PUR԰n9=K-Alg f(K[x];A)."EacrhelementaUR2A2+isOmappSedtothehomomorphismofalgebrasa]:K[x]3p(x)7!p(a)2A.ThisOisahomomorphismۂofalgebrassincea(p(x)+qn9(x))UR=p(a)+q(a)UR=a(p(x))+a(qn9(x))and9a(p(x)qn9(x))R2=p(a)q(a)=a(p(x))a(q(x)).Another9reasontoseethisisthatK[x]$isthefree(commrutative)$K-algebraorver$fxgi.e.esinceeacrhmapfxgl!AcanuSbSeuniquelyextendedtoahomomorphismofalgebrasK[x]!A. ThemapA$3a7!a (2K-Alg f(K[x];A)WisbijectivrewiththeinversemapK-Alg f(K[x];A)$3fQ7!URfG(x)2A.8FinallythismapisnaturalinAsinceO_MBK-Alg f(K[x];B)J\32fd6"Ѝά-|6,-fHAHDK-Alg f(K[x];A),{fd60ά-6,-fHʟǠ*FfeǠ?' ~gH Ǡ*Ffe<Ǡ?`tK-Alg(K[*x]\t;gI{)commrutesforallgË2URK-Alg f(A;B).\ERemark2.3.6./oSinceGA2+ dhasthestructureofanadditivregroupthesetsofho-momorphismsofalgebrasK-Alg f(K[x];A)arealsoadditivregroups.Lemma2.3.7.The7functorGm by=]U:K-35Alg >*!`G.r >de neffd7byGmĹ(A)]:=U@(A),the[]underlyingmultiplicffativegroupofunitsofthealgebraA,isarepresentablefunctor-m7 aK3.pAFFINE!ALGEBRAICGR9OUPS!O45VrffepresentedVvbythealgebrffaK[x;x21 \|]UR=K[x;yn9]=(xy.W1)VvtheringofLffaurentVvpolynomialsin35onevariablex.]Proof.@_WVe0harvetodeterminenaturalisomorphisms(naturalinA)GmĹ(A)P԰=K-Alg f(K[x;x21 \|];A).Eacrh`[elementa2GmĹ(A)ismappSedtothehomomorphismofalgebrasa ,:=((K[x;x21 \|]3x7!a2A).Thisde nesauniquehomomorphismofalgebrassinceeacrhhomomorphismofalgebrasffromK[x;x21 \|]_r=K[x;yn9]=(xyT1)towAiscompletelydeterminedbrytheimagesofxandofy butinadditiontheimagesharvetosatisfyfG(x)f(yn9)UR=1,-@i.e.fG(x)mrustbSeinvertibleandfG(yn9)mustbSetheinversetofG(x).8Thismappingisbijectivre.FVurthermoreitisnaturalinAsincePAQ}`BAQK-Alg f(K[x;x21 \|];B)Ԟ32fd*ά-|6&-ΟȊ~'AȊsK-Alg f(K[x;x21 \|];A)l{fd+7ά-6&-ΟH[BǠ*FfetǠ?'ygH Ǡ*Ffe>Ǡ?4K-v2Algp(K[*x;x-:1 ] N4;gI{)]forallgË2URK-Alg f(A;B)commrute.v]Remark2.3.8.ySinceFU@(A)hasthestructureofa(mrultiplicative)FgroupthesetsK-Alg f(K[x;x21 \|];A)arealsogroups.tLemma2.3.9.wThe efunctorMn :K-35AlgJX!"K-Algpwith eMnP(A)thealgebrffaofn7n-matricffesXjwithentriesinAisrepresentablebythealgebraKhx11 ;x12;:::ʜ;xnn Рi,the35noncffommutativepolynomialringinthevariablesxijJ.]Proof.@_TheMcpSolynomialringKhxijJiisfreeorverMcthesetfxijginthecategoryof(nontocommrutative)algebras,i.e.7foreachalgebraandforeachmapfչ:?fxijJgYa!9Athereexistsauniquehomomorphismofalgebrasg:xKhx11 ;x12;:::ʜ;xnn Рix?! AعsucrhthatthediagramH*>fxijJg>4KhxijJi?{fdά-lH`}f<ׁ @< @< @< @M>@M>RHARǠ*FfeǠ?'8gv commrutes.t SoSeachmatrixinMnP(A)de nesauniqueahomomorphismofalgebrasKhx11 ;x12;:::ʜ;xnn РiURn!1A꨹andconrverselyV.G]Example2.3.10.1.8Theanealgebraicgroupcalledadditive35grffoup=GaY!:URK-c Alg f!*Abwith|GaϹ(A)Nf:=A2+ ufromLemma2.3.5isrepresenrtedbytheHopfalgebraK[x].WVedeterminecomrultiplication,counit,andantipSode..j7 46242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVByuCorollary2.2.7thecomrultiplicationis=1w2 2K-c Alg f(K[x];K[x] K[x])PUR԰n9=GaϹ(K[x] K[x]).8HenceS3x((x)UR=1(x)+2(x)UR=x 1+1 x:Thecounitis"UR=eK 4=02K-c Alg f(K[x];K)P԰n9=GaϹ(K)henceD"(x)UR=0:TheanrtipSodeisS)=URidW 1؍ K[x] 2URK-c Alg f(K[x];K[x])P԰n9=GaϹ(K[x])hencebf[S׹(x)UR=x:2.8Theanealgebraicgroupcalledmultiplicffative35groupGm Z:URK-c Alg f!*AbwithXGmĹ(A):=A2 =U@(A)fromLemma2.3.7isrepresenrtedbytheHopfalgebraK[x;x21 \|]UR=K[x;yn9]=(xy1).8WVedeterminecomrultiplication,counit,andantipSode.ByCorollary2.2.7thecomrultiplicationisS3 UR=1j2V2K-c Alg f(K[x;x 1 \|];K[x;x 1] K[x;x 1])PUR԰n9=GmĹ(K[x;x 1] K[x;x 1]):Hencel΍(x)UR=1(x)2(x)UR=x x:Thecounitis"UR=eK 4=12K-c Alg f(K[x;x21 \|];K)PUR԰n9=GmĹ(K)henceD"(x)UR=1:TheTdanrtipSodeisSo= idW1эK[x;x1 ]62 K-c Alg f(K[x;x21 \|];K[x;x21])P ԰=kGaϹ(K[x;x21])|hencel΍bS׹(x)UR=x 1 \|:3.8Theanealgebraicgroupcalledadditive35matrixgrffoupfhM +ڍn qʹ:URK-c Alg f!*Abf;withjM2+RAnx(A)theadditivregroupofn n-matricesjwithcoSecientsinAisrepresentedbry2thecommutativealgebraM2@+RAn -=2K[xijJj1i;j}n]2(Lemma2.3.9).qThisalgebramrustbSeaHopfalgebra.The#comrultiplicationisUR=12V2K-c Alg f(M2@+RAn]\;M2@+RAn _v M2@+RAn)PUR԰n9=M2+RAnx(M2@+RAn _v M2@+RAn).Hencel΍e(xijJ)UR=1(xij)+2(xij)UR=xij 1+1 xij:Thecounitis"UR=eK 4=(0)2K-c Alg f(M2@+RAn]\;K)P԰n9=M2+RAnx(K)henceS3"(xijJ)UR=0:TheanrtipSodeisS)=URidW 1 ,썑 Mi@"+Ln 2URK-c Alg f(M2@+RAn]\;M2@+RAn)PUR԰n9=M2+RAnx(M2@+RAn)hencewS׹(xijJ)UR=xij:4. ThematrixalgebraMnP(A)alsohasanoncommrutativemultiplication,thema-trixRmrultiplication, de ningthebialgebrapropSertryandtheantipSode>canbSetransferredeasilyV, soHV2 isbDagainaHopfalgebra. jHencethefunctor-L2bB:URveffcDe]!)Dveffc>Ifrom nitedimensionalvrectorEbspacestoitselfinducesaduality-0 2͹:K-hopft)!!1}uK-hopf̹fromthecategoryof nitedimensionalHopfalgebrastoitself.AnRanealgebraicgroupiscalled niteiftherepresenrtingHopfalgebrais nitedimensional.xAformalgroupiscalled niteiftherepresenrtingHopfalgebrais nitedimensional.Thrusthecategoryof niteanealgebraicgroupsisequivXalenttothecategoryof niteformalgroups.TheBcategoryof nitecommrutativeBanealgebraicgroupsisselfdual.AThecat-egory:of nitecommrutative:anealgebraicgroupsisequivXalenrt(anddual)tothecategoryof nitecommrutativeformalgroups.x5.QuantumGroupsDe nition^2.5.1.魹(Drinfel'd)͂A{quantumjgrffoupisanoncommrutative͂noncoScom-mrutativeHopfalgebra.Remark2.5.2.FWVeMshallconsiderallHopfalgebrasasquanrtumgroups. aOb-servre,?however,thatythecommrutativeyHopfalgebrasmarybSeconsideredasanealgebraicgroupsandthatthecoScommrutativeHopfalgebrasmarybeconsideredasformal_groups.TheirpropSertryasaquantumspaceorasaquantummonoidwillplaysomerole.NButoftenthe(pSossiblynonexisting)dualHopfalgebrawillharvethegeo-metrical*vmeaning.ThefollorwingexamplesSL#qj!(2)andGLΟq̹(2)willhaveageometricalmeaning.Example2.5.3. TheSsmallestpropSerquanrtumgroup,i.e.ythesmallestnoncom-mrutativenoncoScommutativeHopfalgebra,isthe4-dimensionalalgebraqH4V:=URKhgn9;xi=(g 21;x 2;xg+gn9x)3{7 k5.pQUANTUM!GR9OUPS+51Vwhicrhwas rstdescribSedbyM.Sweedler.8Thecoalgebrastructureisgivenby#9ʍp(gn9)UR=g g;PY(x)UR=x 1+g x;}[g"(gn9)UR=1;"(x)UR=0;hS׹(gn9)UR=g21 ʵ(=g);US׹(x)UR=gn9x:Sinceitis nitedimensionalitslineardualH2VRA4 isalsoanoncommrutativenoncoScom-mrutativeHopfalgebra.3ItisisomorphicasaHopfalgebratoH4.InfactH4isuptoisomorphism8theonlynoncommrutative8noncoScommutativeHopfalgebraofdimension4.%Example}2.5.4.p̹ThenjanealgebraicgroupSLr9(n)N:K-c Alg3Mb!"G.r qde nednjbrySL(n)(A),Lthe8groupofn߮n-matrices8withcoSecienrtsinthecommutativealgebraAandwithdeterminanrt1,YGisrepresentedbythealgebraOUV(SL(n))%=SL(n)=K[xijJ]=(detQ(xij)1)i.e.McSLq(n)(A)PUR԰n9=K-c Alg f(K[xijJ]=(detQ(xij)1);A):SincebSL(n)(A)bhasagroupstructurebrythemultiplicationofmatrices,1therepresent-ingcommrutativealgebrahasaHopfalgebrastructurewiththediagonal=12hence荒(xikl)UR=Xxij xjvk ; thecounit"(xijJ)UR=ij 'andtheanrtipSodeS׹(xij)UR=adjӹ(X)ij 'whereadj(X)istheadjoinrtmatrixofXFչ=UR(xijJ).8WVelearvethevreri cationofthesefactstothereader.fvWVenconsiderSLU(n)URMn=A2n-:2 Aasnasubspaceofthen22-dimensionalanespace.XExample2.5.5.3LetOMq(2)-=Kq qʍWa$dbcc$Xd*Zq2q==IA"asin1.3.6withItheidealgen-čeratedbryM abqn9 1 ʵba;acqn9 1ca;bdqn9 1db;cdqn9 1dc;(adqn9 1bc)(daqn9cb);bccb:WVe rstde nethequantumKdeterminantdetWCqoQ=adqn921 ʵbc=daqn9cbinMq(2).Itisacenrtralelement.#TVoshowthis,6itsucestoshowthatdet՟q8commuteswiththegeneratorsa;b;c;d:9ʍ<(adqn921 ʵbc)aUR=a(daqn9bc);(adqn921 ʵbc)bUR=b(adqn921bc);8l(adqn921 ʵbc)cUR=c(adqn921bc);q'(daqn9bc)dUR=d(adq21 ʵbc):WVecanformthequanrtumdeterminantofanarbitraryquantummatrixinAbyNdet_(qeqʍna205b20o!(c20)d20udq:=URa 09d 0xqn9 1 ʵb 0c 0#=d 0a 0xqn9c 0b 0#='(detQq溹)Jifa'UR:Mq(2)n!1AisthealgebrahomomorphismassoSciatedwiththequanrtummatrixtqʍXa20\b20 cc20Βd20$͟q-%.4*7 52242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVfoGivren!twocommutingquantum222-matricesqʍ ya20Rb20 Yc20ijd20'q2D;qʍ Va200 b200 cc200 d200+T=q4).^ThequanrtumčdeterminanrtpreservestheproSduct,sincebM(3)ןqI-detZq`qʍi=a200Zb200jvc200~̰d200$qz:bInparticularwrehave(detQq溹)\1=detqB detQqjand"(detQq)=1.DThequanrtumdetermi-nanrtisagrouplikeelement(see2.1.6).Norwwede neanalgebra-SLq(2)UR:=Mq(2)=(detQqb1):ThealgebraSLq(2)represenrtsthefunctorHZSLWq[(2)(A)UR=fqʍXa20\b20 cc20Βd20$͟q0w2Mq(2)(A)jdetQq渟qʍa203CVb20 JIc202Jd20;qG/=1g:_ThereisasurjectivrehomomorphismofalgebrasMq(2)[!-SLq(2)andSLq(2)isasubfunctorofMq(2).LetIBXJg;Y岹bSecommrutingquantummatricessatisfyingdetq/(X)=1=detӟqѹ(Yp).SincedetKZqX(X)detQq渹(Yp) ==det~q$|(XYp)forcommrutingquantummatriceswegetdetQq溹(XYp)=1,henceO{SL(q&(2)O{isaquanrtumsubmonoidofMq(2)andSLq(2)isabialgebrawithdiagonalݍ"qʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq- qʍ a9 b 9cd!q,Y;ލandy$"qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,:WTVoshorwthatSLq(2)hasanantipSodewe rstde neahomomorphismofalgebrasT:URMq(2)n!1Mq(2)2op ŹbrypTğqʍ wa0)b Ucd"q.ɹ:=URqʍMd5qn9b *qn921 ʵc>$ Mq(2)>xMq(2)[t]t{fd@ά->>MGq(2)񥔟{fdЍά-H0A`ğׁ @ğ @ğ @ğ @D>@D>RH#rǠ*FfeVǠ?H`Մׁ Մ Մ Մ >> withtUt?7!deteq&aqʍ#a207b20$c206d20?.qH<1V:ThrusGLɭq^(2)(A)isasubsetofMq(2)(A).ObservethatfpMq(2)URn!1GLq(2)isnotsurjectivre.Since lthequanrtumdeterminantpreservesproSductsandtheproductofinrvertibleelemenrtsAisagaininvertiblewegetGL(q(2)isaquantumsubmonoidofMq(2),hencetUR:GLq(2)n!1GLq(2) GLq(2)hwithqʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq+ qʍ \a"b Ncdq+Uعand(t)UR=t t.fpWVeconstructtheanrtipSodeforGLq(2).8Wede neT:URMq(2)[t]n!1Mq(2)[t]2op Źbry  {Tğqʍ wa0)b Ucd"q.ɹ:=URtqʍyd49qn9b Vqn921 ʵc; 1aH1şqnqandwTƹ(t):=detq< qʍ#ba3ob#c3ThenslC(2)isasubspaceofthealgebraM@(2)of2f2-matricesorverK.WVeeasilyverify[XJg;Yp]UR=XYYXFչ=HV,m[HF:;X]=HXtXHB=2X,mand^[HF:;Yp]=HVYYHB=2Y,sooathatslC(2)bSecomesaLiesubalgebraofM@(2)2LGع, whicrhistheLiealgebraofmatricesoftracezero.uyTheunivrersalenvelopingalgebraU@(slC(2))isaHopfalgebrageneratedasanalgebrabrytheelementsXJg;Y;HwiththerelationsKNf[XJg;Yp]UR=HF:; [H;X]UR=2XJg; [HF:;Yp]=2Y:AsaconsequenceoftheProincarse-Birkho -WittTheorem(thatwredon'tprove)theHopfalgebraU@(slC(2))hasthebasisfX2iV]Yp2jzHV2kji;j;k 2Ng.mFVurthermoreonecanprorvethatall nitedimensionalU@(slC(2))-moSdulesaresemisimple.7`<7 k5.pQUANTUM!GR9OUPS+55VExample2.5.9.AWVe_Lde netheso-calledqn9-deformedvrersionUq(slC(2))ofU@(sl(2))foranryq]2y$K,q6=1;1andqminrvertible.xLetUq(slC(2))bSethealgebrageneratedbrytheelemenrtsE;FS;K5;Kܞ20withtherelations+ōKܞK20)=URK20K1=1;LKܞEK20)=URqn922.=E;KFK20)=URqn922 ʵFS;!EFLnFE i=ōKFKܞ20[z'0L ΍qqn91,:,?SinceKܞ20YistheinrverseofK inUq(slC(2))wrewriteKܞ21;=Kܞ20׹.,oTherepresentationtheoryofthisalgebraisfundamenrtallydi erentdepSendingonwhetherq\isarootofunitryornot.WVeshorwthatUq(slC(2))isaHopfalgebraorquantumgroup.8WVede ne,YčʍH5n(E)UR=1 E^+E K5;(Fƹ)UR=Kܞ21 FLn+F 1;(Kܞ)UR=KF K5;("(E)UR="(Fƹ)=0;"(Kܞ)=1;XvS׹(E)UR=EKܞ21 9; S׹(Fƹ)=KܞFS;S׹(K)UR=Kܞ21 9:$)i.First WwreshowthatcanbSeexpandedinauniquewaytoanalgebrahomomor-phism5:Uq(slC(2))! Uq(slC(2))R Uq(slC(2)).WVrite5Uq(sl(2))5astheresidueclassalgebraKhE;FS;K5;Kܞ21 9i=I+whereIisgeneratedbry,v-NKܞK211;K21 9KF1;y^KܞEK21qn922.=E;KFK21qn922 ʵFS;EFLnFE^ōKFKܞ21۟[z/ ΍7qqn914ϝ:,Since`Kܞ21=zmrustbSemappedtotheinrverse`of(Kܞ)wremusthave(Kܞ21 9)=Kܞ21D Kܞ21 9.Norw canbSeexpandedinauniquewaytothefreealgebraUR:KܞhE;FS;K5;K21 9i!kUq(slC(2)) Uq(slC(2)). 2WVe=harve(KܞK21 9)=(Kܞ)(K21)=1=andsimilarly(Kܞ21 9Kܞ)r=1.mFVurthermorewrehave(KܞEK21 9)r=(Kܞ)(E)(K21 9)r=(K! Kܞ)(1 E+E Kܞ)(K21< K21 9)UR=KܞK21 KEK21+KEK21 K22K21l=URqn922.=(1 E+E, xKܞ)UR=qn922.=(E)=(qn922E)Eandsimilarly(KܞFK21 9)UR=(qn922 ʵF).0FinallyEwrehaveO<ǎ􍍑1(EFLnFE)|=UR(1 E^+E Kܞ)(K20U FLn+F 1)'(Kܞ20U FLn+F 1)(1 E^+E Kܞ)|=URKܞ20U EFLn+F E^+EKܞ20U KܞF+EF K'Kܞ20U FE^Kܞ20E FKFFLn EFE K|׹=URKܞ20U (EFLnFE)+(EFLnFE) KQq|׹=ōKܞ20U (KFKܞ20׹)+(KKܞ20׹) K[zs ΍?Bqqn91|׹=URqō KFKܞ20 [z'0L ΍qqn914lq8r57 56242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVhencevXanishesonIandcanbSefactorizedthroughauniquealgebrahomomorphismvUR:Uq(slC(2))n!1Uq(slC(2)) Uq(slC(2)):InasimilarwrayV,actuallymruchsimpler,onegetsanalgebrahomomorphismh;"UR:Uq(slC(2))n!1K:TVocrheckthatiscoassoSciativeitsucestocheckthisforthegeneratorsofthealgebra.WVe, K K`0=(1 )(1 E> +E Kܞ)=(1 )(E). Similarly2wreget(c 1)(Fƹ)UR=(1c )(F).FVorNK[theclaimisobrvious.Thecounitaxiomiseasilycrheckedonthegenerators.Norw&weshowthatSڹisanantipSodeforUq(slC(2)).TFirstde neS):URKhE;F;K5;Kܞ21 9i!nUq(slC(2))2op Źbrythede nitionofSonthegenerators.8WVehave<ܧI$K4S׹(KܞK21 9)UR=1=S(Kܞ21 9Kܞ);P1S׹(KܞEK21 9)UR=KܞEK21 9K21l=URqn922.=EK21l=URS׹(qn922.=E);US׹(KܞFK21 9)UR=KܞKFK21l=URqn922 ʵKF=S׹(qn922 ʵFƹ);6fS׹(EFLnFE)UR=KܞFEK21EK21 9KF=URKFK21KEKFEFd=ōKܞ21K[z/ ΍7qqn918ϙ=URS՟qō `KFKܞ21 `[z/ ΍7qqn91="qHx:ޒfd^pά-f$#`<Bհfi?ku:\9P:\ŽjP:\P:\9P:\̎iP:\P:\9P:\֎hP:\P:\9P:\gP:\P:\9P:\fP:\P :\9P:\ePǑPǑq`հ]fVHVHVHVHVHVHVHVH VHVHVH'\ǗH'\Ǘj`" tfǟ-:0rܟ@rܟ@rܟ@rܟ@%rܟ@/VǗ@/VǗR`A[Ǡ;feAǠ?h-㎍HrFAf|I`\Corollary2.6.2.The35cffategoryofbialgebrashas nitecoproducts.Proof.@_The%coproSduct`zBiofbialgebras(BidjiUR2I)%inK-Alg isanalgebra.FVorthediagonalandthecounitwreobtainthefollowingcommutativediagramsO'Bk: Bk_`شBi `BisD32fd `ά-荒pji?k ji?kHBkH[`Bi{fdC+ά-e˓ji?kHZҟǠ*FfeǠ?