; TeX output 1998.02.24:1922%7 YRXQ cmr12CHAPTER2nNff cmbx12CommutativeffandNoncommutativeffAlgebraicGeometry"5PnN cmbx121.FnThePrinciplesofCommutativeAlgebraicGeometryWVeMwillbSeginwithsimplestformsof(commrutative)Mgeometricspaces.aConsideraHsetofpSoinrtswithoutanyadditionalgeometricstructure.WVeintroSducethenotionofitsalgebraoffunctions. C>Example2.1.1.oQLet?g cmmi12X¹bSeaset.kThen( msbm10K22cmmi8X ):=Map(XJg;K)isaK-algebrawithcompSonenrtwiseNadditionandmrultiplication:(f+lgn9)(x)UR:=fG(x)+gn9(x)Nand(fGg)(x)UR:=fG(x)gn9(x).8WVestudythisfactinmoredetail.TheԝsetK2X &consideredasavrectorspacewiththeaddition(f+rgn9)(x)UR:=fG(x)+gn9(x)andthescalarmrultiplication( fG)(x)z:= fG(x)de nesarepresenrtablecontravXariantfunctorK2-:URSet!", cmsy10!+fVec@=.;ThisfunctorisarepresenrtablefunctorrepresentedbyK.;InfactK2hC:URK2Y  !K2X &3isalinearmapforevrerymaph:XF``!YsinceK2he( f% + Ogn9)(x)=( f+ Ogn9)(h(x))o;= fG(h(x))+ Ogn9(h(x))o;=( fGh+ Ogn9h)(x)o;=( K2he(fG)+ OK2h(gn9))(x)henceK2he( f+ Ogn9)UR= K2he(fG)+ OK2h(gn9).Considerthehomomorphismo:URK2X =Q K2Y  !K2X K cmsy8Y,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=(f+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.BWAsvaspSecialexamplewreobtainamultiplicationrUR:K2X NK2X2up r4!"K2XX2)ppmsbm8K-:Aacmr6p!5!0WK2Xwhere::X ![ X<˹X inSetWisthediagonalmap(x):=(x;x).ʖFVurthermorewre{get#aunitË:URK2fg2QK-:;cmmi6p^!'@ K2X Awhere:XF``!fgistheuniquemapinrtotheoneelementset.OneXvreri eseasilythat(K2X;n9;r)XisaK-algebra.TwropropSertiesareessentialhere,theassoSciativitryandtheunitofKandthefactthat(XJg;;)isa\comonoid"inthecategorySetӋ:YowX+XoX+XXl32fd&H`O line10-'|{Ycmr81Z&XZiX+XM{fd?ά-̍ȕHKʟǠ*Ffe~Ǡ?` H=ʟǠ*FfepǠ?p#|1/^+o cmr937&*7 &e3862.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY Ԡt XԠX+X:2fdnά-ζ vzҟǠ@fez8Ǡ??AҬ>UH'<w ׁ w w w S̟>S̟> #K2Y#5lK2Xܞ32fd"€ά-VK-:fcommrute.ThrusK2-:URSet!+fK-cAlg"isacontravXariantfunctor.ByLthede nitionoftheset-theoretic(cartesian)proSductwreknowthatK2X ͠="u cmex10Q UWXrK.0This`idenrtitydoSesnotonlyholdonthesetlevel,բitholdsalsoforthealgebraRJstructuresonK2X 1resp.8Q7XK.FVor;eacrhpSointxUR2X-there;isamaximalidealmxofQXzKde nedbymx9:=URffQ2Map+Q(XJg;K)jfG(x)=0g.IfYIXJ̹isa nitesetthentheseareexactlyallmaximalidealsofQHVXeK.QTVoshorwthisweobservethefollowing.QThesurjectivehomomorphismpxk:Q UWXrKi!Khaskrernelmx ?rhencemxisamaximalideal.\IfmiQXmKisamaximalidealC{anda삹=( 1;:::ʚ; nP)2mC{thenforanry iQ\6=0weget(0;:::ʚ;0;1id;0;:::;0)=L(0;:::ʚ;0; 1 i p ;0;:::;0)( 1;:::; nP)UR2mahencethei-thfactor0-~:::%K:::0aisin[em. SotheelemenrtsaUR2m[emusthaveatleastonecommoncompSonent j\=UR0sincem}6=K.ButmorethanonesucrhacompSonentisimpSossiblesincewewouldgetzerodivisorsintheresidueclassalgebra.)ThrusmUR=mx wherex2XQisthejӹ-thelemenrtsoftheset.OnewcaneasilyshorwmorenamelythattheidealsmxarepreciselyallprimeidealsofMap(XJg;K).With)eacrhcommutativealgebraAwecanassoSciatethesetSpec(A)