|/i?kHLҟǠ*FfeǠ?D29q1*XH"BkH`cBi|{fdά-e̓nji?kH㨤ú"i?kcLׁ @cL @cL @cL @ީ̟>@ީ̟>RH""7KǠ*Ffe,Ǡ?D9q1*"x~sincejinbSothcases`BiύisacoproductinK-Alg f. Thenitiseasytoshorwthatthesehomomorphismsde neabialgebrastructureon`Bitandthat`Bitsatis estheunivrersalpropSertyforbialgebras.\TheoremN2.6.3.4mLffet>Bbeabialgebra.ThenthereexistsaHopfalgebraHV(B)andahomomorphismofbialgebrffasUR:BXV!HV(B)suchthatforeveryHopfalgebraHandforheveryhomomorphismofbialgebrffasfȹ:BS:!sHVAthereisauniquehomomorphismof35HopfalgebrffasgË:URHV(B)!H such35thatthediagramMΜHBH⑹HV(B)|{fd i@ά- ޏH`ņ5f|ׁ @Ƚ| @ҽ| @ܽ| @>@>RHH *Ǡ*Ffe<\Ǡ? g|cffommutes.:<7 58242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVProof.@_De neasequenceofbialgebras(BidjiUR2N)bryIʍB0V:=URB;Bi+1:=URBOopcopái;i2N:LetB20;bSethecoproductofthefamily(BidjiUR2N)withinjectionsi,:URBiӷ!) B20i?.GBecauseB20ιisacoproSductofbialgebrasthereisauniquehomomorphismofbialgebrasSן20b:URB20!nB20@^opcop6sucrhthatthediagramsL%ӍBOopcopái+12B20^opcopnj,32fdά-͍ϫ8:i+1ȥ>Biȥ*B20\{fd33@ά-8:iH Ǡ*Ffe<Ǡ?k}`idH"Ǡ*FfeUǠ?nHencewrehaveJIʍBz((1Sן20qYNu")idڹ(x))qYN=URPidڹ(x(1) \|)Sן20i(x(3)) (1Sן20)idڹ(x(2))u"idڹ(x)qYN=URPidڹ(x(1) \|)Sן20i(x(3)) ((1Sן20)u")idڹ(x(2))>+Pidڹ(x(1) \|)Sן20i(x(3)) u"idڹ(x(2))u"idڹ(x)qYN=URPidڹ(x(1) \|)Sן20i(x(3)) (1Sן20+u")idڹ(x(2))>+Pidڹ(x(1) \|)Sן20i(x(2)) 1Bd0 u"idڹ(x) 1Bd0qYN=URPidڹ(x(1) \|)Sן20i(x(3)) (1Sן20+u")idڹ(x(2))>+(1Sן20+u")idڹ(x) 1Bd0qYN2URB20 I++I B20i?:M5ThrusI+isacoidealandabiidealofB20i?.;f7 zU6.pQUANTUM!A9UTOMORPHISMGROUPSq59VNorwletHV(B):=B20i?=I{andlet~ǹ:B20&? ?!HV(B)bSetheresidueclasshomomor-phism.>WVedshorwthatHV(B)isabialgebraand&isahomomorphismofbialgebras.HV(B)CisanalgebraandQ isahomomorphismofalgebrassinceIƹisatrwoCsidedideal.SinceIFURKerBm(")thereisauniquefactorizationNHIB20H B20i?=ID{fd!wЍά- {&H ; "20ׁ @ʭ @ԭ @ޭ @>@>RH""K2Ǡ*Ffe,dǠ?9"whereM"UR:B20i?=IF``!Kisahomomorphismofalgebras.Since(I)B20K IF+I B20Ker( F:2B20' B20q !0B20i?=IQ B20=I)andthrusIv2KerrM(( ǹ))wehaveauniquefactorizationE|⍍B20 B20ⵡB20i?=I+ B20=I̞32fd#ά-*i~ HDLB20H68B20i?=I{fdDά-H@>RHB20i?=IǠ*FfeDǠ?S8ԍThisholdsifIFURKerBm(Sן20).8SinceKer()UR=I+itsucestoshorwSן20(I)URI.8WVeharve:-QNP gSן20((Sן20+1)idڹ(x))UR=􍍒 g=URrW(Sן20 ^2{i Sן20idڹ)i(x) g=URrW(Sן20+ 1)(i+1 i+1AV)idڹ(x) g=URr(1 Sן20)(i+1 i+1AV)Widڹ(x) g=URr(1 Sן20)(i+1 i+1AV)i+1(x) g=UR(1Sן20)i+1AV(x)9andrL^Sן 0(u"idڹ(x))UR=Sן 0(1)"idڹ(x)=Sן 0(1)"i+1AV(x)=Sן 0(u"i+1AV(x));hencewreget^8Sן 0((Sן 0+1u")idڹ(x))UR=(1Sן 0u")i+1AV(x)UR2I:<u7 60242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVThis{shorwsSן20(I)KI.gSo{thereisauniquehomomorphismofbialgebrasS:KHV(B)!nHV(B)2opcop[sucrhthatthediagramJvՍ ٍB20+^opcop ٍޛHV(B)2opcop`32fdά-H $B20H1HV(B)t{fd*ά- H"Ǡ*FfeKTǠ?nI+Sr}-:0HǠ*FfeԟǠ?`sTS!commrutes.NorwnweshowthatHV(B)isaHopfalgebrawithantipSodeS׹.ByProposition2.1.3it5sucestotestongeneratorsofHV(B)henceonimagesidڹ(x)ofelemenrtsx82Bi.WVeharveeʍ:j(1S׹)idڹ(x)UR=r(lo Sǹ)idڹ(x)=r(lo )(1 Sן20)idڹ(x)UR=uF=URǹ(1Sן20)idڹ(x)=u"i(x)=u"i(x):eByPropSosition2.1.3SisananrtipodeforHV(B).WVe(prorvenowthatHV(B)togetherwithrW:=0 2[:B ]&!AHV(B)isafreeHopfalgebra orverB..LetH`bSeaHopfalgebraandletfQ:URBX !_7HbSeahomomorphismofbialgebras.4MWVewillshorwthatthereisauniquehomomorphismWk*fD:URHV(B)n!1HEsuchthatBpHBH⑹HV(B)|{fd i@ά-pH`ǡf|ׁ @Ƚ| @ҽ| @ܽ| @>@>RHH *Ǡ*Ffe<\Ǡ?6ݍ㎍rfO2commrutes.WVede neafamilyofhomomorphismsofbialgebrasfi,:URBiӷ!) Hbryʍf0V:=URf;fi+1:=URSHDfid;i2N:gWVeharveinparticularfi,=URS2ibHDfɹforalli2N.-ThrusthereisauniquehomomorphismofbialgebrasfG20k:URB20=`Bi,ӷ!) HsucrhthatfG208i,=URfiOforalli2N.WVeshorwthatfG208(I)UR=0.8Letx2Bidڹ.Then=eʍ[fG208((1Sן20)idڹ(x))=URfG208(r(1 Sן20)(i idڹ)i(x))=URPfG208idڹ(x(1) \|)fG20Sן20i(x(2) \|)=URPfG208idڹ(x(1) \|)fG20i+1AV(x(2))=URPfidڹ(x(1) \|)fi+1AV(x(2))=URPfidڹ(x(1) \|)Sfi(x(2))=UR(1S׹)fidڹ(x)=u"fi(x)=u"i(x)=URfG208(u"idڹ(x)):=eThistogetherwiththesymmetricstatemenrtgivesfG208(I)UR=0. Hencethereisauniquefactorization ythroughahomomorphismofalgebrasW#*f r:URHV(B)n!1H Ϲsucrh ythatfG20k=W*URf gǹ.=U7 zU6.pQUANTUM!A9UTOMORPHISMGROUPSq61VTheohomomorphismW2*f:7~HV(B)Q !H\޹isoahomomorphismofbialgebrassincethediagramLt B20i?=Iğ:2fd-6ά-־ɯ ɯ ğ{fdz{0ά-zr9fǟ-:0Rt:2fd.ά-퍍㎍_r%fKK`32fd`K@ά-h2rfǟ-:0 fǟ-:0AAIB20i?=I+ B20=I`tfd 6ά-zf AAԟtfd `ά-^͍㎍?cry)fpH ㎍:rfAz3Hz'*H H8r耄@fe9)耬?=$X.HA `fe$ `?nf-:0ώ+tB20+B20 B20 r耄@fe>耬?`vkcommruteswiththepSossibleexceptionoftherighthandsquareW*f and(W*fg WCj*f)209.Butg isYsurjectivresoalsothelastsquarecommutes.!Similarlyweget"HW *Df==UR"H(Bd).!ThusϢW*f _=isLahomomorphismofbialgebrasandhenceahomomorphismofHopfalgebras. Remark2.6.4.ϹIn@؝>RHaH AŸǠ*FfeǠ?`tf 1΍commrutes.SinceythefQ:URM@(A)n!1HfufactorizesyuniquelythroughW*f :URHV(M(A))n!1HwregetacommutativediagramO񯍍HȠAHPHV(M@(A)) AD{fd sά-ML̟-:0Hk}P@htׁ @ht @ht @ht @Ӯ>@Ӯ>RHH A"Ǡ*FfeTǠ?6ݍ㎍`rf 1΍withauniquehomomorphismofHopfalgebrasWyR*fR:URHV(M@(A))n!1H.ThisEcproSofdependsonlyontheexistenceofaunivrersalalgebraM@(A)forthealgebraA.8HencewrehaveLCorollary̌2.6.5."LffetprX2Ebeaquantumspacewithuniversalquantumspace(andquantumFmonoid)M(Xӹ).[Thentherffeisaunique(uptoisomorphism)quantumgroupH(M(Xӹ))35actinguniversallyonX.>f7 62242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSVThis2TquanrtumgroupH(M(Xӹ))canbSeconsideredasthe\quantumsubgroupofinrvertibleelements"ofM(Xӹ)orthequantumgroupof\quantumautomorphisms"ofXӹ.C>7.DualityofHopfAlgebrasInނ2.4.8wrehaveseenthatthedualHopfalgebraHV2 ܹofa nitedimensionalHopfalgebraHPsatis escertainrelationsw.r.t.theevXaluationmap.ThemrultiplicationoffHV2 isderivredfromthecomultiplicationofHandthecomultiplicationofHV2 isderivredfromthemultiplicationofHV.Thiskindofdualitryisrestrictedtothe nite-dimensionalsituation.*Neverthelessone1wrantstohaveaproScessthatisclosetothe nite-dimensionalsituation.I{ThisshortsectionisdevrotedtoseveralapproachesofdualityforHopfalgebras.Firstwreusetherelationsofthe nite-dimensionalsituationtogiveageneralde nition.;De nition2.7.1.LetHandLbSeHopfalgebras.8Letv@(evW:URL HB3a h7!ha;hi2KbSeabilinearformsatisfyinge2(4)`ha b;XUTh(1)$ h(2) \|iUR=hab;hi; h1;hiUR="(h)!(5)`nhXUVa(1)$ a(2) \|;h jiUR=ha;hji; ha;1i="(a)(6)Pha;S׹(h)iUR=hS(a);hiSucrhamapiscalledaweffak@dualityofHopfalgebras.eThebilinearformiscalledleft(right$D)nondeffgenerateifha;HVi̹=0impliesa=0 ߒ(hL;hi=0impliesh=0).HEAduality35ofHopfalgebrffas꨹isawreakdualitythatisleftandrightnondegenerate.;Remark-2.7.2.IfiHisa nitedimensionalHopfalgebrathentheusualevXaluationev:URHV2X HB\3!K꨹de nesadualitryofHopfalgebras.Remark2.7.3.Assumethatev\:qL1 H^ x!5Kde nesawreakdualityV. oBy[AdvXancedAlgebra^Q,]p1.22wrehaveisomorphismsHom(L HF:;K)PUR԰n9=Hom(y(L;Homy(H;K))andHomgO(L HF:;K)PZ԰s= Hom("(H;Homy(L;K)).B b;Ph(1) N h(2) \|iUR=Pha;h(1)ihb;h(2)iUR=P'(a)(h(1))'(b)(h(2))=('(a)>'(b))(h)brythede nitionofthealgebrastructureonHomd1(HF:;K).ev :URL HB\3!K꨹satis estheleftequationof(4)i '(1)(h)=h1;hi="(h).ThesecondpartoftheLemmafollorwsbysymmetryV.ލExample2.7.5.%There/isawreakdualitybSetweenthequantumgroupsSLܟq8ڹ(2)andUq(slC(2)).8(Kassel:ChapterVISI.4).0Prop`ositionD2.7.6.{LffetȹevK:URLAD HB!gKbeaweakdualityofHopfalgebras.VLetI:=sKer('s:Le!eSHom.(HF:;K))andJ:=sKer( m:sHS4!RHom/2(L;K)).zLffet\-z ӍL:=L=Iand~\-z ӍHX:=HF:=Jr. HThen~\-z ӍLand~\-z ӍHarffe~Hopfalgebrasandtheinducedbilinearformdz 1Kev:UR\-z ӍL \-z ӍH!"eOK35isaduality.ލProof.@_FirstobservrethatIVandJEaretwosidedidealshence\-z ӍL5and\-z ӍHO i Pa2IBMordz+Ka~=0.kGThrusthebilinearformdz 1Kev%:\-z ӍL 1\-z ӍH|!#K-hde nesadualitryV.|ލProblem2.7.15.(inLinearAlgebra)0(1)+FVor|U6URVrde neU@2? :=ffQ2Vp2\tjfG(U@)=0g.FVorZ1Vp2 vde neZܞ2? Nh:=fvË2+VpjZܞ(vn9)UR=0g.8Shorwthatthefollowinghold:/((a)CѬU6URV=)LU=U@2??Թ;/' (b)CѬZ1URVp2 GanddimZ<1꨹=)Z=Zܞ2??;0uD(c)CѬfUhVpjdimDVN8=U<1gP԰k=p_fZj"Vp2\tjdimDZ<1g8yunderthemapsCѬU67!URU@2? HandZ17!Zܞ2?.(2)+Let?V=21RAi=1AVKxibSeanin nite-dimensionalvrectorspace.69Findanelement+gË2UR(VG Vp)2thatisnotinV2  V2l(UR(VG V)2).0De nition2.7.7. LetAbSeanalgebra.;WVede neA2o0ѹ:=ff2A2j9꨹ideal\A&@IA rAUR:dim(A=I)<1꨹and|fG(I)=0g.Lemma2.7.8.Lffet35AbeanalgebraandfQ2URA2.fiThefollowingareequivalent:(1)+fQ2URA2o;(2)+therffe35existsIA 36URAsuchthatdim{A=IF<1andfG(I)=0;(3)+AfQURA 34Hom$K+e(:AA;:K)35is nitedimensional;(4)+AfA35is nitedimensional;@7 64242.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSV(5)+r2(fG)UR2A2j A2.ffProof.@_1. #=)$d2.8and4. #=)3.aretrivial.2. }=)$3.ELetIA fAwithfG(I)=0anddimA=Iz<1.EWVriteA20 ,A'!*Kashgn9;ai.8Thenhaf;iiUR=hf;iai=0henceAfQURI2? anddimAf<UR1.3. 8=)$82.N&LetdimAfc<ad1.ThenIA ?H:=(AfG)2? 7isanidealof nitecoSdimensioninAandfG(I)UR=0holds.2. d=))ʹ1. z-LetUIA @\AwithdimA=IA<\1andfG(I)=0bSegivren. z-ThenrighrtRmultiplicationinduces'q:A!Hom/uK6-(A=I:;A=I:)anddim&End/3WK59(A=I)q<1.Thrus[J|= Ker%(') Aisatrwosidedidealof nitecoSdimensionandJ| I޹(since'(jӹ)(n1)UR=0=n1 j%=L)Nj =impliesj2I).8FVurthermorewrehavefG(Jr)URf(I)=0.1. #=)$d4.8hafGb;iiUR=hf;biai=0impliesAfAUR)A qI2?bA hencedimAfGA<1.3. =)'5.WVeúobservrethatr2(fG)=fr2(A>r A)2.WVeúwranttoshowthatr2(fG)2A2C ?A2.Let&Fg1;:::ʜ;gn ΖbSeabasisofAfG.Thenthereexisth1;:::ʜ;hn c2A2sucrhLthatbf=pPhidڹ(b)gi.,LetLa;bp2A.ThenLhr2(fG);a bip=hf;abi=hbf;ai=Phidڹ(b)gi(a)UR=hPgi hid;a bi꨹sothatr2(fG)UR=Pgi hi,2A2j A2.5. I=)$O*3.^LetYr2(fG)j=Pgi& Lhi2A2sP A2.^ThenYbf=Phidڹ(b)gi\3forallb2AasbSefore.8ThrusAf2isgeneratedbytheg1;:::ʜ;gnP.V)ffProp`ositionPb2.7.9.5Lffet@O(A;m;u)bffeanalgebra.sThenwehavem2(A2o)URA2o* A2o.Furthermorffe35(A2o;;")isacffoalgebrawithUR=m29and"=u2.Proof.@_Letf2yA2oandletg1;:::ʜ;gn HbSeabasisforAfG.xThenwrehavem2(fG)y=Pgi Rhi)ѹforsuitablehi,2URA2asintheproSofofthepreviousproposition.Sincegi,2URAfwregetAgimfAf<ʹanddim(Agidڹ)<1andhencegim2A2o.WIChoSosea1;:::ʜ;an 2AsucrhthatCgidڹ(ajf )j=ijJ.^Then(fGajf )(a)=f(ajf a)=hm2(f);ajG =ai=Prgidڹ(ajf )hi(a)j=hj(a)implies_fGaj=hj2fGA.GObservre_thatdim(fA)<1_ʹhencedim(hjf A)<1,}so_thathj\2URA2o.8Thisprorvesm2(fG)UR2A2oFh A2o.OnecrheckseasilythatcounitlawandcoassoSciativityhold.gwWTheoremr2.7.10.2(TheSweedlerdual:)ULffet+(B;m;u;;")bffeabialgebra.Then笹(B2o6;2;"2;m2;u2)againisabialgebrffa. IfB=}HisaHopfalgebrawithantipffode35S,thenS2 isanantipffode35forB2o=URHV2o.Proof.@_WVe$wknorwthat(B2[ ;2;"2)$wisanalgebraandthat(B2o6;m2;u2)$wisacoal-gebra.?WWVeB%shorwnowthatB2o !(G KR;GM KR)PUR԰n9=R-Hopf-Alg2"(K[t;t21 \|] RJ;H5_ H R)PUR԰n9=R-Hopf-Alg2"(R[t;t21 \|];H5_ H R)PUR԰n9=fxUR2U@(H H RJ)j(x)=x x;"(x)=1g,since(x)UR=x xand"(x)UR=1implyxS׹(x)="(x)=1.Considerdx%U2HomޟR#q((H2 RJ)2;R)%U=HomޟR(HV26 RJ;R).Thend(x)%U=x xdݹi x(vn92.=wR2)L=hx;vn92wR2i=h(x);vn923 swR2i=x(vn92.=)x(wR2)Sand"(x)L=1Si hx;"iL=1.HencexUR2RJ-c 7AAlg f((H RJ)2;R)PUR԰n9=K-c Alg f(HV2Z;R)UR=DS(G)(RJ).a%B P7 VC-a7 VRCHAPTER3[RepresentationffTheoryf,ReconstructionandTannakaDuality"Intro`ductionOne+ofthemostinrterestingpropSertiesofquantumgroupsistheirrepresentationtheoryV.It.hasdeepapplicationsintheoreticalphrysics.Themathematicalsidehastodistinguish[bSetrweentherepresentationtheoryofquantumgroupsandtherepresen-tationtheoryofHopfalgebras.7InbSothcasestheparticularstructureallorwstoformtensorproSductsofrepresenrtationssuchthatthecategoryofrepresentationsbSecomesamonoidal(ortensor)categoryV.The~problemwrewanttostudyinthischapteris,thowmuchstructureofthequanrtum׉grouporHopfalgebracanbSefoundinthecategoryofrepresentations.2WVewillAshorwthataquantummonoidcanbSeuniquelyreconstructed(uptoisomorphism)from1itsrepresenrtations.|TheadditionalstructuregivenbytheantipSode1isitimitelyconnectedѝwithacertaindualitryofrepresentations. WVewillalsogeneralizethisproScessofreconstruction.OnCtheotherhandwrewillshowthattheproScessofreconstructioncanalsobeusedto?DobtaintheTVamrbaraconstructionoftheuniversalquantummonoidofanoncom-mrutative=geometricalspace(fromcrhapter1.).2ThuswewillgetanotherpSerspectiveforthistheorem.Arttheendofthechapteryoushoulds2 +〹understandrepresenrtationsofHopfalgebrasandofquantumgroups, +〹knorwqthede nitionand rstfundamentalpropSertiesofmonoidalortensor+categories, +〹bSefamiliarwiththemonoidalstructureonthecategoryofrepresenrtations+ofHopfalgebrasandofquanrtumgroups, +〹understandOuwhrythecategoryofrepresentationscontainsthefullinformation+abSoutothequanrtumgroupresp. utheHopfalgebra(TheoremofTVannakXa-+Krein), +〹knorwptheproScessofreconstructionandexamplesofbialgebrasreconstructed+fromcertaindiagramsof nitedimensionalvrectorspaces, +〹understandbSettertheTVamrbaraconstructionofauniversalalgebrafora nite+dimensionalalgebra./^67D-7 68# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVltT1.}TRepresentationsofHopfAlgebrasLet&AbSeanalgebraorver&acommrutative&ringK.ELetA-M0offd`bethecategoryofA-moSdules.8AnA-moduleisalsocalledarffepresentationofA.ObservreuthattheactionAJ M6!MAYsatisfyinguthemoSduleaxiomsandanalgebrahomomorphismBAc!=End*(M@)areequivXalenrtdescriptionsofanA-moSdulestructureontheK-moSduleM@.ThepfunctorUc:URA-M0offd ! K-M0offdչwithpU1(AM@)=MandU1(fG)=fiscalledtheforffgetful35functor꨹ortheunderlyingfunctor.IfB isabialgebrathenarffepresentation֨ofBisalsode nedtobSeaB-module.PItwill=turnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofrepresenrtationsinacanonicalwayV.LetCQbSeacoalgebraorveracommrutativeringK.LetCܞ-FC5omoffd!bethecategoryofCܞ-comoSdules.8AC-comoSduleisalsocalledacfforepresentation꨹ofC.The/functorUݹ:Cܞ-FC5omoffd!~,#!3bK-M0offdifwithU1(2CM@)=MpandU1(fG)=fwiscalledtheforffgetful35functorortheunderlyingfunctor.IfYB_isabialgebrathenacfforepresentation`oofYBisalsode nedtobSeaB-comodule.ItpawillturnoutthatthepropSertryofbeingabialgebraleadstothepossibilitryofbuildingtensorproSductsofcorepresenrtationsinacanonicalwayV.UsuallyznrepresenrtationsofaringareconsideredtobSemodulesorverznthegivrenring.fTheroleofcomoSdulescertainlyarisesintheconrtextofcoalgebras.Butitisnot{quiteclearwhatthegoSod{de nitionofarepresenrtationofaquantumgrouporitsrepresenrtingHopfalgebrais.FVor(thispurpSoseconsiderrepresenrtationsMijofanordinarygroupG.|Assumefore1thesimplicitryoftheargumentthatGis nite.|RepresentationsofGarevectorspaces?YtogetherwithagroupactionGM!M@. 6EquivXalenrtly?YtheyarevectorspacesztogetherwithagrouphomomorphismG'!۹Aut-(M@)ormoSdulesorverzthegroupalgebra:K[G]k M6!M@.!InthesituationofquanrtumgroupsweconsidertherepresenrtingHopfalgebraHasalgebraoffunctionsonthequantumgroupG.Then`thealgebraoffunctionsonGistheHopfalgebraK2G,thedualofthegroupalgebradK[G].AneasyexerciseshorwsthatthemoSdulestructureK[G]4 MJ!=MtranslatestothestructureofacomoSduleM6!K2G uAMandconrverselyV.ܛ(ObservethatGis nite.)Sowreshouldde nerepresentationsofaquantumgroupascomoSdulesorvertherepresenrtingHopfalgebra.UUProblemv3.1.16.LetgGbSea nitegroupandK2G t:=URK[G]2'thedualofthegroupalgebra.YShorwthatK2G 5isaHopfalgebraandthateachmoSdulestructureK[G] M!nMtranslates.tothestructureofacomoSduleM6!K2G 0 MandconrverselyV.Showthatthisde nesamonoidalequivXalenceofcategories.DescribSethegroupvXaluedfunctorK-c Alg f(K2G;)intermsofsetsandtheirgroupstructure.De nition%3.1.1.ýLetGbSeaquanrtumgroupwithrepresentingHopfalgebraHV.ArffepresentationofGisacomoSduleorvertherepresentingHopfalgebraHV.E77 oSN1.pREPRESENT:ATIONS!OFHOPFALGEBRASjs69VFVromthisde nitionwreobtainimmediatelythatwemayformtensorproSductsofrepresenrtationsofquantumgroupssincetherepresentingalgebraisabialgebra.WVex comenorwtothecanonicalconstructionoftensorproSductsof(co-)represen-tations.wLemma3.1.2.LffetDBbeabialgebra.=LetM;N2OB-;Modx"betwoB-modules.ThenMy 8N͡isaB-moffdulebytheactionb(m n)UR=Pb(1) \|m b(2)n..IffQ:URM6!QM@20andng2:Nk3!/N@20~arffehomomorphismsofB-modulesinB-;ModProof.@_WVeharvehomomorphismsofK-algebras qʹ:^;BA!q End*}(M@)and :BA!End (N@);de ningtheB-moSdulestructureonM@andN.gThruswegetahomomorphismof~algebrascan( s  O)UR:BX !_7Bh BX !End*k(M@) Endڣ(N)URn!1End)(M N).ThrusM' NN^is zaB-moSdule.&Thestructureisb(m n)UR=canF( q 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,Gthenwrehave(f Гgn9)(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.3G>Corollary3.1.3.Lffet8wB}beabialgebra.v/Then _:B-;MoffdB-Moffd`!!WB-Moffdwith35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.Proof.@_ThefollorwingareobviousfromtheordinarypropSertiesofthetensorproSduct0orverK.1M  ~1N n=UR1M" Nܹand(f gn9)(fG20W g20beabialgebra.QkLetM;N62URB-;C5omod!betwoB-comodules.ThenMs 2NWisaB-cffomodulebythecffoactionM" N(m n)UR=Pm(1) \|n(1) ) m(M")S n(N").If%PfQ:URM6!QM@204mandgË:N6!QN@204marffehomomorphismsofB-comodulesinB-;C5omodthen35f gË:URM N6!QM@20 N@20BRisahomomorphismofB-cffomodules.Proof.@_ThecoactiononM@g N¹maryalsobSedescribedbry(rB M 1M s 1ND)(1B  j1ND)(M N)UR:M jN6!B M B N6!B B M N6!B M N:AlthoughadiagrammaticproSofofthecoassociativitryofthecoactionandthelawofthecounitisquiteinrvolveditallorwsgeneralizationsowegiveithere.Considerthenextdiagram.Square(1)commrutessinceM+andNarecomoSdules.Squares u(2)and(3)commrutesinceݹ:M Nʤ/!mN MJYfor uK-moSdulesMandN+isanaturaltransformation.Square(4)represenrtsaninterestingpropSertyofAŹnamely.񍍟ʍ8"(1 1 W)(Bd Ma>;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)FIH7 70# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVthatusesthefactthat(1is gn9)(fr 1)2M=(f is1)(1 gn9)holdsandthatBd Ma>;B$=( 1)(1 W)andMa>;Bd B"ƹ=UR(1 )( 1).Square(5)and(6)commrutebythepropSertiesofthetensorproduct.Square(7)commrutessinceBisabialgebra.*Y*Y\BE M@ $g 6:6:BE N0%`09M N099 T,ҲfdDvpά-)wS٠L *Y*Y=BE B $g 6:6:M N0%`0909,ҲfdAά-)A1 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?t0ah 1 1pz4BE M pztBE B M@ jܠ|/2BE N|zJBE B Nvvv='sH2fd4AЍά-o쁍D1 L 1 pzpzBE B B jܠ||%BE M NvvvtsH2fd5npά-n>51-:2* X.MB";BI 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 :M N:xM BE N`$$;fd7Opά-bD1 1 L DDBE B B R ::%BE M N`$$'fd6@ά-j91 1 1 r 1=@"j2*Ffe"d?鰍'Or 1 1=@ز*Ffe ?鰍dr 1 1 1 1=@&R*FfeY?鰍 r 1 1 1 1zzBE B Ǡ x xxM BE NiBE M NE#Ğ32fd0aά-ׁP1 L zz=BE B Ǡ x xgiBE M ND32fd2~ά-n-1 1 r 1WBE B M N-㤞32fd&fpά-0-2l1 r 1 1R@U(1)R@(2)OU(3)O(4)@U(5)@(6)Ln"(7)앍ThelarwofthecounitisKN::\BE M@ `zzBE NΠZM NZ9 T{fdDvpά- AS٠L ::=BE B `zzM NΠZZ{fdAά-).A1 r 1ZZcBE M N&}{fd9Ѝά-24r 1 1H8Jdؤ13$ҁ H=$ׁ HG$܁ HQ$ H[$ He$ Ho$ Hy$ H}$džH}$džjQM NM NyԞ32fdHά-ZУ"1oQM N#T32fdHά-ZF1'زǠ$g`fe Ǡ?.d" 1 " 1' G2Ǡ$g`fe zdǠ?.," " 1 1HǠ*FfeǠ?͞d" 1 1#wherethelastsquarecommrutessince"isahomomorphismofalgebras.Norwletf2andgXbSehomomorphismsofB-comodules.8ThenthediagramMՍ::\BE M@ `zzBE NΠZM NZ9 T{fdDvpά- AS٠L ::=BE B `zzM NΠZZ{fdAά-).A1 r 1ZZcBE M N&}{fd9Ѝά-24r 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 zz=BE B Ǡ k kM@20 N@20KKĞ32fd@xPά-nU1 r 1KKaBE M@20 N@20&}32fd7Pά-03*r 1 1(commrutes.8Thusf gXisahomomorphismofB-comoSdules.sMKCorollary_3.1.5.,LffetBMbeabialgebra. Then :B-;C5omoffd!zBB-C5omoffd#C&5!B-;C5omoffd!with35 (M;N@)UR=M Ntand (f;gn9)=f gnisafunctor.؍Prop`osition3.1.6.uLffet̎Bgbeabialgebra.D2Thenthetensorproduct UR:B-;Moffd B-;Moffd!!DbB-Moffdfsatis es35thefollowingprffoperties:G^ 7 oSN1.pREPRESENT:ATIONS!OFHOPFALGEBRASjs71V(1)+The assoffciativityisomorphism h:UR(M1 M2) M3V {!M1 (M2 M3) with+ ((m n) p)=m (n p)Y"isanaturffaltransformationfromthefunctor+ j( Id .A)tothefunctor j(Id ; )inthevariablesM1,M2,andM3"in+B-;Moffd 3Q.(2)+The02cffounitisomorphismsUR:Kk M6!QMqwith02( m)UR=mand:M kK+.I!=?M'with(mU )UR=marffenaturaltransformationsinthevariableM'in+B-;Moffdffrffom35thefunctorK - resp.fi- DF K35totheidentityfunctorIdr.(3)+The35followingdiagrffamsofnaturaltransformationsarecommutativeQ퍍Ɔ((M1j M2) M3) M4oƹ(M1j (M2 M3)) M4:2fd ά-έ''Ѐl(S (Mq1*;Mq2;Mq3) 12M1j ((M2 M3) M4):2fd ά-έ''Ѐ[ (Mq1*;Mq2 Mq3;Mq4)?Ǡ@fe@&Ǡ?ꃀD؜ (Mq1* Mq2;Mq3;Mq4)qEjǠ@feqxǠ?ꃀ,!J1 (Mq2*;Mq3;Mq4)Ɔ(M1j M2) (M3 M4)2M1j (M2 (M3 M4))32fd`ά-r (Mq1*;Mq2;Mq3 Mq4)Y卍jT(M1j K) M2?M1j (K M2):2fd6ά-̯ (Mq1*;K;Mq2)34[M1j M2ꃀ(Mq1*) 1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1 (Mq2*)L L攴L?^L#0#0+ gProof.@_Thefhomomorphisms ,,andfarealreadyde nedinthecategoryK-M0offdand9satisfytheclaimedpropSerties. ԓSowrehavetoshow,@thatthesearehomomorphismsinB-M0offd'andthatKisaB-moSdule.BKisaB-moSdulebry" 1K ¹:B KURn!1K.Theeasyvreri cationusesthecoassoSciativityandthecounitalpropSertyofB. #Similarlywregeth%Prop`osition3.1.7.Lffet35B;beabialgebra.fiThenthetensorproductMrc UR:B-;C5omoffd ^B-C5omoffd!#o!2qB-C5omoffdsatis es35thefollowingprffoperties:(1)+The assoffciativityisomorphism h:UR(M1 M2) M3V {!M1 (M2 M3) with+ ((m n) p)=m (n p)Y"isanaturffaltransformationfromthefunctor+ j( Id .A)tothefunctor j(Id ; )inthevariablesM1,M2,andM3"in+B-;C5omoffd`.(2)+The02cffounitisomorphismsUR:Kk M6!QMqwith02( m)UR=mand:M kK+.I!=?M'with(mU )UR=marffenaturaltransformationsinthevariableM'in+B-;C5omoffd!frffom35thefunctorK - resp.fi- DF K35totheidentityfunctorIdr.Hx 7 72# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYV(3)+The35followingdiagrffamsofnaturaltransformationsarecommutativeKNƆ((M1j M2) M3) M4oƹ(M1j (M2 M3)) M4:2fd ά-έ''Ѐl(S (Mq1*;Mq2;Mq3) 12M1j ((M2 M3) M4):2fd ά-έ''Ѐ[ (Mq1*;Mq2 Mq3;Mq4)?Ǡ@fe@&Ǡ?ꃀD؜ (Mq1* Mq2;Mq3;Mq4)qEjǠ@feqxǠ?ꃀ,!J1 (Mq2*;Mq3;Mq4)Ɔ(M1j M2) (M3 M4)2M1j (M2 (M3 M4))32fd`ά-r (Mq1*;Mq2;Mq3 Mq4)KjT(M1j K) M2?M1j (K M2):2fd6ά-̯ (Mq1*;K;Mq2)34[M1j M2ꃀ(Mq1*) 1l Ql攴Ql?^QlQz̟0Qz̟0sꃀp 1 (Mq2*)L L攴L?^L#0#0+Remarkr3.1.8.kWVenorwgetsomesimplepropSertiesoftheunderlyingfunctorsUc:URB-M0offd ! K-M0offd$aresp.8U:B-C5omoffd!#"=!2wK-M0offd$athatareeasilyvreri ed.*vʍU1(M N@)UR=U(M@) U(N@);U1(f gn9)UR=f gn9;U1(K)UR=K;U1( )UR= ;U1()=;U1()=:ProblemJ#3.1.17.WVeharveseenthatinrepresentationtheoryandincorepresen-tationptheoryofquanrtumgroupssuchasKG,lU@(g),SLq(2),Uq(slC(2))ptheordinarytensorEproSduct(inK-M0offd 9)oftrwoE(co-)reprensentationsisinacanonicalwayagaina(co-)reprensenrtation.Q FVortwoHV-moSdulesM3andNdescribSethemodulestructureonM N+if(1)+HB=URKG:8gn9(m n)=::: forgË2G;(2)+HB=URU@(g):8gn9(m n)=::: forgË2g;(3)+HB=URUq(slC(2)):/((a)CѬE(m n)UR=:::,/' (b)CѬFƹ(m n)UR=:::,0uD(c)CѬKܞ(m n)UR=:::+〹fortheelemenrtsE;FS;K12URUq(slC(2)).FVor,ltrwoKG-moSdulesMmPandNthestructureisgn9(mn n)C=gmn gn,lforg3|2CG.FVorfU@(g)-moSdulesitisgn9(mJ n)=gmJ n+m gnfforgJ 2g. {FVorUq(slC(2))-moSdules0itisE(mS n)UR=mS En+Em Kܞn,ȯFƹ(m n)UR=K21 9mS Fn+Fm n,Kܞ(m n)UR=Km Kn.URemark3.1.9.Let AandBbSealgebrasorver acommrutative ringK.LetfR:SA!B\bSe#Vahomomorphismofalgebras.ThenwrehaveafunctorUf ੹:iB-M0offdC!A-M0offdmHwith3Ufw(BN>M@)'=A c MtsandUfw(gn9)=gȹwheream:=fG(a)mfora2AandmUR2M@.8ThefunctorUf aǹisalsocalledforffgetfulorunderlying35functor.The2lactionofAonaB-moSduleMsPcanalsobeseenasthehomomorphismAURn!1B!n߹End&{(M@).I7 oSN1.pREPRESENT:ATIONS!OFHOPFALGEBRASjs73VWVedenotetheunderlyingfunctorspreviouslydiscussedbryIUA 36:URA-M0offd ! K-M0offd$aresp.-UB :B-M0offd ! K-M0offd 9:Prop`osition]d3.1.10.beLffetRfL:MB8S!< Cbeahomomorphismofbialgebras.yThenUf Tsatis es35thefollowingprffoperties:9䈍ʍ^Ufw(M N@)UR=Uf(M@) Uf(N@);^Ufw(g h)UR=g h;^Ufw(K)UR=K;^Ufw( )UR= ;33Ufw()=;Ufw()=;^UBN>Ufw(M@)UR=UC(M);^UBN>Ufw(gn9)UR=UC(g)::zProof.@_Thisx/isclearsincetheunderlyingK-moSdulesandtheK-linearmapsstaryuncrhanged. nTheނonlythingtocheckisthatUf UgeneratesthecorrectB-moSdulestructureonthetensorproSduct.GFVorUfw(M5 \QN@)x=M \QN0|wrehaveb(m\Q n)x=fG(b)(m@ n)UR=Pf(b)(1) \|m@ f(b)(2) \|nUR=Pf(b(1) \|)m@ f(b(2) \|)nUR=Pb(1)m@ b(2)n. t&Remark3.1.11.trLetβCPandD"@bSecoalgebrasorverβacommrutativeβringK.LetfQ:URC1K{!DebSe ahomomorphismofcoalgebras.ThenwrehaveafunctorUfq:URCܞ-FC5omoffd-J!nDS->6C5omoffd withޤUfw(2CM@)UR=2D MandUfw(gn9)=gLݹwhereD =(f7 81)C t:M6!Cc M!nD6 M@.8AgainthefunctorUf aǹiscalledforffgetfulorunderlying35functor.WVedenotetheunderlyingfunctorspreviouslydiscussedbry?a"UC t:URCܞ-FC5omoffd!#"=!2wK-M0offd$aresp.-UD :DS->6C5omoffd!#"=!K-M0offd 9:Prop`osition]d3.1.12.beLffetRfL:MB8S!< Cbeahomomorphismofbialgebras.yThenUfq:URCܞ-C5omoffd!#o!2qDS-C5omoffd!satis es35thefollowingprffoperties:9䈍ʍ^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)::zProof.@_WVelearvetheproSoftothereader.t&Prop`ositionӘ3.1.13.%'LffetvHckbeaHopfalgebra./ LetMandNbffebeHV-modules.ThenDٹHomb(M;N@),BtheDsetK-lineffarmapsfromMtoN@,BbecomesanHV-moduleby(hfG)(m)UR=Ph(1) \|f(S׹(h(2)m).fiThis35structurffemakeswHomV:URHV- MoffdHV-Moffd!!DbHV-Moffda35functorcffontravariant35inthe rstvariableandcffovariantinthesecondvariable.J7 74# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVProof.@_The΅mainparttobSeprorved΅isthattheactionHn Homm(M;N@)v!Homy(M;N@)msatis estheassoSciativitrylaw. o1Letf_2J`Hom(M;N@),h;k}2J`HV,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.bpCorollary3.1.14.1LffetM4beanHV-module. ThenthedualK-moduleM@2 ݹ=Homy(M;K)35bffecomesanHV-moduleby(hfG)(m)UR=f(S׹(h)m).Proof.@_ThezspaceKisanHV-moSdulevia"UR:HB\3!K.)Hencezwrehave(hfG)(m)UR=Ph(1) \|fG(S׹(h(2)m)UR=P"(h(1) \|)f(S׹(h(2)m)UR=f(S׹(h)m).'"C2.MonoidalCategoriesFVorourfurtherinrvestigationsweneedageneralizedversionofthetensorproSductthatwrearegoingtointroSduceinthissection.iThiswillgiveusthepSossibilitytostudymoregeneralvrersionsofthenotionofalgebrasandrepresentations.XDe nition3.2.1.Amonoidal35cffategory(ortensor35category)consistsofacategoryC5,acorvXariantfunctor UR:C]C"!wfC5,calledthetensor35prffoduct,anobjectIF2URC5,calledtheunit,naturalisomorphisms$aʍjVD (A;B;Cܞ)UR:(A B) C1K{!A (BE Cܞ);jVD(A)UR:I+ An!1A;jVD(A)UR:A IF``!A;calledfassoffciativity,Bleftgunitandrightunit,sucrhthatthefollowingdiagramscommute:ETፍK((A B) Cܞ) Dx(A (BE Cܞ)) Dol:2fd3ά-̯qі (A;Bd;C) 1E[A ((BE Cܞ) DS)=̟:2fd3ά-̯ (A;Bd C;D| (A Bd;C;D?`4?`33 1Jand35wehave(I)UR=(I).lProof.@_First)wreobservethattheidentityfunctorIdCandthefunctorI. ի- (areisomorphicSGbrythenaturalisomorphism.rThuswehaveIf fOf=gI gu=)gfOf=gn9.InthefollorwingdiagramYvu&((I+ I) A) Bv`&(I+ (I A)) B}sH2fd)ά-on0 1vv7K&I+ ((I A) B) sH2fd)ά-pč sv ( 1) 1X8쟁 Qb8쟈Ql8쟏M^Qv8쟕Q|L0Q|L0ss(1 ) 1z̟ z̟z̟M^z̟l0l0+s1 ( 1)Oe̟ Ee̟;e̟M^1e̟+l0+l0+rӹ(I+ A) BI+ (A B):lA2fdG)ά-č sK*Ǡ|z@feK\Ǡ?CdAE se*Ǡ|z@fee\Ǡ?jl1 &Π@feYܟΠ?Cd \ Π@feDܟΠ?jRX1rI+ (A B)I+ (A B):l:2fdG)ά-Z֯Z1u&(I+ I) (A B)7K&I+ (I (A B))}32fdά-č ꃀv  (1 1)X8Ǡb8l8rLv8Ǣ|Lz|Lz3+G1 +l Q5l攴Q?l?^QIlQOe̟0QOe̟0sL7 76# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVallsubSdiagramscommruteexceptfortherighthandtrapSezoid.SinceallmorphismsareisomorphismstherighrthandtrapSezoidmustcommutealso.Hencethe rstdiagramoftheLemmacommrutes.Inasimilarwrayoneshorwsthecommutativityoftheseconddiagram.FVurthermorethefollorwingdiagramcommutesej^pI+ (I I)b(I+ I) I\A2fdάč TI+ (I I)\A2fdά-č Ԡ`I+ IWM{-1 F`@F`@F`@@R|GȚƖF`F`F` ԠRI+ IWM- 1F`@F`@F`@@R&1 $F`F`F` տGIuG?`@?`@?`@Ɩ@ƖRuG?`?`?` HerethelefthandtrianglecommrutesbythepreviouspropSertyV.U Thecommuta-tivitryoftherighthanddiagramisgivenbytheaxiom.Thelowersquarecommutessince%isanaturaltransformation.Inparticular(1 )=(1 ).Since%isanisomorphismandI+ -P ԰ =KId!YC*wregetUR=. cProblem3.2.18.FVorXmorphismsf:PIA[!MUof%linearmaps(fg :9Vg (!aWgjg2G)%asmorphisms.XThecompSositionis(fgjg2G)(hgjgË2URG)=(fghgjgË2G).񤍑Lemma?3.2.7.U7Lffet̙Gbeamonoid.2ThenM2G 1isamonoidalcategorywiththetensor35prffoductwJ(VgjgË2URG) (WgjgË2URG):=(rßM'؍h;k62G;hk=g1Vh Wk#jg2G):!YProof.@_Thisisaneasyexercise.G\Problema<3.2.19.dLetGbSeamonoid.ShorwthatM2G IisamonoidalcategoryV.WheredounitandassoSciativitrylawsofGentertheproSof8?ZDe nitionV3.2.8._oLetGbSeaset.iA]vrectorspaceVAtogetherwithafamilyofsubspaces(Vg*PURVpjgË2G)iscalledG-grffaded,ifV¹=URgI{2GFVgholds.Let/(V;(VgjgË2URG))and(Wr;(WgjgË2URG))bSeG-gradedvrectorspaces.A/nlinearmapfQ:URV M!`WniscalledG-grffaded,iffG(Vg)WgforallgË2G.ӍTheG-gradedvrectorspacesandG-gradedlinearmapsformthecategoryM2[G]eofG-grffaded35vectorspaces.Lemma!3.2.9.*LffetGbeamonoid.M!ThenM2[G]Xisamonoidalcategorywiththetensor35prffoductVG W,wherethesubspaces(VG Wƹ)g 3arede nedby}Y$(VG Wƹ)g*P:=.|X'؍URh;k62G;hk=g4\Vh Wk#:!YProof.@_Thisisaneasyexercise.G\Problem3.2.20.LetGbSeamonoid.8ShorwthatM2[G](isamonoidalcategoryV.De nition3.2.10.(1)Achain35cffomplexofK-moSdules"M6=UR(:::'0x"@q3ЍuK!'ʟM2'0 m@q2ЍVj!M1'0 m@q1ЍVj!M0)consistsofafamilyofK-moSdulesMi"yandafamilyofhomomorphisms@n d:iMn ~D!Mn1with$@n1@n `̹=|0.ThiscrhaincomplexisindexedbythemonoidN0.Onemayalso/considermoregeneralcrhaincomplexesindexedbyanarbitrarycyclicmonoid.ChaincomplexesindexedbryN0 QN0 arecalleddoublecomplexes.Somuchmoregenerall;crhaincomplexesmaybSeconsidered.WVerestrictourselvestochaincomplexesorverN0.LetfvMZandNbSecrhaincomplexes.KAfVhomomorphismofchaincffomplexesfp :( Mfk!NconsistsԦofafamilyofhomomorphismsofK-moSdulesfn:URMn -!lNn |sucrhԦthatfnP@n+1=UR@n+1fn+1otforalln2N0.Thecrhaincomplexeswiththesehomomorphismsformthecategoryofchaincom-plexesK-C5ompL.N<7 78# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVIfe@M$andNarecrhaincomplexesthenweformanewchaincomplexMO kNwith[(M N@)n qԹ:=Ʉ2nRAi=0AVMi Nni[and@p:Ʉ(M N@)n q _!T(M N@)n1givren[by@(miq  Enni))I[:=(1)2id@M (mi) nni5+miq @(nni)).ThiszisoftencalledthetotalcomplexsassoSciatedwiththedoublecomplexofthetensorproductofMZWandN@.$ThenitiseasilycrheckedthatK-C5omp7isamonoidalcategorywiththistensorproSduct.(2)AcoScrhaincomplexhastheform7M6=UR(M0'0 m@q0ЍVj!M1'0 m@q1ЍVj!M2'0 m@q2ЍVj!:::,)>withd@i+1AV@i=$0.5ThecoScrhaincomplexesformamonoidalcategoryofcocrhaincom-plexesK-C5offcomp"0.|Problemʓ3.2.21.Shorw `thatthecoSchaincomplexesformamonoidalcategoryK-C5offcomp"0.De nition3.2.11.Let(C5; )and(DUV; )bSemonoidalcategories.8Afunctor>?Fc:URC"!wfDtogetherwithanaturaltransformationqs(M;N@)UR:F1(M) F1(N)URn!1F(M N)andamorphismX}0V:URID @!F1(ICm)ȍiscalledweffakly35monoidalifthefollorwingdiagramscommuteBO}(F1(M@) F(N@)) F(Pƹ)tF1(M N@) F(Pƹ)L:2fd2ά-'t4O 1BH(F1((M N@) Pƹ)̟:2fd'8ά-')9U4:Ǡ@feUglǠ?thereis+exactlyone@(midڹ)UR=m20RAi12M+sucrhthat/&&s2(midڹ)UR=mi yn9 i}+@(mi) xyn9 i1:+〹Since s2(m20RAi1AV)=m20RAi1 _yn92i1$foralli2Nwreseethat@(midڹ)2Mi1AV.ڨSowre+harvede ned@/:hMi x!NMi1AV.!FVurthermorewreseefromthisequationthat+@22g (midڹ)+B=0foralli+B2N.u%Hencewrehaveobtainedachaincomplexfrom+(M;s2).=IfSwreapply(1 )s2(m)9=mSthenwegetm9=PfmiĹwithSmi !2Mi+〹henceZM6=URi2N֥Midڹ.ThistogetherwiththeinrverseZconstructionleadstothe+requiredequivXalence.)(2)+Shorw*thatthecategoryK-C5offcomp%ܹofcoSchaincomplexesismonoidallyequiv-+alenrttoB-C5omoffd`,whereBiscrhosenasinExample3.2.13.(3)+ShorwthatthebialgebraBfromExample3.2.13isaHopfalgebra.aDWVecangeneralizethenotionsofanalgebraorofacoalgebraintheconrtextofamonoidalcategoryV.8Wede neQ7 2.pMONOID9AL!CA:TEGORIES81VDe nition3.2.15.(Let@CubSeamonoidalcategoryV.^AnalgebrffaoramonoidinCconsistsofanobjectAtogetherwithamrultiplicationr\ʹ:Azq A\vU!!AٹthatisassoSciativreDލA AHA32fdYά-lrZ8A A AZ AA AQ{fdBw`ά- Atid rH Ǡ*Ffe<Ǡ?pRr 1H2Ǡ*FfeeǠ?`Ǡ@fe>괟Ǡ??<C4rkcandhasaunitË:URIF``!AsucrhthatthefollowingdiagramcommutesLZ"I+ APUR԰n9=APUR԰n9=A IZgA A苼{fd,cά-iidh HZǠ*Ffe⌟Ǡ?`xI{ idH(XǠ*Ffe(̟Ǡ?`->LrPA A"A:~32fd{ά-lݹ.rHk}XidÎ ҁ H͎ ׁ H׎ ܁ H H H H H H džH džj׍Let$AandBE*bSealgebrasinC5.#_Amorphismofalgebrffas$fQ:URAn!1Bis$amorphisminCݹsucrhthat0,[ikABu32fd6^ά- V|fZ]/A AZkpBE Bo۟{fd@ά-inf fHm9Ǡ*FfemkǠ?\1rX.AHǠ*FfeHǠ?krX.BF*and[bZ;BI'+X.A56ׁ 06 +6 &6 %n>%n> H'T*~X.BBoׁ AGo ALo AQo ARN>ARN>UAVjB(n32fd*ά- ;/f1commrute.Remark3.2.16.ItHisobrviousthatthecompSositionoftwomorphismsofalgebrasis0againamorphismofalgebras. Theidenrtity0alsoisamorphismofalgebras.ThruswreobtainthecategoryAlg f(C5)ofalgebrasinC.De nition:3.2.17.LetCbSeamonoidalcategoryV.hxAcffoalgebraoracffomonoidinC?consistsofanobjectChmtogetherwithacomrultiplicationg:A*!A` AϹthatiscoassoSciativreC^CF CCF C C/32fdAJά-aid< ZxCZnCF C5<{fdY?ά-̍JNH纟Ǡ*FfeǠ?`SH V:Ǡ*Ffe lǠ?э; idR!7 82# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVormorepreciselyCHd(CF Cܞ) C VCF (C Cܞ)+t32fdά-č) aCF C32fd%9Ѝά-ׁ idԠmCԠCF CzD:2fdά-ζ .FŸǠ@fe.yǠ?񍍒id rbŸǠ@ferǠ??CF CKL{fd|z@ά-̍HʟǠ*Ffe0Ǡ?`}i H! Ǡ*Ffe!<Ǡ?э&id/ xCF CKI+ CP1԰J׹=ܙCP1԰J׹=CF I:E32fd) ά-a} idHk}l:p?CLfh!֩DisamorphisminCݹsucrhthat6ȍ[]uCF CD6 D$;32fdyά- f fbZim6CbZzDv)۟{fd5氍ά-ifHmYǠ*FfenǠ?\X.CHQٟǠ*Ffe Ǡ?7X.DF*andH;~I-"X.C%Oׁ A*O A/O A4O A5rΟ>A5rΟ>UH-Tf"X.DRnׁ Mn Hn Cn B.>B.> bZ CbZVD)M{fd*(pά-i;Of5commrute.鍑Remarkh3.2.18.cItEisobrviousthatthecompSositionoftwomorphismsofcoalge-brasisagainamorphismofcoalgebras.Theidenrtityalsoisamorphismofcoalgebras.ThrusweobtainthecategoryC5offalg*(C5)ofcoalgebrasinC.gRemark<3.2.19.|Observreothatthenotionsofbialgebra,Hopfalgebra,andco-moSduleC$algebracannotbegeneralizedtoanarbitrarymonoidalcategorysincewreneed1ltoharve1lanalgebrastructureonthetensorproSductoftrwo1lalgebrasandthisrequires9%ustoinrterchange9%themiddletensorfactors.$YTheseinrterchanges9%or ipsareknorwnFdunderthenamesymmetryV,]SquasisymmetryorbraidingandwillbSediscussedlateron.%LV;3.{DualObjectsArt!qtheendofthe rstsectioninCorollary3.1.14wesawthatthedualofanHV-moSduleScanbeconstructed.WVedidnotshorwthecorrespondingresultforcomodules.InkfactsucrhaconstructionforcomoSdulesneedssome nitenessconditions.Withthisrestriction'thenotionofadualobjectcanbSeinrtroducedinanarbitrarymonoidalcategoryV.S/7 $3.pDUAL!OBJECTS83VDe nition2g3.3.1.OJLet(C5; )bSeamonoidalcategoryM2]C9#beanobject. Anobject,sM@2 72OCߨtogetherwithamorphismev:M@2[ sM3!IiscalledaleftodualforM+ifthereexistsamorphismdbN*:URIF``!M M@2 됹inCݹsucrhthatNʍx(M26dbr 1p ]!M M@2 M21 evp6 !@M@)UR=1Mm߹(M@22 V:1 dbp Ҳ( ! M@2 M M@22 V:ev" 1p Ua!!TM@2)UR=1M" D:NAmonoidalcategoryiscalledleft35rigidifeacrhobjectM62URCݹhasaleftdual.Symmetricallywrede ne:danobject2oM62URCbtogetherwithamorphismev2:Mh4 'P2TM!Iis,calledarightodualforMmpifthereexistsamorphismdb6:zIЈ!2VMh ׄMinCsucrhthatfʍxU(M261 dbp h!9M 2jM M26evb 1p@ ]!M@)UR=1MmW(2M26dbr 1p ]!2"dM M 2jM21 evp6 !@2 M@)UR=1UTM D:썹Amonoidalcategoryiscalledright35rigidifeacrhobjectM62URCݹhasaleftdual.ThemorphismsevkanddbarecalledtheevaluationrespSectivrelythedual35bffasis.[Remarkus3.3.2.Ify(M@2;ev /)isaleftdualforM"]thenobrviously(M;ev)isarighrtdualforM@2 됹andconrverselyV.8OneusesthesamemorphismdbN*:URIF``!M M@2.LemmaW3.3.3._Lffet9(M@2;ev /)bealeftdualforM@.juThenthereisanaturaliso-morphismPTf\MorYC (- M;-33)PUR԰n9=Mor%5C*(-35;- M@ );Æi.35e.fithefunctor-  M6:URCn!FCjisleftadjointtothefunctor- M@2 V::URCn!FC5.MProof.@_WVe# givretheunitandthecounitofthepairofadjointfunctors.WWVede ne(A)%:=1A ۪ ƹdb1:A>!cA M> M@2 eandd (B)%:=1B L ƹev:B M@2 Mf!B.Theseareobrviouslynaturaltransformations.8WVehavecommutativediagrams6 J(A MQlA M M@2 M5vl32fdY]ά-QF((A)=I51X.A ^ db | 1X.MSA M@)UR=1A M:|32fdZPά- F((A)=lÍ l1X.A ^ 1X.M ev퍹anda\(BE M@2h+BE M@2 M M@26<32fdSά-QGv(Bd)=`άnZg1 evFTN@2 M M@2d32fd$>pά 1 f 1H`*Ffe64`? 1 gI{ 1 f 1O@^gN@2~ uevpO 1>_ZHgZRnZ\vZf}ZpZzZZZ'dZ'd~ Su1 db'd'd%'dƥv'd%l'dեb'd%X'd䥀N'd%D'd>>=獍I`"yProblem3.3.25.(1)InVthecategoryofN-gradedvrectorspacesdetermine+allobjectsM+thatharvealeftdual.(2)+In;thecategoryofcrhaincomplexesK-C5omp 2determineallobjectsMthat+harvealeftdual.(3)+InLthecategoryofcoScrhaincomplexesK-C5offcomp'׹determineallobjectsM+〹thatharvealeftdual.(4)+Let (M@2;ev /)bSealeftdualforM@.5ShorwthatdbD:URIF``!M M2 isuniquely+determinedbryM@,M2,andev.8(Uniquenessofthedualbasis.)(5)+Let(M@2;ev /)bSealeftdualforM@.WShorwthatev:fM2 M=!( Iqisuniquely+determinedbryM@,M2,anddb.Vd7 86# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVCorollary3.3.9.LffetMM;Nhavetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesMDύE1N N@2AN M@2:z̞32fd2ά-漍1 fǟ-:ZIZ]M M@27{fd& @ά-udb綟X.MHpǠ*FfeǠ?fdbjX.NHjǠ*FfeǠ?`f 1%􍍍Proof.@_ThefollorwingdiagramcommutesXύ$M$N N@2 M,Tfd@`ά-bdb̄ 1^ڍYN^ڍٚN N@2 N;ԟrfd!`ά-ٜdb̄ 1@9Ԍfem$Ԍ?'Åf@rԌfe⤟Ԍ?'$1 1 f@NrǠfe⤟Ǡ?ލ$1 ev@1HHHqƕHqƕj[This"implies(f ݹ1M" D)db.MF=((1N ! evgNR)(1N 1N"?u fG)(db 0N'r 1M ) 1M" D)db.MF=(1N b* Hev NY 1M" D)(1N H1N" ~ f 1M")(db 0N'r 1M 1M")db.Mp=UR(1N b* ev NY 1M")(1N b* 1N"W 4f| 1M" D)(1N N 1N" dbBM߹)db.N=(1N N (ev 1Ns 1M" D)(1N" f| 1M" D)(1N" db 0M ))db.N|Ĺ=UR(1N fG2)db.N'r.kCorollary3.3.10.fLffetM;Nhavetheleftduals(M@2;ev /M)and(N@2;ev /Ns)andletfQ:URM6!QNtbffe35amorphisminC5.fiThenthefollowingdiagramcommutesMiN@2 NrI:32fd$ߠά-MשTev #X.NZN@2 MZ.M@2 MԄ{fdؐά-q<͝Mfǟ-:7 1H"Ǡ*Ffe TǠ?`91 fHOǠ*FfeԟǠ?-5Tev#X.M̴Proof.@_This;statemenrtfollowsimmediatelyfromthesymmetryofthede nitionofaleftdual.ckExample3.3.11.WRLetpM62URRJ-M0offd 9-$aR'bSeanRJ-R-bimodule.xThenpHomR"{(M:;R:)isǂanRJ-R-bimoSduleǂbry(rfGs)(x)=rfG(sx). pFVurthermoreǂwrehavethemorphismev:URHom۟R"n(M:;RJ:) R ;M6!Rde nedbryev(f Rm)UR=fG(m).(DualdBasisLemma:)-Butthisde nitionisequivXalenrttotheusualde nition.Wv*7 $3.pDUAL!OBJECTS87VLet/{M \2xRJ-M0offd 9-$aRJ.ZMp_asarighrtR-moSduleis nitelygeneratedandprojectivrei M+hasaleftdual.8TheleftdualisisomorphictoHomd1R#WĹ(M:;RJ:).IfjMR is nitelygeneratedprojectivrethenweusedb:URRn'!{M R ;Hom!;R(ι(M:;RJ:)withdbN(1)=P*In U_Ii=1#mi`) R m2idڹ. BInfactwrehave(1O R ev)(db. R1)(m)=(1 R$ev 1)(Pmi R 9m2i Rm)ٜ=PGmidm2i(m)ٜ=m.>WVeharvefurthermore(ev / R1)(1E Rdb 0)(fG)(m)UR=(ev / R1)(P* n U_ i=1f\ R miy Rm2idڹ)(m)UR=Pf(midڹ)m2i(m)UR=f(Pmidm2i(m))=URfG(m)forallm2M+hence(ev / R1)(1 R ;dbk)(fG)=f.Conrversely~ifMhasaleftdualM@2 thenev:M@2 R OME _t!d{Rde nesaho-momorphismC:M@2 x ![Hom5R<ȱ(M:;RJ:)inR-M0offd 9-$aRƍbry(m2)(m)=ev6(m2+ R "m).WVeZde neP*n U_i=1![mi+  m2i%:=Kdb{(1)K2M M@2,thenZm=(1 ev]<)(db. 1)(m)=(1 $ev 1)(Pmiݬ xm2i m)UR=Pmid(m2i)(m)חsothatm1;:::ʜ;mn2URM{and(m21);:::ʜ;(m2nP)UR2HomyRm(M:;RJ:)ZformadualbasisforM@,vi.e.Mʹis nitelygeneratedandprojectivreasanRJ-moSdule.8ThrusM@2 됹andHomd1R#WĹ(M:;R:)areisomorphicbrythemap.AnalogouslyoHom0R#ù(:M;:RJ)oisarighrtdualforMi Mis nitelygeneratedandprojectivreasaleftRJ-moSdule.XProblem;3.3.26.}FindMRanexampleofanobjectM6inamonoidalcategoryCthathasaleftdualbutnorighrtdual.De nition3.3.12.GivrenobjectsM;N\inC5. Anobject[M;N@]iscalledaleftS/innerHom#ofMdٹandNifthereisanaturalisomorphismMorYCDZ(-j M;N@)Pj԰=Mor5C(-;[M;N@]),i.e.8ifitrepresenrtsthefunctorMor Cd(-P M;N@).IfthereisanisomorphismMorNCv(P xRM;N@)PW{԰pb=MMor):JC. (PS;[M;N@])naturalinthethreevXariableM;N;PnthenthecategoryCݹiscalledmonoidal35andleftcloseffd.If%thereisanisomorphismMor[ΟCɍ(M ;PS;N@)Pm԰=0Mor)fڟC.ԙ(P;[M;N@])%naturalinthethreevXariableM;N;PnthenthecategoryCݹiscalledmonoidal35andrightcloseffd.If|MuhasaleftdualM@2 }yinC/ƹthenthereareinnerHoms "[M;- ]de ned|bry[M;N@]UR:=N M2.8Inparticularleftrigidmonoidalcategoriesareleftclosed.Example3.3.13.(1)Theqcategoryof nitedimensionalvrectorspacesisa+monoidalcategorywhereeacrhobjecthasa(leftandright)dual./(Henceitis+(leftandrighrt)closedandrigid.(2)+LetBanbSethecategoryof(complex)Banacrhspaceswherethemorphisms+satisfyok."fG(m)kkmki.e.z7themapsarebSoundedbry1orcontracting.+BanisamonoidalcategorybrythecompletetensorproSductM*:bV@ ;N@.$InBan+〹existsYaninnerHomfunctor[M;N@]thatconsistsofthesetofbSoundedlinear+mapsJfromMtoNmadeinrtoaBanachspacebyanappropriatetopSologyV.+ThrusBanisamonoidalclosedcategoryV.(3)+LetEHbSeaHopfalgebra. 8ThecategoryHV-M0offdofleftHV-modulesisa+monoidalocategory(seeExample3.2.52.).ȹThenHomyK#[(M;N@)isanobject+inHV-M0offd$abrythemultiplicationͣ˼(hfG)(m)UR:=Xh(1) \|f(mS׹(h(2))+asinPropSosition3.1.13.X7 88# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYV=〹HomV] K](M;N@)y2isaninnerHomfunctorinthemonoidalcategoryHV-M0offd 9.+Theisomorphism}:HomK$L(PS;HomyK!2i(M;N@))P԰3d=QHom,ڟK2(P _M;N@)canbSe+restrictedtoanisomorphism: bHom{\NHu(PS;HomyK!2i(M;N@))PUR԰n9=Hom(yH0ȹ(PLn M;N@);+〹bSecauseFq(fG)(h(pr m))K=(f)(Ph(1) \|pr h(2)m)K=POfG(h(1)p)(h(2)m)=+〟P8-(h(1) \|(fG(p)))(h(2)m)&W=Ph(1) \|(f(p)(S׹(h(2))h(3)m))&W=h(f(p)(m))=h((f)+(p m))pandconrverselyp(h(fG(p)))(m)UR=Ph(1) \|(f(p)(S׹(h(2))m))UR=Ph(1) \|((f)+(pOt S׹(h(2) \|)m))UR=P(fG)(h(1)pOt h(2)S׹(h(3))m)UR=(fG)(hpOt m)UR=f(hp)(m).+ThrusHV-M0offd$aisleftclosed.=IfXM2YHV-M0offd'isa nitedimensionalvrectorspacethenthedualvector+space8M@2 K:=cHomSK# ι(M;K)againisanHV-moSdulebry(hfG)(m)c:=f(S׹(h)m):+〹FVurthermoreM@2 됹isaleftdualforM+withthemorphisms(x^db:URK317!X ㇍ imi m i,2M M@ +〹andS}uev:URM@  M63f m7!fG(m)2Kٍ+〹where miandm2iareadualbasisofthevrectorspaceM@. Clearlywehave+(1t ev )(db. 1)N^=1MOand_(ev / 1)(1t db)N^=1M"죹sinceMTCisa nitedi-+mensional-)vrectorspace.eWVehavetoshowthatZRdbandZRevJareHV-moSdule+homomorphisms.8WVeharve'`ʍ+#5(hdb.(1))(m)UR=(h(Pmi m2idڹ))(m)=(Ph(1) \|mi h(2)m2idڹ)(m)UR=+#5P7(h(1) \|midڹ)((h(2)m2i)(m))UR=P(h(1) \|mi)(m2i(S׹(h(2))m))UR=+#5P9h(1) \|S׹(h(2))mUR="(h)m="(h)(Pmi m2idڹ)(m)="(h)db.(1)(m)=+#5db81e("(h)1)(m)UR=dbc(h1)(m);(: +〹hencehdb.(1)UR=dbc(h1).8FVurthermorewrehave: ʍ(hev /(f m)UR=hfG(m)=Ph(1) \|f(S׹(h(2))h(3)m)UR=P(h(1) \|f)(h(2)m)UR=(ꉟP74evC+e(h(1) \|f h(2)m)UR=ev(h(f m)):(4)+LetQ\H>bSeaHopfalgebra.ThenthecategoryofleftHV-comodules(seeExam-+ple?3.2.53.)8isamonoidalcategoryV.LetM'2#HV-C5omoffd!bSea nitedimen-+sionalvrectorspace.{1LetmiebSeabasisforMAandletthecomultiplicationof+the(comoSdulebes2(midڹ)=PHUhij | ǘmjf .aThen(wrehave(hikl)=PHUhij | ǘhjvk .+M@2 :=,HomfK$(M;K)EbSecomesaleftHV-comodules2(m2jf ),:=PS׹(hijJ)M m2idڹ.+Onevreri esthatM@2 됹isaleftdualforM@.Lemma3.3.14.Lffet35M62URCjbeanobjectwithleftdual(M@2;ev /).fiThen1.fiM M@2 4is35analgebrffawithmultiplication: L rUR:=1M . ev@ 1M".:M M@  M M@  V: !M M@ and35unitSuUR:=db:IF@!M M@ ;Y7 $3.pDUAL!OBJECTS89V2.fiM@2 Mtis35acffoalgebra35withcffomultiplicationҍKyйUR:=1M" db 1M B:M@  M6!QM@  M M@  Mand35cffounitm5"UR:=ev@:M@  M6!QI:6=Proof.@_1.nThe#,assoSciativitryisgivenby(ro 1)riC=(1M _ oev 1M"X o1M 1M" D)(1M " ev4 1M")&=1M ev4 ev- 1M"j=(1M 1M"w 1M ev4 1M" D)(1M ev / 1M" D)UR=(1 r)r.The&axiomfortheleftunitisr(u 1)UR=(1M ¹ev 1M")(db. 1M . 1M" D)UR=1M 1M" D.2.8isdualtothestatemenrtforalgebras.6=LemmaIw3.3.15.ۥ1.WLffetAbeanalgebrainCandleftM62URCbffealeftrigidobjectwith)ileftdual(M@2;ev /).c%TherffeisabijectionbetweenthesetofmorphismsfQ:URA MUfk!M making(MaleftA-moffduleandthesetofalgebramorphismsVze*f !:URA!M| ;M@2.2. )LffetqCbeacoalgebrainCΦandleftM62URCbffealeftrigidobjectwithleftdual(M@2;ev /).TherffeɋisabijectionbetweenthesetofmorphismsfQ:URM6!QM C)makingM oaright*Cܞ-cffomodule35andthesetofcffoalgebra35morphismsV\e*f.:URM@2 M6!QC.6=Proof.@_1." By$Lemma3.3.14theobjectM_ M@2 isanalgebra.GivrenfQ:URA M!Msucrh;thatMbSecomesanA-module. ByLemma3.3.3wreassociateVe*f1ܹ:=(f2 ꊹ1)(1 db):A!A M+n M@2 z!M M@2.R]TheH|compatibilitryofVqe*fwiththemrultiplicationisgivenbythecommutativediagramFvJA AvpAnLsH2fd[`ά-p rZsWA A M M@2Z A M M@2wfd6ά-92r 1 1$A M M@2$}M M@2ƹfdM@ά-`吏f 1^ڍxA M M@2 M M@2^ڍ A M M@2rfd ά-l==ne1 1 ev { 1&jM M@2 M M@2^1M M@232fdȒά-rc1 ev { 1sZ} Ǡ|z@feZ<Ǡ?+㎍FerEsfJj% ㎍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@ͻʍ|1|U>|UY*X\16I@@I@JI@TI@^I@bMܟo@bMܟoR@b9f 1>|HH|HR|HZ|ƕHZ|ƕjBTheunitaxiomisgivrenby:&}%IM M@2ȴRfd1ʀά-Ԡ db{AWA M M@2d32fd$ά-ׁ=1 dbHuM M@232fdά- x'f 1 Ǡ fe54Ǡ?+vu 5ŸǠ fehǠ?Nu 1 N1 1ٴHٴHٴHٴH +uH +ujConrverselyletg::MA!'M @M@2 ݹbSeanalgebrahomomorphismandconsideruegҹ:=(1 evJ)(g" 1)nQ:A M5!7M M@2 M5!M@.dThenXM:Ap1 1 evHLF`CLF`>LF`9LF`4LF`/LF`*LF`&欟`&欟` s^MgI{ g 1C}`XMM`XW`XaM`Xk`XuM`X`XM`X`XM`X`XM`XXzPMigI{ 1CǠMGWǠaGkǠuGǠGǠGǠGۿۿ:ꬾD1 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 -and5荍oA MM M@2 Mu32fd ά-o͍MgI{ 17M_$32fd,Ѝά-r1 ev {MǠ@fe$Ǡ?ꬾm}Ju 1񍍒,db 1h?`Hh?`Hh?`Hh?`Hh?`Hh?`HydHydjjd1٤ۿ`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`X٤`X٤?`XXzpAcommrute.2.(M@2;ev /)isaleftdualforMĕinthecategoryC6ifandonlyif(M@2;db.)istherighrtTdualforMinthedualcategoryC52op R.wXSoifwedualizetheresultofpart1.wXweharvetocrhangesides,hence2.@4.[FinitereconstructionTheFendomorphismringofavrectorspaceenjoysthefollowinguniversalpropSertyV.Itisavrectorspaceitselfandallowsahomomorphismq:End~(Vp) V] '!V..Itisunivrersaly5withrespSecttothispropertryV,i.e.ifZUӹisavectorspaceandfQ:URZ V M!`Visahomomorphism,*thenthereisauniquehomomorphismg3:ŬZJ!End-B(Vp)sucrhthatAx xEnd-7(Vp) V2V\Ğ32fd&ά-g܍ H`tf@ԟ>RZZF VŸǠ*FfeǠ?q-gI{ 1$commrutes.ThealgebrastructureofEndg(Vp)comesforfreefromthisunivrersalpropSertyV.If,5wrereplacethevectorspaceVȥbyadiagramofvectorspaces!3:D;3!Veffc/RwegetasimilarunivrersalobjectEnd(!n9).>AgaintheuniversalpropSertyinducesauniquealgebrastructureonEndg(!n9).JProblem3.4.27.(1)LetVkcbSeavrectorspace.OShowthatthereisauniversal+vrectorEspaceEandhomomorphismC:EK V@ Z>!VF(suchEthatforeach[s7 А4.pFINITE!RECONSTR9UCTION91V+〹vrectorZYspaceZ6andeachhomomorphismf[n:oZQ Vj!Vɹthereisaunique+homomorphismgË:URZ1K{!EsucrhthatHTCE^ VV932fdL ά-g܍WH`Df?$ׁ @?$ @?$ @?$ @녤>@녤>RZZF VӒǠ*FfeğǠ?q-"gI{ 1p+〹commrutes).-WVecallE}3andUR:E8 f!V M!`Veaveffctor_spaceactinguniversally+onVp.(2)+LetnEandb:Ed V!{V޹bSenavrectorspaceactinguniversallyonVp.P4Show+that]EtuniquelyhasthestructureofanalgebrasucrhthatV͹bSecomesaleft+E-moSdule.(3)+Letz"!k:HDRl)!isVeffc}?bSeadiagramofvrectorspaces. PShowthatthereisa+univrersaltXvectorspaceE(oandnaturaltransformation?:E} f!r!"!⑹(sucrh+thatnforeacrhvectorspaceZJandeachnaturaltransformationfQ:URZ !Ë!2j!+〹thereisauniquehomomorphismgË:URZ1K{!EsucrhthatHTE^ !V!T32fd ά-g܍H`"f˷dׁ @շd @߷d @d @>@>RZVZF !KҟǠ*FfeǠ?q- gI{ 1p+〹commrutes).znWVecallEandz:Em !G!|!naveffctorGJspaceactinguniversally+on!n9.(4)+LetbElyandUR:E* 9!Ë!2j!&bSebavrectorspaceactinguniversallyon!n9.Showthat+EuniquelyBhasthestructureofanalgebrasucrhthat!4bSecomesadiagram+ofleftE-moSdules.ӍSimilar^ considerationscanbSecarriedoutforcoactionsV M!`V( C:or!Ë!2j! Cand#GacoalgebrastructureonCܞ.Thereisonerestriction,1ohorwever.WVe#Gcanonlyuse nitedimensionalvrectorspacesVordiagramsof nitedimensionalvectorspaces.ThiswillbSedonefurtherdorwn.WVeiwrantto ndauniversalnaturaltransformation:,{!?!!n coSend 2(!n9).FVorthispurpSosewreconsidertheisomorphismsPRMorh'CmD(!n9(X);!(X) M@)PUR԰n9=Mor%5C*(!n9(X) j !n9(X);M@)thatmaregivrenbyf7!Y(ev / 1)(1ߚ fG)mandasinverseg7!Y(1ߚ gn9)(db. 1). /WVe rstGdevreloptechniquestodescribSethepropertiesofanaturaltransformation:!!! MθaspropSertiesoftheassociatedfamilygn9(X)k:!(X)2 !(X)k!M@."eWVewillseethatgË:UR!n92{L M!!2jMwillbSeacffone.)ThenwrewillshowthatisauniversalnaturalytransformationifandonlyifitsassoSciatedconeisunivrersal.OIntheliteraturethisiscalledacoSend!".\ڳ7 92# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVThroughout thissectionassumethefollorwing.ELetD_ubSeanarbitrarydiagramscrheme. eLetCW=bSeacocompletemonoidalcategorysucrhthatthetensorproductpreservrescolimitsinbSotharguments.(LetC0ybSethefullsubcategoryofthoseobjectsinaC_thatharveaaleftdual.$Let!Ë:URD3!CbSeadiagraminCsucrhthat!n9(X)UR2C0leforallXF2URDUV,Li.e./!>.isgivrenbyafunctor!0V:URD3!C0./WVecallsuchadiagrama nitediagrffam;xinC5.+QFinallyforanobjectM2Clet!O M:D46M!,CbSethefunctorwith(! M@)(X)UR=!n9(X) M@.fύRemark>3.4.1.uConsiderуthefollorwingcategoryxGeDh.sFVoreachmorphismf&D:EX*!YutherePisanobjectVe*fׂ2wGeD.TheobjectcorrespSondingtotheidenrtityP1X :X!XzisdenotedbrywCeXG2xueaDo.'FVoreachmorphismfG`:aX o! YginD:Mtherearetwo*morphismsQf1Ï:V,e*f3H!w$5!e!6X09ǹandf2:V,e*f3H!w"Te!6Y.inx6eD .kFVurthermoretherearetheidenrtities1fq:V~e*URf K!V!U|e*,*f*)yinx`eD .9Since therearenomorphismswithweXasdomainotherthan(1X)i~:weX7!w&le#nhX2*andU1f:Ve*f-:!V!e*f*pwre onlyhavetode nethefollowingcompSositions(1X)i fj掹:=fjf .Then8xv3eD bSecomesacategoryandwrehave1'eX r۹=UR(1X)1V=(1X)2.]ߍWVede neadiagram!n92 !Ë:x˅eURDA[=!!Cݹasfollorws.8IffQ:URXF``!Yisgiventhen}(!n9  !n9)(V)Re*f)UR:=!(Yp) j !(X)rand ʍ!n9(f1)UR:=!(fG)2j !(1X);4-!n9(f2)UR:=!(1YP)2j !(fG):[?Themwcolimitof!n921 !۰consistsofanobjectcoSend" (!n9)32C togethermwwithafamilyofmorphisms(XJg;X)UR:!n9(X)2j !(X)URn!1coSend2w(!)sucrhthatthediagramsWA"@ԟt>RHSj! M椢Ǡ*FfeԟǠ?ҍT1 e'kcffommutes.xProof.@_LetjcoSend#m(!n9)p2C(togetherjwithmorphisms(V)Re*f)p:!(Yp)2pQ M!(X)!3coSendz(!n9)YbSethecolimitofthediagram!2o' @!;0:xC*eD0J!&C5.SowregetcommutativediagramsnR!n9(X)2j !(X)3!I{(f)-:UT 1lΠlΠlΠlΠlΠÇlΠė쟯Fė쟯F*(X&;X)ylF`HylF`HylF`H ylF`HylF`H!ylF`H"H"j!n9(Yp)2j !(Y)ꃀ1 !I{(f)l?`Hl?`Hl?`Hl?`Hl?`HÇl?`HėHėjꃀ(Yx;Y)ylǠylǠylǠ ylǠylǠ!ylǠ"?"?*\n!n9(Yp)2j !(X)&coSendDc(!n9)@ :2fdx`ά-̰荒o@(㎍erf)ӼforeacrhfQ:URXF``!YinC5.FVor@X2xCwrede neamorphisms2(X):!n9(X)!P}!n9(X) coSend!F(!)@bry(1 (XJg;X))(db. 1)UR:!n9(X)n!1!n9(X)_ !n9(X)2c !n9(X)URn!1!(X)_ coSendu(!n9).ThenxwregetasinCorollary3.3.5(XJg;X)UR=(1 ev@)(1 s2(X)).WVeshorwthat]ڹisanaturaltransformation.8ForeacrhfQ:URXF``!YthesquareS)ߍs!n9(Yp) !(Yp)2h!n9(Yp) !(X)2:\32fd/逍ά-W`J(1 !I{(f)-:HGIH !n9(X) !(X)2ܟ{fdQِά-HDdbǺX.XH*Ǡ*Ffe \Ǡ?f~dbX.YHǠ*Ffe9Ǡ?`!I{(f) 1^7 94# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVcommrutesbyCorollary3.3.9.8Thusthefollowingdiagramcommutes|21O!n9(X)!n9(X) !(X)2j !(X)PܟA2fd=ά-偍ddboyL 1,!n9(X) coSend ^"(!) !A2fdά-1 (X&;X)L!n9(Yp) !(Yp)2j !(X)!n9(Yp) !(X)2j !(X)al:2fd ά-έ''ЀJJ1 !I{(f)-:UT 12+q!n9(Yp)R!n9(Yp) !(Yp)2j !(Yp)P32fd@<0ά-ׁe3 dbp 1-`!n9(Yp) coSend ^"(!): L32fdά-1 (Yx;Y)? zǠM@fe?SǠ?){!I{(f)XzǠM@feY)Ǡ?],!I{(f) 1񍍑i\dbt 1K< QU@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͞T1 q0cffommute.V]Proof.@_BecauseloftheunivrersalpropSertyofcoSend"(!n9)therearestructuremor-phisms}[):coSend"(!n9)t!߹coSend4Y(!)d coSend (!)}and[):coSend"(!)t!I._This}impliesthecoalgebrapropSertrysimilartotheproofofCorollary3.3.8.kMV]Observre,thatbythisconstructionallobjectsandallmorphismsofthediagram!۹::D!'C0 CQcare.comoSdulesormorphismsofcomodulesorver.thecoalgebracoSendz(!n9). EInDYfactC~Z:=coSend#U6(!)istheunivrersalcoalgebraoverwhichthegivendiagrambSecomesadiagramofcomodules.r0Corollary3.5.2.Lffet](DUV;!n9)beadiagramCwithobjectsinC0.tThenallobjectsofthediagrffamarecomodulesoverthecoalgebraC|*:=coSend"S(!n9)andallmorphismsarffemorphismsofcomodules.KIfD7OisanothercoalgebraandallobjectsofthediagramarffeDS-comodulesby'(X):!n9(X)@ԟt>RH֘[! D椢Ǡ*FfeԟǠ?ҍT1 e'cffommutes.ɍProof.@_The͈morphisms'(X)UR:!n9(X)n!1!n9(X)o* D!de ne͈anaturaltransforma-tionsinceallmorphismsofthediagramaremorphismsofcomoSdules..Sotheexistenceandtheuniquenessofamorphismge':URcoSend!(!n9)URn!1Dgfisclear.FTheonlythingtoshorwisthatthisisamorphismofcoalgebras.sThisfollorwsfromtheuniversalpropSertyofC1=URcoSend!(!n9)andthediagram_2R!_2M! CzD[˲fdCgά-YR퍒&`[ujĠm@jĠw@jğ@X@XR[uf1 \Ġm@\Ġw@\ğ@X@XR[2Πdfe2dΠ?㪍}?1[2Πdfe$dΠ?[A n1 e'+T! C+Np! CF CmIJfd+s ά-8@iL 12Ǡdfe+dǠ?@ҍn1 e'2ǠdfedǠ?e21 e' e'ԠR!Ԡ! DzD:2fdBά-`@W''uG 'jğ?`@jğ?`@jğ?`@@RԐg,1 \ğ?`@\ğ?`@\ğ?`@@R! D~! D6 D32fd*(pά-o͍շ' 1wheretherighrtsideofthecubSecommutesbytheuniversalpropSertyV.-eSimilarlywegetthati7e'ƹpreservresthecounitsincethefollowingdiagramcommutes+!+Ł9! C$IJfd - ά-K퍒Z!Z ! D${fdά-i'Hn`*FfeD`? 1H%R`*FfeX`?@ҍ?1 e'HܒǠ*FfeğǠ?8JD1H^1 "ݐ䟜@䟦@䟰@䟺@d@dRH͞1 "ݐׁ @ @ @ @d>@d>R⍒0#! K HHHHHHHHFHFj`Z 1H"0#! K8Jɔ1ҁ Hׁ H܁ H H H H H HdžHdžjI`b?o7 98# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVI.6..ThebialgebracoSendLet!Q:uDn!YCsand!n920 :DUV20 2!'LCsbSediagramsinC5. |WVecallthediagram(DUV;!n9)v (D20#;!n920@ԟt>RH! B椢Ǡ*FfeԟǠ?`T1 fHcffommutes.>Proof.@_ThemrultiplicationofcoSend!"(!n9)arisesfromthefollowingdiagramN#H>x!n9(X) !(Yp)Hxe !n9(X) !(Yp) coSend ^"(!) coSend ^"(!)Y8t{fdά- A_L !n9(X+ Yp)v!n9(X+ Yp) coSend ^"(!)P32fdK(ά-mtn0H3S2Ǡ*Ffe3dǠ?H׺Ǡ*FfeǠ?HP԰Թ=!n9(X) !(Yp) coSend ^"(!)`&ğׁ @0ğ @:ğ @Dğ @FD>@FD>RFVoritheconstructionoftheunitwreconsiderthediagramD0u=-q(fIg;fid ʤg)togetherwith+!0kP:LD0 !0)Veffc ,)!0(I)=K,the+monoidalunitobjectinthemonoidalcategoryofCndiagramsinVeffc .C4Then(Km!gK K)m=(!0q !k!0 coSend (!0))Cnistheunivrersalmap.8ThefollorwingdiagramtheninducedtheunitforcoSend!"(!n9)M3⍑j 9K⍒:K Kv4{fdAά-‚.⍍cxcx=c4i!n9(I)!n9(I) coSend ^"(!)}Ԟ32fd!wЍά-HnǠ*FfenǠ?HtǠ*FfȩǠ?HPN԰35=K coSend ^"(!n9)`t+ʬQttQtQt+QtrQ tQf4Qf4sCByusingtheunivrersalpropSertyonechecksthelawsforbialgebras.TheuWabSorvediagramsshowinparticularthatthenaturaltransformationhQ:!!n! coSend ^"(!n9)ismonoidal.ې ͍T7.eThequantummonoidofaquantumspaceProblemb3.7.29.dܹIf+Aisa nitedimensionalalgebraand:A!dM@(A)1" AtheunivrersalcoSoperationoftheTVambarabialgebraonAfromtheleftthenW:iA!fAtK M@(A)(withthesamemrultiplicationonM(A))isaunivrersalcoSoperationofM@(A)onAfromtherighrt.Thecomultiplicationde nedbythiscoSoperationisWUR:M@(A)n!1M(A)nW M(A)./ Thrus!wehavetodistinguishbSetweentheleftandtherighrtTVambarabialgebraonAandwehaveMrb(A)UR=Ml!ȹ(A)2cop .NorwconsiderthespSecialmonoidaldiagramschemeD:=DUV[X;m;u].ǔTVomakethingsRsimplerwreassumethatVeffcUѹisstrictmonoidal.qThecategoryDUV[X;m;u]RhastheobjectsX !::: !Xn=}mX2 nforalln2N(andIn:=X2 0 M)andthemorphismsm):XѴ 1X̬7!bX,Lu:I̬7!X*̹and9Iallmorphismsformallyconstructedfromm;u;idbrytakingtensorproSductsandcompositionofmorphisms.er7 ] 7.pTHE!QUANTUMMONOIDOFAQUANTUMSP:A9CEO101VLetAbSeanalgebrawithmrultiplicationmA 4ѹ:VAL Apx!gA뙹andunituA 4ѹ:Kpx!gA.Then_!A 36:URD3!CNde nedbry!n9(X)=A,:!(X2 n 6K)=A2 n Dȹ,:!(m)=mA yCand_!(u)=uAisϲastrictmonoidalfunctor./IfAis nitedimensionalthenthediagramis nite.WVeget+Theorem3.7.1."LffetqzAbea nitedimensionalalgebra.!8ThenthealgebraM@(A)cffoactingEuniversallyfrffomtherightonA(therightTambarabialgebra)M@(A)andcoSendz(!A)35arffeisomorphicasbialgebras.卍Proof.@_WVeSharvestudiedtheTVambarabialgebraforleftcoactionfQ:URAn!1M@(A)' A!butherewreneedtheanalogueforuniversalrightcoactionfQ:URAn!1A' M@(A)!(seeProblem(44).LetrBs׍l5gm 1BHA BAfǠG᤟<᤟<:胀j d1 1󤤟;XXӤeXӤezpiA 4HA B|T32fd0ά-W`'(X)commrute.8FVurthermorethefollowingcommuteqk&SA2 r ق A2 s&5A2 r ق A2 s \ BE BLA2fdf*`ά-΍'(Xҟ-: r [) '(Xҟ-: s h)&SA2 r ق A2 s \ B2 rt B2 s@l`XlF`XlQXlQz@lR l lƛlƛ:nǠM@fenǠ?`̃l fe̶l?*uǠM@fe*Ǡ?0獒 A2 (r@ >RHSimilarreconstructionscanbSegivenformorecomplicatedquanrtumspacessuchassocalledquadraticquantumspaces.gG7 t}8.pRECONSTR9UCTION!AND8 cmsy9Cf-CA:TEGORIESf103Vq`8.ReconstructionandC5-categoriesNorwweshowthatanarbitrarycoalgebraC\canbSereconstructedbythemethoSdsinrtroSducedabovefromits(co-)representationsormorepreciselyfromtheunderlyingfunctor !:C5omoffd`-!CmU!Veffc .InthiscaseonecannotusetheusualconstructionofcoSendz(!n9)thatisrestrictedto nitedimensionalcomoSdules.TheJfollorwingTheoremisanexamplethatshowsthattherestrictionto nitedimensional)comoSdulesingeneralistoostrongforTVannakXareconstruction.4dTheremarygbSeuniversalcoSendomorphismbialgebrasformoregeneraldiagrams. OntheotheruhandthefollorwingTheoremalsoholdsifoneonlyconsiders nitedimensionalcorepresenrtationsƇofCܞ.׀HowevertheproSofthenbecomessomewhatmorecomplicated.De nitionϢ3.8.1.#dLet/C,bSeamonoidalcategoryV.A/categoryDMtogetherwithabifunctor) UR:CT D3!D~չandnaturalisomorphisms :(A  B) M6!A (B% M@),Ë:URI+ M6!M+iscalledaC5-cffategory꨹ifthefollorwingdiagramscommuteF؍ G((A B) Cܞ) M(A (BE Cܞ)) Mv<:2fd+7ά-̯t& (A;Bd;C) 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") G(A B) (CF M@)>7A (BE (CF M@))v<32fd@ά-} i>(A;Bd;C M")NOSqVr(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+эAC5-categoryiscalledstrictifthemorphisms O;Xaretheidenrtities.Leth(DUV; )and(D20#; )bSeC5-categories.AhfunctorF]8:,'D}!1D20ntogetherwithanaturaltransformation(A;M@)UR:A F1(M)URn!1F(A M)iscalledaweffakTC5-functorifthefollorwingdiagramscommuteE(X(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$ά- [1 9F1(A (BE M@))L32fd2$ά- )M^Ӎ3I+ F1(M@)3F1(I+ M@)n:2fd"€ά-'֪F1(M@)uG}?`@?`@?`@Ɩ@ƖRꃀF((I{)?`?`?` TUIf,ʫinaddition,߹isanisomorphismthenwrecallFaC5-functor.RThefunctoriscalledastrict35C5-functorifҩistheidenrtitymorphism.h-7 104# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVAonaturaltransformation'p:FL f !~F120 $bSetrween(weak)C5-functorsiscalledaC5-trffansformation꨹ifIWA F120J(M@)F120J(A M@)>32fdά-h2Qq-:0HNA F1(M@)H@F1(A M@){fdά-iةH:Ǡ*FfelǠ?`1X.A ^ '(M")H :Ǡ*Ffe lǠ?`y'(A M")commrutes.f?Example*w3.8.2.PLetfCbSeacoalgebraandC:=URVeffc .ThenthecategoryC5omoffd`-!Cof.righrtCܞ-comoSdulesisaC5-categorysinceN2Comoffd`-!C andVd2C{E=Veffc1.impliesthatVG N+isacomoSduleswiththecomodulestructureofN@.Thedunderlyingfunctor!4:#C5omoffd`-!C$!>!Veffcg/isastrictC5-functorsincewrehaveVe C!n9(N@)Ԓ=!(Ve CN@).\Similarly!. Mv:ԒC5omoffd`-!C0ʻ!OVeffcisaC5-functorsinceVG (!n9(N@) M)PUR԰n9=!n9(VG N) M.f?Lemma3.8.3.{LffetCKbeacoalgebra. Let!w:>C5omod`-!Cj G!_V^ecbbetheun-derlyingKBfunctor.Lffet'[:!$0!|!Q zM&beKBanaturaltransformation.Then'isaC5-trffansformation35withC=URV^ec .S<Proof.@_Itcsucestoshorw1V ~* 'b'(N@)UR='(V N@)cforanarbitrarycomoSduleN.WVeshorwthatthediagramO.{VG NVG N M32fd2Bά-W`1X.Vn '(N")Z{VG NZVG N M{fd2Bά-`'(V N")HjǠ*Ffe'Ǡ?8J51HǠ*FfeܟǠ?8J \1"commrutes.ZuLet(vidڹ)bSeabasisofVp.FVoranarbitraryvrectorspaceWletpi9:h_VN GW!nWxbSetheprojectionsde nedbrypidڹ(t)UR=pi(P jvjw mwjf )=wi;йwherePjvj mwj=isctheuniquerepresenrtationofanarbitrarytensorinVG Wƹ.8SowegetrtUR=X ㇍ ivi pidڹ(t)8forralltUR2Vw ڒWƹ.$NorwrweconsiderVw ڒNHVasacomoSdulebythecomoSdulestructureofN@.BThenvthepi,:URVZt N6!NarehomomorphismsofcomoSdules.HencealldiagramsoftheformGۍNo1N M:,32fdT0ά-W`'(N")Z[VG NZVG N M/{fd>ά-`'(V N")HJǠ*FfeH|Ǡ?'"-p8:iH ʟǠ*Ffe Ǡ?i|p8:i,r Mi٠7 t}8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESf105Vcommrute. oExpressedinformulasthismeans'(N@)pidڹ(t)E=pi'(Vy  N@)(t)foralltUR2VG N@.8Hencewrehavehrʍ2e(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)hrSowrehave1V p '(N@)UR='(VG N@)asclaimed.?WVe!prorvethefollowingTheoremonlyforthecategoryCYV=!Veffc>ofvectorspaces.The`Theoremholdsingeneralandsarysthatinanarbitrarysymmetricmonoidalcategory]CthecoalgebraC:orepresenrtsthefunctorC5-Natބ(!n9;!gI M@)PV԰2==Mor&C,+(C5;M)ofnaturalC5-transformations.8Theorem3.8.4.M(Reconstructionofcoalgebras)'LffetCbeacoalgebra.@Let!o6:C5omoffd`-!C1[!V^effc/bffe35theunderlyingfunctor.fiThenCP1԰J׹=ܙcoSend.(!n9).?Proof.@_Let MBinVeffc*andlet'}%:!^!!( MbSe anaturaltransformation.WVede ne)thehomomorphismAe':URC1K{!M bryAe'=UR(Y 1)'(Cܞ))usingthefactthatCǹisacomoSdule.LetNbSeaCܞ-comodule.*ThenNisasubcomoduleofN U.CmbryȄ:URN6!N U.CsincethediagramFNN CN CF C<32fd(eά-a^L 1ZaNZ~N Cܟ{fd@Z@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?p}1 3commrutes.8Thusthefollorwingdiagramcommutesf+`ΉN+E`N CnIJfdWά-K퍒ՐZS)N MZN CF M|D{fd<@ά- AgRL 1He"`*Ffef#T`? M'(N")H^`*Ffeۑԟ`? ^'(N" C)=1X.N2 '(C)H>N M8JU1ԟ Xԟҁ Xԟ Xԟׁ Xԟ Xԟ܁ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ Xԟ X ԟ Xԟ Xԟ X&ԟdžX&ԟdžzH͞t1  1ҁ Hׁ H܁ H  H H H( H2 H6džH6džje21 e'4@4@4@4@ 4@4@4@)4@34@=4@>P@>PRhrInparticularwrehaveshownthatthediagramK5&Z!Zݲ! C{fd - ά--(H'[' ׁ @ @ @ @Q>@Q>RH! MV2Ǡ*FfedǠ?ҍ;1 e'commrutes.jO7 106# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVTVo,shorwtheuniquenessofe'ιletgr:G9C#=b!MPbSeanotherhomomorphismwith(1 gn9)+=m'. mFVorQc2C.Mwrehavegn9(c)m=g( 1)(c)m=( 1)(1 g)(c)m=( 1)'(Cܞ)(c)UR=e' ȹ(c).ThecoalgebrastructurefromCorollary3.5.1istheoriginalcoalgebrastructureof&TCܞ.ThiscanbSeseenasfollorws.Thecomrultiplication.:!)B!!A HCisanaturaltransformationnhence(w\ *1C)J:5!Q!!rc C CJisnalsoanaturaltransformation.As^inCorollary3.5.1thisinducedauniquehomomorphism1:Cb|Z!C 4Cso^thatthediagramPfe! coSend ^"(!n9)! coSend ^"(!n9) coSend(!n9)32fdά-aL 1H2!H|! coSend ^"(!n9)-{fdSά--0HǠ*FfeԟǠ?k}#H bǠ*Ffe Ǡ?p1 commrutes..JInasimilarwaythenaturalisomorphism!P԰=.! sKinducesauniquehomomorphismUR:C1K{!KsothatthediagramUсH@!HW! coSend ^"(!n9)ht{fd 0ά--0H}id]!Xׁ @X @X @X @ԟt>@ԟt>RHؒ! I椢Ǡ*FfeԟǠ?͞T1 ͍commrutes. BecausewoftheuniquenessthesemustbSethestructurehomomorphismsofCܞ._l!iWVeneedamoregeneralvrersionofthisTheoreminthenextchapter.'SoletC'bSeaQcoalgebra.ULet!:eC5omoffd`-!CBb[!VeffcnbSetheunderlyingfunctorand:!!SN!u @َ>RHm!n92n1 MBǠ*FfetǠ?ҍx1 e'kԍTVoshorwtheuniquenessof?(e'3letgr:u9Cܞ2n '!!M}bSeanotherhomomorphismwith(1!I{n ,gn9)s22(n)P='. 5WVe>harvegU=gn9("2n :| ,1Cn =Ϲ)W2n @l=g("2n :| ,1Cn =Ϲ)s22(n) (C5;:::ʜ;Cܞ)=Z΍("2nR 1M )(1Cnw gn9)s22(n) (C5;:::ʜ;Cܞ)UR=("2n 1M )'(C5;:::ʜ;Cܞ)=e' ȹ.dNorwweprovethe nitedimensionalcaseofreconstructionofcoalgebras.[Prop`osition3.8.7.(Reconstruction)ZdLffetC7beacoalgebra.LetC5omod`0"sd-&C7bethexcffategoryof nitedimensionalCܞ-comodulesand!:C5omod`0"sd-&C&!V^ecubetheunderlying35functor.fiThenwehaveCP1԰J׹=ܙcoSend.(!n9).Proof.@_LetNMmbSeinVeffcQandlet'S:!m!l!\ MbSeNanaturaltransformation.WVebde nethehomomorphisme':^Cʇ!Masfollorws. 4Letc2Cܞ. 4LetNbSea nite)MdimensionalCܞ-subScomodule)MofCconrtainingc.Thenwede negn9(c):=(jN 1)'(N@)(c).If>N20[isanother nitedimensionalsubScomoduleofCܹwithc2N@20[andwithN6URN@20ŹthenthefollorwingcommutesBЌԠNԠiN Ml:2fdzά-̯'(N")Ǡ@fe׼Ǡ?[ʟǠ@feǠ?KN@20K.LN@20 M,32fd ά-*'(N"-:0)胀@?`Hά-c  1HThrusκthede nitionofMIe' 0(c)isindepSendentofthechoiceofN@.FVurthermoreMIe'Z:هN!}[M`۹isobrviouslyalinearmap.FVoranytwoelementsc;c2022cCthereisa nitedimensionalusubScomoduleN6URCwithc;c20#2NYe.g.thesumofthe nitedimensionalsubScomodulesconrtainingcandc20separatelyV.8Thusi7e'ƹcanbSeextendedtoallofCܞ.mZ7 t}8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESf109VTheBrrestoftheproSofisessenrtiallythesameastheproofofthe rstreconstructiontheorem.{>SԍTherepresenrtationsallowtoreconstructfurtherstructureofthecoalgebra.EWVeprorve|areconstructiontheoremabSoutbialgebras. RecallthatthecategoryofB-comoSdulesorverabialgebraB͹isamonoidalcategoryV,:furthermorethattheunderlyingfunctor!:]C5omoffd`-!B[!p'Veffc%isamonoidalfunctor. FVromthisinformationwrecanreconstructthefullbialgebrastructureofB.8WVeharveHTheorem3.8.8.\Lffeto%B +beacoalgebra.%LetC5omod`-!B +beamonoidalcategorysuchthatj]theunderlyingfunctor!):uC5omoffd`-!BV{!x]V^effcgisamonoidalfunctor. ThentherffeisauniquebialgebrffastructureonBthatinducesthegivenmonoidalstructureonthecfforepresentations.SԍProof.@_Firstwwreprovetheuniquenessofthemultiplicationr\$:B5 z/B*!lBandKboftheunith0:K! {B.[TheKbnaturaltransformationm):!!{!Z BhbSecomesamonoidaljnaturaltransformationwithrUR:B@ BX !_7BandjË:Kn!1BWVejshorwthatr꨹andXareuniquelydeterminedbry!ands2.Letr20<:B 9BV o!*BWӹandn920u:BV o!*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)yandL⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jk!n9(K)ᅢ!n9(K) BL32fdH8pά-L(K)卹commrute.8InparticularthefollowingdiagramscommuteVHAFe!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)n z7 110# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVandSI?⍒K⍒JRK K̟{fdY?ά-‚.⍍PP=HSJǠ*Ffe|Ǡ?HʟǠ*FfeǠ? |1 n920jk!n9(K)ᅢ!n9(K) BL32fdH8pά-L(K)"BFHence uwregetP b(1) j$  c(1) b(2) \|c(2)ι=URPb(1) c(1) r209(b(2) c(2) \|) uand1 1UR=1 n920M"έ!BM Bعisahomomor-phism[tofofBzcomoSduleswherethecomodulestructureonM8X tBzisonlygivrenbythe|qdiagonalofBwthatistheC5-structureon!:MwComoffd`-!B}!OVeffcwregetthatalsos2(M@)%} (N@)UR:Mfa %}N6!M N BD|isvacomoSdulehomomorphism.#%Hencethe rstsquareinthefollorwingdiagramcommutesQ\Ԡ$M NԠLM BE N BN=:2fdSά-̯aL(M") (N)M N B3"M BE N BE BZV 32fd;kά-Y(OL(M") (N) 1X.BԠԠ=M N BE B:2fd;kά-n1 r 11bM N BE B B 0,32fd#:ά-n )1 r 1 17zǠ@fe7ϬǠ?ꃀ7`L(M" N)EǠ@fexǠ?ꃀ 4L(M" Bd N B)hǠ@fei",Ǡ?ꃀ/Ѡ1 1 L(Bd B)"DThesecondsquarecommrutesbyasimilarreasoningsincethecomoSdulestructureonM~ =BPpresp.'!N BPpisjgivrenbythediagonalonBPphenceM~ =NNcanbSefactoredoutofthenatural(C5-)transformation.8Norwweattach;1M . 1N   1B :URM N BE B BX !_7M N Bto>thecommrutative>rectangleandobtains2(M +N@)UR=(1M / 1No r)(1 H 1)(s2(M@) s2(N@). sXThrusthecomoSdulestructureonMO kN鴹isinducedbythemultiplicationrUR:BE BX !_7Bde nedabSorve.oϠ7 t}8.pRECONSTR9UCTION!ANDCf-CA:TEGORIESf111VSothefollorwingdiagramscommute6ڍv:BE BvBE B B B_sH2fd@Z@ά-o0u vv$bBE B B B8sH2fd()`ά-on1 r 1:BE B0BE B B_A2fd͟pά-)"L(Bd B)ԠuBE B BԠ]BE B B Bh:2fd(ά-041 1 FB=1BE BSŒ32fdPά- sKVՠ@feK,ՠ?ySP<1X.BX 1X.BsMzՠ@feM㬟ՠ?␍R,1 rv*L(Bd B)W޼ Qa޼Qk޼F^Qu޼Q{0Q{0slL(Bd B) 17 -#F^¼0¼0+KVǠM@feK,Ǡ?¼?PrMzǠM@feM㬟Ǡ?XR,r 1ꬾv*  1{ q攴g?^]W޼0W޼0+ꬾ<  1 1¼ Q¼攴Q'¼?^Q1¼Q70Q70ssd7p BE B8 B:dA2fdZЍά- Nr"p BE B"0K:d32fdAά-@? ԠBE B BfL(Bd B)ZTF`HZTF`HZTF`HZTF`HZTF`HZTF`HjԟHjԟjP  1LTΠLTΠLTΠ LTΠLTΠ!LTΠ"\ԟF"\ԟF*bǠM@fe锟Ǡ?Ӛꍑw1ꬾf1 1 jԟ?`jԟ?`jԟ?`jԟ?`jԟ?`jԟ?`ZTZTҿ䍒t*4F` 4F`4F` 4F`4F`4F`4F``` qčw7"ChKw7"]%@BP-sH2fd Pά-o67ҙԩ"AAK Bԩ"K BE B8:2fd)tά-̯UL(K) 1ԩ"ԩ"Q BE B$B:2fd)Vά-@n⍍5H5H=CO@BBE BPK32fdZЍά-)͍5"I{ 1s"3AK BL(K)hԠ}`XrԟM`X|ԟ`XԟM`Xԟ`XԟM`Xԟ`XԟM`Xԟ`XԟM`Xԟ`XԟM`XԟXԟzs~T4L(K)PKM`@ZKM`@dKM`@nKM`@xKM`@KM`@KM`@`@`Rsמ⍍:tT:tT=hՠhՠ(hՠ2hՠl"Π@feTΠ?ې1 "Ǡ@feTǠ?ɞ⍍=sa"ΠM@feaTΠ?ɼfӍandgՁ0"v2K0"2KA2fd`ά-Z֯Z1ԩ" IK BZ\L(K)NF`HNF`HNF`HNF`HNF`HNF`H^H^j81 @Π@Π@Π@Π@Π@ΠPFPF*ǍF`@F`@F`@F`@F`@F`@F`@Ɩ`@Ɩ`Rҿ䍒<ǠǠǠǠ ǠǠǠ$G$G HB؜*Ǡ@fe\Ǡ?ɞ⍍݁݁=ӍHenceXandrarecoalgebrahomomorphisms.p(7 112# 3.pREPRESENT:ATION!THEORY,RECONSTR9UCTIONANDTANNAKADUALITYVTVo}shorwtheassoSciativityofrweidentifyalongthemaps c:O(MO N@) PP԰ =Mm (N Pƹ)^andfurthermoresimplifytherelevXanrtdiagramby xingthatIrepresentsasuitablepSermrutationofthetensorfactors.8Thenthefollowingcommute+BE B B+)BE B B B B BNIJfdV0ά-: U9jI{(L(Bd) (B) (B)++fBE B B/LIJfd3pά-q7   1ZBE B BZ)BE B B B B B\z\zNfdV0ά-kX\I{(L(Bd B) (B))h:h:N҄fdV0ά-&X\I{(L(Bd) (B B))ZZr8BE B/L{fd?ά-).=    1BE B B\rBE B B BN32fdnά-p7L(Bd B B)~QBl32fdc0ά-n7   1H.U`*Ffe.`? 3;<1H.UǠ*Ffe.Ǡ?8J3;<1H泪`*Ffeܟ`? ɐ1 (r 1)Hj`*Ffe򜟽`? 1 (1 r)H\ꟽ`*Ffe`?>0hr 1Hh`*Ffeܟ`?>0N\1 rH9Ǡ*FfelǠ?p<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-onv 1"#BE K"__BE B K B'ПA2fd4 ά--L(Bd) (K)"".BE K B B0A2fd@ά-nA1 r 1""J~mBE B B*(A2fd`ά-7͍+ 1 1uBE BސA2fdά-n 1 1ԩ"#BE Kԩ"BE K B'П:2fd_ά-̯stL(Bd K)#BVBE B p32fd8ά-VAL(Bd)B{ 32fd/pά-nv 1sޟՠ@feՠ?"}-:1s#^ՠ@feVՠ? 1 1 L(K)sgBՠ@feguPՠ?^MH|p1 1 sՠ@feՠ?^M1 ޟΠ@feΠ? =ޟΠ@feΠ?ېw1 1 rgBǠM@feguPǠ?XP1 rǠM@feǠ?¼rޟǠ@feǠ?uG =s)vL(Bd) 1/ՠ9ՠCՠMՠWՠaՠb'Mb'M*PM:!0 1?`H?`H(?`H2?`H<?`HF?`HG%pHG%pjeI`q;7 VRCHAPTER4TheffIn nitesimalTheory"n71.7IntegralsandFourierTransformsAssumeforthiscrhapterthatKisa eld.ШLemma4.1.1. LffetDC!Qbea nitedimensionalcoalgebra.EveryrightCܞ-comoduleMvis5aleftCܞ2-moffdulebyc2mUR=Pm(M") hhc2;m(1) \|i5andcffonverselybys2(m)UR=Pidc2RAimw ciwherffe35Pc2RAij ciis35thedualbasis.㍍Proof.@_WVecrheckthatM+bSecomesaleftCܞ2-moduleQʍ=W(c2c20 ̟^\й)mk=URPm(M") hhc2c20 ̟^\;m(1) \|iUR=Pm(M") hhc2;m(1) \|ihc20ȟ^ ;m(2)ik=URc2Pjm(M") hhc20ȟ^ ;m(1) \|i=c2(c20ȟ^m):Itiseasytocrheckthatthetrwoconstructionsareinrversesofeacrhother.InparticularassumelethatMIisarighrtCܞ-comoSdule.Chooselemi?suchthats2(m)UR=Pmi cidڹ.Thenc2RAjmUR=Pmidhc2RAj;ciiUR=mjPandꨟPSc2RAim ci,=URPmi ci=URs2(m).dQ㍑De nition4.1.2. 1.:LetLAbSeanalgebrawithaugmenrtation" ':A#!-K,+uanalgebraބhomomorphism.tLetMhbSealeftA-module.tThen2A hM5J=ffm2M@jam="(a)mg꨹iscalledthespfface35ofleftinvariants꨹ofM@.Thisde nesafunctor2A Ȍ-:URA-M0offd ! Veffc .2. LetX3C4ѹbSeacoalgebrawithagrouplikreelement1Æ2Cܞ. LetX3MbSearightCܞ-comoSdule.2ThenM@2coCg:=KGofK2G isthemap"v?:KG! KrestrictedtoG,hence"(x)v?=1=hx;1>KG iforallxUR2G.8TheanrtipSodeoffQ2URK2G @isgivrenbyS׹(fG)(x)UR=hx;S(fG)i=f(x21 \|).tm+7 1164.pTHE!INFINITESIMALTHEOR:YVTheDelemenrtsofthedualbasis(x2jx}2G)Dwithhx;yn92.=i}=x;y$consideredDasmaps-JfromGtoKformabasisofK2G.8TheysatisfytheconditionsAhHx yn9 =URx;y "hx and!~X!x2G4x V=1>KG  sincehz;x2yn92.=iUR=hz;x2ihz;yn92.=iUR=zV;x zV;y=x;y "hhz;x2iandhz;Px2G"=x2iUR=1=hz;1>KG i.MHencesthedualbasis(x2jxY2G)sisadecompSositionoftheunitinrtoasetofminimalorthogonalidempSotenrtsandthealgebraofK2G @hasthestructureRvQ$K G t=URx2G/Kx PV԰.==K:::K:InparticularK2G @iscommrutativeandsemisimple.ThediagonalofK2G @isAhf/(x )UR=XSyI{2Gyn9  (yn9 1 ʵx) V=XyI{;zV2G;yz=x3MUyn9  z  ՍsinceG>)hz3 u;(x2)iUR=hzu;x2i=x;zVu19=zV1 HEx;u"=PyI{2G#GRyI{;z /yI{1 ;;x;uGϹ=URPyI{2G!GThz;yn92.=ihu;(yn921 ʵx)2iUR=hz3 u;PyI{2G!yn92 (yn921x)2i:Leto]a752KG.Thenade nesamap/eaѓ:GP!K2K2G brya=Px2G$5:e$ua*(x)x.FVorӒarbitraryfQ2URK2G @anda2KGthisgivresAhy?tha;fGiUR=f(X!x2GeUVaW(x)x)=Xvfx2Gjeaթ(x)f(x): LThecounitofK2G @isgivrenby"(x2)UR=x;eswheree2Gistheunitelemenrt.TheanrtipSodeis,asaborve,S׹(x2)UR=(x21 \|)2.WVeconsiderHpF=K2G sasthefunctionalgebraonthe nitegroupGandKGasthedualspaceofHB=URK2G @henceasthesetofdistributionsonHV.Then(7)UQR\G:=URXvfx2GxUR2HV  =KG  isa(trwosided)inrtegralonHsincePUx2G$(yn9xUR=Px2G#,x="(y)Px2G"=x=Px2G#,yx.WVewriteo甆ZfG(x)dxUR:=hUQR;fi=Xvfx2Gf(x):wWVei.harveseenthatthereisadecompSositionoftheunit1l2K2G ƹinrtoasetofprimitivreorthogonalidempSotentsfx2jx|2Ggsuchthateveryelementf{2|K2G hasauniquerepresenrtationf86=7PfG(x)x2. SinceUQRyn92 t=7Px2G#.hx;yn92.=iwegetUQRfGyn92 t=RP x2G=hx;fGyn92.=iUR=Pf(x)yn92.=(x)=f(yn9)hence!قfQ=URX(甆ZfG(x)yn9 .=(x)dx)yn9 :uݠ7 pe1.pINTEGRALS!ANDF9OURIERTRANSFORMSbk117VProblemh4.1.30.DescribSeրthegroupvXaluedfunctorK-c Alg f(K2G;)intermsofsetsandtheirgroupstructure.De nitionB9andRemark4.1.11.ѹLetKbSeanalgebraiclyclosed eldandletGbSea niteabeliangroup(replacingRaborve).+AssumethatthecrharacteristicofKdoSesnotdividetheorderofG.DLetH=9K2G.WVeidenrtifyK2G ѹ=9Hom(KG;K)alongthelinearexpansionofmapsasinExample2.1.10.0Let=usconsiderthesetx?^G):=s@f:G! K2j꨹grouphomomorphismtB.g.mSince=K2Ais]܍anabSeliangroup,thesetx^Gisanabeliangroupbrypoinrtwisemultiplication.Thegroupx^Giscalledthecharffacter35group꨹ofG.ObrviouslygthecharactergroupisamultiplicativesubsetofK2G $=tHom(KG;K).Actually>itisasubgroupofK-c Alg f(KG;K)URHom(KG;K)>sincetheelemenrtsUR2xT^GexpandeWtoalgebrahomomorphisms:.?(ab)=(P xHxP y yn9)=P x y (xyn9)=(a)(b)and(1)=(e)=1.~]Conrverselyanalgebrahomomorphismf 2K-c Alg f(KG;K)KrestrictstoacrharacterfQ:URGn!1K2.ThusxCM^KG0I=K-c Alg f(KG;K),*thesetofrationalpSoinrtsoftheanealgebraicgrouprepresentedbyKG.ThereisamoregeneralobservXationbSehindthisremark.Lemmae4.1.12.CLffetQH>bea nitedimensionalHopfalgebra.4ThenthesetGr(HV2Z)of35grffouplikeelementsofHV2 isequaltoK-Alg f(HF:;K).Proof.@_Infactf:PH=Wg!Kisanalgebrahomomorphismi hf uf;a bi=hf;aihf;biUR=hf;abi=h(fG);a bi꨹and1UR=hf;1i="(fG).|fHencethereisaHopfalgebrahomomorphism'UR:Kx^G !K2G @bry2.1.5.*Prop`osition4.1.13.CTheTHopfalgebrffahomomorphism'7w:Kx^G |#! K2G isTbijec-tive.Proof.@_WVegivretheproSofbyseverallemmas.LemmaK4.1.14.A2nysetofgrffouplikeelementsinaHopfalgebraHislinearlyindepffendent.Proof.@_AssumerTthereisalinearlydepSendenrtsetfx0;x1;:::ʜ;xnPgrTofgrouplikreelemenrtsHinHV.ChoSosesuchasetwithnminimal.ObviouslynUR1Hsinceallelementsarenonzero.8Thrusx0V=URP*n U_i=1 AS idxiOandfx1;:::ʜ;xnPglinearlyindepSendent.8WVegetVHX ㇍Z,Hi;jij id jf xi xj\=URx0j x0V=(x0)=X ㇍ i idxi xi:!?Sinceall i,6=UR0andthexia |xj:arelinearlyindepSendenrtwegetnUR=1and 1V=UR1sothatx0V=URx1,aconrtradiction._CorollaryD4.1.15.*(Deffdekind's8Lemma)A2nysetofcharactersinK2G islinearlyindepffendent.v 7 1184.pTHE!INFINITESIMALTHEOR:YV]܍Thrusd3'$5:Kx^G hl!K2G ˹isinjectivre.Nowweprovethatthemap'$5:Kx^G hl!K2G ˹issurjectivre.ILemma4.1.16.`(Pontryagin[duality)Theevaluationx^G%Gy<ߧ!XK2 isanon-deffgenerate35bilinearmapofabeliangroups.LProof.@_FirstwreobservethatHom (CnP;K2)P԰׹=7Cn OUforacyclicgroupofordernsinceKhasaprimitivren-throSotofunity(char(K)6u tjGj).Since;thedirectproSductandthedirectsumcoincideinAb ¹wrecanusethefunda-menrtaltheoremfor niteabSeliangroupsGPUR԰n9=Cnq1 :::t0Cnt togetHom|(G;K2)PUR԰n9=Gfor anryabSeliangroupGwithchar(K)6u tjGj.sThusx^GPۢ԰=Gandύ^31x^G=G.sIn particular(x)?=1Eforallx?2GEi ?=1.H+ByEthesymmetryofthesituationwregetthatthe]܍bilinearformh:;:iUR:xT^GDGn!1K2isnon-degenerate.TThrusjx^G DjUR=jGjhencedim(Kx^G D)=dim(K2G).8ThisprorvesPropSosition2.1.13.LDe nitionҲ4.1.17.LetqHǹbSeaHopfalgebra.=A%K-moduleMRUthatisarighrtHV-moSduleUbryUR:M1 HB\3!M9andUarightHV-comoSdulebyȄ:URM6!M1 H|iscalledaHopf35moffduleifthediagramA1ԠM HԠzH :2fdzά-(73ԠԠ M H :2fdzά-mSqLM H H H8M H H H㌞32fdנά-n˙1 r 1Ǡ@feǠ? L SjǠ@fe?`6x9 r,2commrutes,ui.e.+vifhs2(mh)UR=Pm(M") hh(1) Xpm(1) \|h(2)holdsforallh2Handallm2M@.ݍObservreythatHfisanHopfmoSduleoveritself.FVurthermoreeachmoSduleoftheformV %'HisaHopfmoSdulebrytheinducedstructure.TMoregenerallythereisafunctorVeffcXo3URV7!VG HB2Hopf-Mo`d@1ƹ-DnHV.Prop`ositionV4.1.18.^The twofunctors-EA2coH~:URHopf-Mod@1-DdHB!gV^effcand- 8 `HB:V^effcR3URV7!VG HB2Hopf-Mo`d@1-DdH arffe35inverseequivalencesofeachother.LProof.@_De nenaturalisomorphisms~w/ h:URM@ coH HB3m h7!mh2MwithinrversemapO  1]:URM63m7!Xm(M") hS׹(m(1) \|) m(2)2URM@ coH HandX, :URV3vË7!v 12(VG HV) coH$withinrversemap(VG HV) coH83URv h7!vn9"(h)2V:w7 pe1.pINTEGRALS!ANDF9OURIERTRANSFORMSbk119VObrviouslyLthesehomomorphismsarenaturaltransformationsinMeandVp.^lFVur-thermore 7isahomomorphismofHV-moSdules.8 21Ziswrell-de nedsince'ʍʍRs2(Pm(M") hS׹(m(1)))@7=URPm(M") hS׹(m(3) \|) m(1)S׹(m(2))@7(sinceM+isaHopfmoSdule)@7=URPm(M") hS׹(m(2) \|) n9"(m(1))@7=URPm(M") hS׹(m(1) \|) 1)AhencexP#m(M") hS׹(m(1) \|)Gg2M@2coH$a.FVurthermorex 21isahomomorphismofcomoSdulessinceovʍ%s2 21 p (m)T=URs2(Pm(M") hS׹(m(1) \|) m(2))UR=Pm(M") hS׹(m(1)) m(2)$ m(3)T=URP 21 p (m(M") h) m(1)ι=UR( 21 1)s2(m):Finally 7and 21Zareinrversetoeacrhotherby^h+D{  1 p (m)UR= (XUVm(M") hS׹(m(1) \|) m(2))UR=Xm(M") hS׹(m(1))m(2)ι=m^iandovʍ' 21 p (m h)n6=UR 21 p (mh)=Pm(M") hh(1) \|S׹(m(1)h(2)) m(2) \|h(3)n6=URPmh(1) \|S׹(h(2)) h(3)G$(brys2(m)UR=m 1){=URm h:Thrus 7and 21ZaremutuallyinversehomomorphismsofHopfmoSdules.The4qimageof isin(VyT HV)2coHbrys2(vK 1)=vK (1)=(vK 1) 1.:Hy&!Hhisanalgebraanrtihomomorphism,άthedualHV2 isanHV-moSduleinfourdi erenrtways:ʍ(8)ʍL1xh(fQ*URa);gn9i:=ha;gn9fGi;ޯXh(aUR(fG);gn9i:=ha;fGgn9i;L1xh(fQ+URa);gn9i:=ha;S׹(fG)gn9i;ޯXh(aUR)fG);gn9i:=ha;gn9S׹(fG)i:ovIfHis nitedimensionalthenHV2 aisaHopfalgebra.Theequalitryh(f*a);gn9i=ha;gn9fGiUR=Pha(1) \|;gn9iha(2);fGi꨹implies󍍍(9)خ(fQ*URa)=Xa(1) \|ha(2);fGi:^iAnalogouslywrehave^h(10)ح(aUR(fG)=Xha(1) \|;fia(2):Prop`osition4.1.19.dLffetWHE@bea nitedimensionalHopfalgebra.ԊThenHV2 Disaright35HopfmoffduleoverHV.xy7 1204.pTHE!INFINITESIMALTHEOR:YVProof.@_HV2 zisK aleftHV2Z-moSdulebryleftmultiplicationhenceby2.1.1arightHV-comoSdule,Nbrys2(a)UR=Pidb2RAia% bidڹ.mLet,Nf;gË2URHanda;bUR2HV2Z.mThe,N(left)mrultiplicationofHV2 satis es-Or?abUR=Xb(H) ha;b(1) \|i:tWVeusetherighrtHV-moSdulestructureP(aUR)fG)=Xa(1) \|hS׹(f);a(2)i:SonHV2 =URHom(HF:;K).NorwwechecktheHopfmoSdulepropertryV.2P ^G"h;xi=CP 2P ^Gh;xi=h;xiP2P ^G h;xi.Since'Wforeacrhx2Gnfeg'Wthereisasuchthat>h;xiUR6=1andwregetuȍSX-K2P ^G`h;xiUR=jGje;x :"HenceꨟPUx2G;2P ^G6`h;xiUR=jGj= 21Zandjk(19)UȄ=URjGj 1 zX- \z2P ^G :$׍LetYHbSe nitedimensionalfortherestofthissection."qInCorollary1.22wrehaveseen&thatthemapHM3f7!(UQR (fG)2HV2 Xis&anisomorphism.zZThismapwillbSecalledtheFourier35trffansform. Theorem4.1.26.The35FouriertrffansformHB3URfQ7!V~e*f K2HV2 is35bijectivewith(20)Vse*Jf=UR(UQR US(fG)=XhUQR*(1)\};fiUQR*(2)The35inverseFouriertrffansformisde nedby(21)ne]Caݖ=URXSן 1 S((1) \|)ha;(2)i:Sincffe35thesemapsareinversesofeachotherthefollowingformulashold$`(22)GN hV)Re*f;gn9iUR=甆ZUTfG(x)g(x)dxUha;V*e*biUR=甆ZUTMXSן21 S(a)(x)b(x)dx*N fQ=URPSן21 S((1) \|)hV)Re*f;(2)iUaUR=PhUQR*(1)';)ea*iUQR*(2)#:)ZProof.@_WVe:usetheisomorphismsH ' !G7HV2 de nedbryV!b*f+ :=VI|e* *fR= *(UQR +(fG)=U^P hUQR*(1)';fGiUQR*(2)͹andHV2  7!qHde nedbryba j:=UR(a*s2)=P(1) \|ha;(2)i.8Becauseof“(23)|ha;V*b*biUR=ha;(b*s2)i=hab;s2iand(24)hV)Re*f;gn9iUR=h(UQR US(fG);gn9i=hUQR;fGgn9iwregetforallaUR2HV2 andfQ2H) ha;UUObVvb*Lf _pi*=URhaV)Rb*f;s2i=Pha;(1) \|ihV)Rb*f;(2)iUR=Pha;(1) \|ihUQR;fG(2)i(bryLemma1.25)*=URPha;S׹(fG)(1) \|ihUQR;(2)iUR=ha;S׹(fG)ihUQR;s2iUR=ha;S(fG)i:|֠7 1244.pTHE!INFINITESIMALTHEOR:YVUVThis_&givresUU5bVb*If =S׹(fG).[SotheinversemapofH"|!>HV2 withVxb*fh=(UQR (fG)=VDe*fis_&HV2-P!HwithSן21 S(+ba+)&=P>Sן21((1) \|)ha;(2)i&=TQea '.Thenthegivreninversionformulasareclear.WVenoteforlateruseha;V*e*biUR=ha;Sן21 S(V*b*b)i=hSן21(a);V*b*bi=hSן21(a)b;s2i.1i *IfGisa nitegroupandHB=URK2G @thenVsje*Jf=URXvfx2GfG(x)x:!y6Since.(s2)UR=Px2G#,x215 <) 1+x2o2wherethex2V2K2G ƹarethedualbasistothex2G,wregetXw8e a7`=URXvfx2Gha;x ix : ]If )Gisa niteAbSeliangroupthenthegroupsGandwSbG\areisomorphicsotheFVouriertransforminducesalinearautomorphisme- *:URK2G t u!K2G @andwrehaveeaw߹=URjGj 1 zX- \z2P ^Gha;i 1#BysubstitutingtheformrulasfortheintegralandtheDiracs2-function(7)and(19)wreget۲(25)VSe*Q_;f[4=URPx2G#,fG(x)x;W)eߖaQ=URjGj21 \zP'2P ^G.a()21 \|;㓍Q_;fQ=URjGj21 \zP'2P ^GV0Ae*.f5+4()21 \|;ߖaUR=Px2G#SWe#,a)-(x)x:ThisimpliesM}(26)VzAe*xCf*()UR=Xvfx2GfG(x)(x)=甆ZURf(x)(x)dxwithinrversetransformˍ(27)e?aj(x)UR=jGj 1 zX- \z2P ^G (a) 1 \|(x):"Corollary4.1.27.TheC%FouriertrffansformsoftheleftinvariantintegralsinHand35HV2 arffe{(28)Vq*_e*qz=UR"ǟ 1s2HV !andQX3ePUQR\۹=12HF::.Proof.@_WVePharvehV%e*q;fGi.=hUQR;s2fGi=hUQR;ǟ21 C(fG)s2i="ǟ21(fG)hUQR;s2i="ǟ21(fG)_henceV'e* k=UR"ǟ21 C.8FVromnPe꨹1 =(UQR US(1)=UQR?wregetSeUQR=UR1. *Prop`osition4.1.28.De ne35aconrvolutionmultiplication35onHV2 byxfchab;fGiUR:=Xha;Sן 1 S((1) \|)fGihb;(2)i:}7 pe1.pINTEGRALS!ANDF9OURIERTRANSFORMSbk125VThen35thefollowingtrffansformationruleholdsforf;gË2URHV:/ٍ(29)V ;f*zfGg)=V~e*URf (eg8:ۍIn\ipffarticularHV2 withtheconvolutionmultiplicationisanassociativealgebrawithunitenf1HN=URUQR US,35i.e.9(30)ɖUQRt?aUR=aUQR =a:iProof.@_Givrenf;gn9;hUR2HV2Z.8Then%8et,(ChVf*fGg !7;hiR=URhUQR;fGgn9hi=hUQR;fGSן21 S(1HD)gn9hihUQR;s2iR=URPhUQR;fGSן21 S((1) \|)gn9hihUQR;(2) \|iUR=PhUQR;fGSן21 S((1) \|)hihUQR;gn9(2) \|iR=URPhV)Re*f;Sן21 S((1) \|)hiheg;(2)iUR=hV)Re*f O(eg8;hi:) FVrom(28)wregetnPe1H9:=URUQR US.8SowehaveVe*fR=VΣf*UR1fG=neUR1 Ve*f =URUQR UQV)Re*f.j.IfGisa niteAbSeliangroupanda;bUR2HV2 =K ^G.8Then͍yE(ab)()UR=jGj 1LX- \z;2P ^G;=@!a()b():##Infactwrehave__F'G(ab)()[{=URhab;i=Pha;Sן21 S((1) \|)ihb;(2)i[{=URjGj21 \zP'2P ^G,ha;21 \|ihb;i=jGj21 \zP';2P ^G;=Ma()b():rOnexofthemostimpSortanrtformulasforFVouriertransformsisthePlancherelfor-mrulaontheinvXarianceoftheinnerproSductunderFVouriertransforms.8Weharve퍑Theorem4.1.29.(The35Plancherffelformula)0(31)6ha;fGiUR=hV)Re*f;ǹ(+ea+)i:iProof.@_Firstwrehavefrom(22)/Ȃ=ha;fGi6չ=URPhUQR*(1)\};)ea*ihUQR*(2);Sן21 S((1) \|)ihV)Re*f;(2)iUR=PhUQR;)ea*Sן21 S((1))ihV)Re*f;(2)i6չ=URPhUQR;Sן21 S((1) \|)ǹ(+ea+)ihV)Re*f;(2)iUR=PhUQR;Sן21 S(S׹(ǹ(+ea+))(1))ihV)Re*f;(2)i6չ=URPhUQR;Sן21 S((1) \|)ihV)Re*f;ǹ(+ea+)(2)iUR=PhUQR;Sן21 S(s2)(2)ihV)Re*f;ǹ(+ea+)S׹(S21 S(s2)(1))i6չ=URhUQR;Sן21 S(s2)ihV)Re*f;ǹ(+ea+)i:0g\ApplythistohUQR;s2i.8Thenwreget:%1UR=hUQR;s2i=hUQR;Sן 1 S(s2)ihV%e*q;ǹ(eUQR)iUR=hUQR;Sן 1 S(s2)i"ǟ 1 Cǹ(1)=hUQR;Sן 1 S(s2)i:/ߍHencewregetha;fGiUR=hV)Re*f;ǹ(+ea+)i: iCorollary4.1.30.If35H isunimoffdularthen=URSן22r.~Z7 1264.pTHE!INFINITESIMALTHEOR:YVProof.@_HunimoSdularmeansthat]ڹisleftandrighrtinvXariant.8Thusweget.PȂ/ha;fGiNI=URPhUQR*(1)\};)ea*ihUQR*(2);Sן21 S((1) \|)ihV)Re*f;(2)iNI=URPhUQR;)ea*Sן21 S((1) \|)ihV)Re*f;(2)iUR=PhUQR;Sן21 S((1) \|S׹(+ea+))ihV)Re*f;(2)iNI=URPhUQR;Sן21 S((1) \|)ihV)Re*f;(2)Sן22r۹(+ea+)i (since]ڹisrighrtinvXariantQ)NI=URhUQR;Sן21 S(s2)ihV)Re*f;Sן22r۹(+ea+)i=hV)Re*f;Sן22r۹(+ea+)i:0HenceSן22-=URǹ._6WVe4?alsogetaspSecialrepresenrtationoftheinnerproductHV2 Hv! Kbrybothinrtegrals:Corollary4.1.31.(32)a6ha;fGiUR=甆Z}eURaS(x)f(x)dx=甆ZUTMVSן 1 S(a)(x)V)Re*f(x)dx:󍍍Proof.@_WVe"9harvetherulesfortheFVouriertransform.ߓFrom(24)wregetha;fGi=UhUQR;)ea*fGiz=UQR:Geza(x)f(x)dxandfrom(23)ha;fiz=hSן21 S(a)V)Re*f;s2i=UQR z: Sן21(a)(x)V)Re*f(x)dx:I`6TheFVouriertransformleadstoaninrterestingintegraltransformonHbydoubleapplication.eProp`ositionh4.1.32. The doubletrffansformWH*f¹:=)(C((UQR)(fG)) de nesanautomorphism35HB!gH with0ۍW$*pzf!(yn9)UR=甆ZURfG(x)s2(xy)dx:GProof.@_WVeharve1W<ꍍE/hyn9;W*f ie==URhyn9;(Ȅ((UQR US(fG))i=h(UQR(fG)yn9;s2ie==URPh(UQR US(URfG);(1) \|ihyn9;(2)iUR=PhUQR;fG(1)ihyn9;(2)ie==URPhUQR*(1)';fGihUQR*(2);(1) \|ihyn9;(2)iUR=PhUQR*(1)';fGihUQR*(2)%yn9;s2iye==URPhUQR*(1)';fGihUQR*(2);(yË*URs2)i=hUQR;fG(yË*s2)ie==URUQR UQfG(x)s2(xyn9)dx4W;sincehx;(yË*URs2)i=hxyn9;s2i.L#2.Deriv@ationsDe nitionހ4.2.1.ǹLetAbSeaK-algebraandA MAbeanA-A-bimodule(withidenrticalK-actiononbSothsides).8AlinearmapD:URAn!1M+iscalledaderivationif)v]DS(ab)UR=aD(b)+D(a)b:ThesetofderivXationsDerKgg(A;AMA)isaK-moSduleandafunctorinA ȌMA.$B7 2.pDERIVA:TIONS127VByinductiononeseesthatD>6satis esKgDS(a1:::anP)UR= knX ㇍Si=1a1:::ai1AVDS(aidڹ)ai+1AT::: anP:׍Let (AbSeacommrutative (K-algebraandA Ma beanA-module.`ConsiderMa asanqA-A-bimoSdulebryma::=am.ͧWVeqdenotethesetofderivXationsfromAtoMybyDerݟK|(A;M@)c.y.dProp`ositionc4.2.2.ְ1. LffetAbeaK-algebra. ThenthefunctorDeryK2f(A;-33) :A-35Moffd 3Q-fAUR!V^effc/isrffepresentable35bythemoSduleofdi erenrtials35 A.2.LffetBIAbeacommutativeK-algebra.ThenthefunctorDer&K(A;-33)c yǹ:KNA-35Modfk!V^effc/is35rffepresentablebythemoSduleofcommrutativedi erentials 2cbA.GProof.@_1.jRepresenrtAasaquotientofafreeK-algebraA4:=KhXidji2Jri=IwhereB7o=iKhXidji2JriisthefreealgebrawithgeneratorsXidڹ.y_WVe rstprorvethetheoremforfreealgebras.a)ArepresenrtingmoSduleforDerKgg(B;-)is( BN>;dUR:BX !_7 B)withPɍf B :=URBE Fƹ(dXidji2Jr) Bwhere\Fƹ(dXidjiM2Jr)isthefreeK-moSduleonthesetofformalsymrbolsfdXidjiM2Jrgasabasis.WVeMharvetoshowthatforeveryderivXationD:OB!TyMU1thereexistsauniquehomomorphisms'UR: B !oM+ofB-B-bimoSdulessucrhthatthediagramK}{0B{/ B餟{fd'Ѝά--80dH`Ǐ`D餟ׁ @餟 @餟 @餟 @0$>@0$>RHM5RǠ*FfehǠ?''Pɍcommrutes._TheY'moSdule B eisaB-B-bimoduleY'inthecanonicalwrayV._TheY'productsX1:::Xn -ofPthegeneratorsXi*ofB VformabasisforB.FVoranryproSductX1:::Xnwre&de ned(X1:::XnP)UR:=P*n U_i=1 ASX1:::Xi1h; &dXi Xi+1AT::: Xn Rvin&particulard(Xidڹ)UR=1eH dXi" 1.TVoK\seethatdisaderivXationitsucestoshorwthisonthebasiselements:(&]/d(X1:::XkiXk6+1 :::!ʪXnP)ɍi=URP*k U_jv=1!BX1:::Xjv1. dXj Xjv+1B:::! "Xk#Xk6+1 :::!ʪXnm'+P*n U_jv=k6+1+-9X1:::Xk#Xk6+1 :::!ʪXjv1. dXj Xjv+1B:::! "Xni=URd(X1:::Xk#)Xk6+1 :::!ʪXnR+X1:::Xkd(Xk6+1 :::!ʪXnP))5NorwhletD~j:*Bm! KMbSeaderivXation.=De ne'by'(1 dXid 1)*:=DS(Xidڹ).=ThismapobrviouslyextendstoahomomorphismofB-B-bimoSdules.FVurthermorewehave%'d(X1:::XnP)d=UR'(P jX1:::Xjv1. dXj Xjv+1B:::! "XnP)d=URPjfX1:::Xjv1B'(1 dXj 1)Xjv+1B:::! "Xn=URDS(X1:::XnP)%hence'dUR=DS.37 1284.pTHE!INFINITESIMALTHEOR:YVTVomHshorwtheuniquenessof'let Ë:UR B !oM,bSeabimodulehomomorphismsucrhthatL n9d!=DS.^Then n9(1a dXiR; 1)!= d(Xidڹ)=DS(Xi)='(1a dXiR; 1).^SinceL and'areB-B-bimoSduleshomomorphismsthisextendsto Ë=UR'.b)NorwletAq:=KhXidji2Jri=I(bSeanarbitraryalgebrawithB =qKhXiji2Jrifree.8De neudG A 36:=UR BN>=(I B + BI++BdB(I)+dB(I)B):-WVei rstshorwthatI B$+] 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 AsucrhSthatFº34hA34螴 A1 32fdά-dX.A: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.F7 2.pDERIVA:TIONS129V2.=If?Aiscommrutative?thenwrecanwriteA ƹ=K[Xidji2Jr]=I¹and? 2cbB Z=B Fƹ(dXidڹ).With* 2cbA =1 2cbBN>=(I 2cbB $ +BdB(I))*theproSofisanalogoustotheproofinthenoncommrutativesituation.$ӍRemark4.2.3.1. A Dҹisfgeneratedbryd(A)asabimoSdule,FhenceallelementsareoftheformPUi-aidd(a20RAi)a200RAir.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)!T withuniquelyde nedcoSecienrtsލōf@f[zF ΍@Xi2URBE B op ~#:EYInthecommrutativesituationwrehaveuniquecoSecientsō`@fWF[zF ΍@Xi2URK[Xidڹ]:4.8WVegivrethefollowingexamplesforpart3:ZsR=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:ZsQThisisobtainedbrydirectcalculationorbytheprffoduct35ruleōf@fGgd[zF ΍@XiJ=UR(1 gn9)ō<@f33[zF ΍@XiT+(f 1)ō@g33[zF ΍@Xif:TheproSductrulefollorwsfrom8C&d(fGgn9)UR=d(f)g+fd(gn9)=X((1 g)ō<@f33[zF ΍@XiT+(f 1)ō@g33[zF ΍@Xif)d(Xidڹ):Y7 1304.pTHE!INFINITESIMALTHEOR:YVf`LetAUR=KhXidi=I.8IffQ2I+then`zQ1 pd(fG)+=dA(: z f)=0henceקXōi@f`0[zF ΍@XiϓdA(\-z %F ӍXi %F)UR=0:>These.arethede ningrelationsfortheA-A-bimoSdule A withthegeneratorsdA(\-z %F ӍXi %F).gAFVorHmotivXationofthequanrtumgroupcaseweconsiderananealgebraicgroupGwithrepresenrtingcommutativeHopfalgebraA.RecallthatHomD(A;RJ)isanalge-brar'withtheconrvolutionr'multiplicationforeveryR2K-c AlgذandthatG(RJ)=K-c Alg f(A;RJ)!HomG(A;RJ)%isasubgroupofthegroupofunitsofthealgebraHomy(A;RJ).΂De nition ZandRemark4.2.4.AKlinearmapT:URAn!1Aiscalledlefttrffanslationinvariant,ifthefollorwingdiagramfunctorialinRn2URK-c AlgQ1commutes:?yfTG(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$Ǡ?ꃀ)^0then35Lie #(HV)isarffestricted35Liealgebrffaorap-Liealgebra./Proof.@_Letka;b2HYbSeprimitivreelements. ;Then([a;b])=(abba)=(aT 1+1 a)(b 1+1 b)(b 1+1 b)(a 1+1 a)UR=(abTba) 1+1 (abba)henceLie #(HV)URH2L isaLiealgebra.$IfthecrharacteristicofKispUR>0thenwehave(aS 1+1 a)2p=URa2p 1+1 a2p].*Thrus Lie #(HV)isarestrictedLiesubalgebraofH2LZ΍withthestructuremaps[a;b]UR=abba꨹anda2[p] ՗=URa2p].w/CorollaryX4.3.3.`VLffetHbeaHopfalgebra.nThenthesetoflefttranslationin-variantderivationsDO:qHP2!LH isaLiealgebrffaunder[DS;D20!ǹ]=DD20uSD20!D.Ifcrhar3|=URp35thenthesederivationsarffearestrictedLiealgebrawithDS2[p] )%=URDS2p.ߝProof.@_Themap m:HV2o !}HV22`p  !#7End8DN(HV)isahomomorphismofalgebrasbryF4.2.6. MHence (dd20fd20d)+=(dd20d20d)+=(d)(d209)(d20)(d).IfdisaprimitivreelementinHV2o ~thenby4.2.7and4.3.1theimageDn.:= (d)inEnd (HV)misalefttranslationinrvXariantmderivationandalllefttranslationinrvariantderivXationsuareofthisform.Since[d;d209]UR=dd20 d20duisagainprimitivrewegetthat[DS;D20!ǹ]==DSD20D20Dzis'alefttranslationinrvXariant'derivationsothatthesetoflefttranslationinrvXariantderivXationsDerxgHgKٹ(HF:;HV)isaLiealgebraresp.darestrictedLiealgebra.И/De nition4.3.4.LetHjbSeaHopfalgebra.%Anelemenrtc2Hiscalledcffocom-mutativeifW(c)=(c),i.e.ifPic(1) ;0c(2)"=PiQc(2) c(1) \|.LetCܞ(HV):=fc2HVjc꨹iscoScommrutativeaTg.LetG(HV)denotethesetofgrouplikreelementsofHV.HLemma4.3.5.LffetH(beaHopfalgebra.8IThenthesetofcocommutativeelementsCܞ(HV),isasubffalgebra,ofHقandthegrffouplikeelementsG(H)formalineffarlyindepen-dentJsubsetofCܞ(HV)._FurthermorffeG(H)isamultiplicffativesubgroupofthegroupofunits35U@(Cܞ(HV))./Proof.@_It3isclearthatCܞ(HV)isalinearsubspaceofH. Ifa;bk2Cܞ(H)3then(ab) =(a)(b)=(W)(a)()(b) =W((a)(b))=(ab)aand(1) =1Ǿ 1 =W(1).8ThrusCܞ(HV)isasubalgebraofH.TheWgrouplikreelementsobviouslyarecoScommutativeandformamultiplicativegroup,henceasubgroupofU@(Cܞ(HV)). TheyarelinearlyindepSendenrtbyLemma2.1.14./Prop`ositionb4.3.6.^LffetHbeaHopfalgebrawithSן22-=URid H9:.^Thenthereisaleftmoffdule35structure lCܞ(HV) Lie #(H)UR3c aUR7!caUR2Lie #(H) 7 eߙ3.pTHE!LIEALGEBRAOFPRIMITIVEELEMENTSW133Vwithc}aUR:=rHD(rH _ }1)(1 W)(1 S/ 1)( 1)(c a)UR=Pc(1) \|aS׹(c(2))suchthat|c[a;b]UR=X[c(1)$a;c(2)b]:؍In35pffarticularG(HV)actsbyLieautomorphismsonLie #(H).%Proof.@_ThegivrenactionisactuallytheactionH= HG !H withha=Ph(1) \|aS׹(h(2)),theso-calledadjoint35action.WVe[` rstshorwthatthegivenmaphasimageinLie #(HV). FVorc/2Cܞ(H)[`anda/2Lie #(HV)wrehave(ca)UR=(Pc(1) \|aS׹(c(2)))UR=P(c(1) \|)(a 1+1 a)(S(c(2) \|))UR=P(c(1) \|)(ah 1)(S׹(c(2)))+Pv(c(2))(1 a)(S׹(c(1)))UR=Pc(1)aS׹(c(4))h c(2)S׹(c(3))+`Pc(3) \|S׹(c(2)) c(4) \|aS(c(1))#=ca 1+1 ca sinceciscoScommrutative,qS22=#id TǟHandaisprimitivre.WVe#]shorwnowthatLie #(HV)isaCܞ(H)-moSdule.(cd)Daٹ=P`c(1) \|d(1)aS׹(c(2)d(2))=Pc(1) \|d(1)aS׹(d(2))S(c(2))UR=c(da).8FVurthermorewrehave1aUR=1aS׹(1)=a.TVoshorwthegivenformulaleta;b7@2Lie #(HV)andc7@2Cܞ(H).GThenckm[a;b]7@=Pc(1) \|(abba)S׹(c(2))UR=Pc(1) \|aS(c(2))c(3)bS(c(4))Pc(1)bS(c(2))c(3)aS(c(4))UR=P(c(1) \|a)(c(2) Ipb)P (c(1)b)(c(2)a)UR=P[c(1)Ipa;c(2)b]dagainsincecUR2Cܞ(HV)discoScommrutative.`Norw^letg;2͚G(HV). GThenga=gn9aS׹(g)=gag21(Թsince^S׹(g)=g21(Թfor^anrygrouplikrefjelement.&FVurthermoregm#[a;b]'=[ga;gb]fjhencegԣde nesaLiealgebraautomorphismofLie #(HV)."H%Problem84.3.31.%ŹShorwthattheadjoint-TactionHR dHB3URh a7!Ph(1) \|aS׹(h(2))2HmakresHanHV-moSdulealgebra.De nition|andRemark4.3.7.'ThealgebraK(s2)G(=K[]=(2236)iscalledthealgebraofdual35numbffers.8ObservrethatK(s2)UR=KK]ڹasaK-moSdule.WVeconsider]ڹasa"smallquanrtity\whosesquarevXanishes.Themapsp :K(s2)!!K⺹withp()=0andj0: K!!K()arealgebrahomomor-phismsatisfyingpj%=URid .LetK(;s220Ak):=K[;s220Ak]=(s22236;s220 u^25).ThenK(;s220Ak)=KCK:uKs220Ks220.ThemapK(s2)37!20L2K(;20Ak)Sisaninjectivrealgebrahomomorphism.FVurthermoreforevrery h2URKwehaveanalgebrahomomorphism' J:URK(s2)3Ȅ7! 2K(s2).ThesemdalgebrahomomorphismsinducealgebrahomomorphismsH" K(s2)URn!1H K(s2)resp.8H K()URn!1H K(;20Ak)forevreryHopfalgebraHV.Prop`osition4.3.8.The35map9ne L- /:URLie #(HV)!H K(s2)H K(;s2 0Ak)with35e2La hɹ:=UR1+a Ȅ=1+s2a35iscffalled35theexpSonenrtialmapandsatis es! 񪝍e2L(a+b)D=URe2La we2Lb2;=Te2L a`=UR' (e2La w); e2L-:0[a;b]p=URe2La we2L̟-:0b K(e2La)21 \|(e2L̟-:0b)21:"pFurthermorffeallelementse2LaF23YH DK(s2)aregrouplikeintheK(s2)-HopfalgebraH K(s2).47 1344.pTHE!INFINITESIMALTHEOR:YVZ΍Proof.@_1.8e2L(a+b)D=UR(1+s2(a+b))=(1+s2a)(1+b)UR=e2La we2Lb2.2.8e2L a`=UR1+s2 a=' (1+s2a)=' (e2La w).ꍑ3.$Sincet(1+s2a)(1a)UR=1twrehave(e2La w)UR=1s2a.$Sotwegete2L-:0[a;b]p=UR1+s2[a;b]=1N+s2(aa)+20Ak(bb)+20Ak(ababba+ab)UR=(1N+a)(1+20Akb)(1a)(120Akb)UR=e2La we2L̟-:0b K(e2La)21 \|(e2L̟-:0b)21.4.3K(L)e(e2La w)@=(1W+a s2)@=1W K(L)ֹ1+(a 1+1 a) yr=@1 K(L)1+s2aQ K(L)1+1 K(L)s2a+a K(L)a=(1Q+a) K(L)(1+a)=e2La e K(L)e2La Jand"K(L)e(e2La w)UR="K(L)(1+s2a)UR=1+"(a)UR=1.+zCorollary 4.3.9.M¹(Lie #(HV);e2L-ݹ)XisthekernelofthegrffouphomomorphismpUR:GK(L)e(H K(s2))!G(HV).֍Proof.@_pUR=1_ pUR:H _K(s2)n!1H _K=HisahomomorphismofK-algebras.WVeshorwthatitpreservesgrouplikeelements.iObservethatgrouplikeelementsinH  K(s2)v3arede nedbrytheHopfalgebrastructureoverK(s2).ہLetg2BGK(L)e(H K(s2)).8Then(H 1)(gn9)UR=g K(L)gXand("H 1)(gn9)=12K(s2).SincepUR:K(s2)n!1KisanalgebrahomomorphismthefollorwingdiagramcommutesAtqU(H K) (H K)AH H K:ۑԞ32fd33@ά-R΍⍍88=ӘZV۹(H K(s2)) K(L)(H K(s2))Ә }_H H K(s2)T:2fdά-@n⍍=ÒǠ@feğǠ?ꃀn'*(1 p) (1 p)1Ǡ@fe1ğǠ?PM6D1 p tWVeidenrtifyelementsalongtheisomorphisms.cThusweget(H ͟ [1K)(1H p)(gn9)m=(1H Hl p)(H 6 1K(L)e)(gn9)t=((1H p) K(L)(1H p))(g K(L)gn9)t=(1H p)(gn9) (1H K p)(gn9),soe/that1H pUR:GK(L)e(H] K(s2))n!1G(HV). cNorwe/wehave(1H K p)(gn9g20+~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)andu("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)).Hence7wrehaveabijectivemap!_:&K-c Alg f(HF:;K(s2))3f%=f0M+f1X7!f0 1#9+f1= Ȅ2URGK(L)e(HV2oO K(s2))."SinceZthegroupmrultiplicationinK-c Alg f(HF:;K())URHomy(HF:;K(s2))istheconrvolutionandthegroupmrultiplicationinGK(L)e(HV2oe NOK())UR݋7 0H4.pDERIVA:TIONS!ANDLIEALGEBRASOFAFFINEALGEBRAICGR9OUPS"h137VHV2o E K(s2)} istheordinaryalgebramrultiplication,wherethemultiplicationofHV2oagain{istheconrvolution,Lit{isclearthat!1isagrouphomomorphism.FVurthermorethĕrighrthandsquareoftheabSovediagramcommutes..ThuswegetanisomorphismexZ:Lie #(HV2o)! ALie'f(G.)(K)onthekrernels.9Thismapisde nedbye(d)xZ=1+d2K-c Alg f(HF:;K(s2)).TVo)*shorwthatthisisomorphismiscompatiblewiththeactionsofKresp.fG(HV2o)let {2h]K,a2HV,andd2Lie #(HV2o).ZpWVeharvee( d)(a)h]="(a)F+ d(a)ۏ=h]u ("(a)+d(a)s2)UR=(u ?(1+d))(a)UR=(u ?e(d))(a)=( e(d))(a)hencee( d)UR= e(d).FVurthermoreIletg2qG(HV2o)=K-c Alg f(HF:;K),qa2HV,andId2Lie #(HV2o).jThenwreharvee(g~d)(a) g =e(gn9dg21 ʵ)(a) g =(1+gn9dg21 ʵs2)(a) g ="(a)+gn9dg21 ʵ(a) ;=Pgn9(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).a'Prop`osition4.4.4.PLffetHbeaHopfalgebraandletI K3:=YKerF("). #Then(vDerݟ"0(HF:;-33)ˣ:V^effcX.!%hV^effcKisrffepresentablebyI=I22 9anddˣ:H21"pR!I2 p&!$I=I22,ܮinpffarticularN-iDer@-"DX(HF:;-33)PUR԰n9=Hom(y(I=I 2;-)3EandK.Lie #(HV o)PUR԰n9=Hom(y(I=I 2;K):Proof.@_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=0hencethereisauniquefactorizationA: sH ұIl:2fdЍά-ށ$idO,"HQHQJI=I22U<:2fd{ά-č3?<D9|?`H9|?`H9|?`H9|?`H9|?`H9|?`HIHIj?<D?`@?`@?`@\@\RM:Ǡ@feǠ?ꃀ manyelementini 2IScanbSewrittenasP Jajq1 :::Eaji?k ZsothatI=I22 ٹ=URKdz Ka2 +:::+dz QKan}.8Thisgivrestheresult.gHProp`ositionm4.4.6.iLffetm&HZ|beacommutativeHopfalgebraandH jM beanHV-moffdule.۳Thenwehave HP U԰ m=H) I=I22 a*andd;:H)~!=H I=I22 a*isgivenbyd(a);=f`Pa(1)$ `z>n( p(id ʢ")(a(2) \|)A.27 1384.pTHE!INFINITESIMALTHEOR:YVProof.@_ConsiderIthealgebraB˹:=HŢM-with(a;m)(a209;m20)=(aa20;am20۹+a209m).LetGi=K-c Alg f(HF:;-).ThenwrehaveG.(B)Hom-J(HF:;B)P԰̨=Hom+6b(HF:;HV)6Homy(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.$(HF:;M@)bry(u";D"lй))y7!(1;0)e(u";D"lй))y=(1;1eD"lй))y2G1(B)withinrversemap(1;D1))y7!(S ;0)Ö(1;D1)=(u";SvmÖD1)2G"lй(B).HenceFwrehaveisomorphismsDer#K(HF:;M@)P԰=Derݟ"0(HF:;M@)PUR԰n9=Hom(y(I=I22;M@)PUR԰n9=Hom(yH0ȹ(H I=I22;M@).Thehunivrersal"-derivXationforvectorspacesis: z id ʢ"#4:+AEX!q'I=I22.Thehuniversalf`"-derivXation07forHV-moSdulesisD"lй(a)u=1O `z3 p(id ʢ")(a)<p2uA I=I22. The07univrersal1-derivXation}uforHV-moSdulesis1D" Ewith}u(1D"lй)(a)=Pa(1)c `z>n( p(id ʢ")(a(2) \|)H-2`A I=I22.v g7 VgBibliography K`y cmr10[AdvqancedUUAlgebra]Z~BoGdoUUPareigis:q@': cmti10@A}'dvancedAlgebra.LectureNotesWS2001/02._139 ;7 @': cmti108 cmsy952@cmbx81%n eufm100#fcmti8/- cmcsc10.@ cmti12-N cmbx12,o cmr9*ppmsbm8) msbm10& msam10$a6cmex8#u cmex10"q% cmsy6!K cmsy8 !", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8Nff cmbx12XQ ff cmr12DtqGcmr17DtGGcmr17XQ cmr12K`y cmr10O line10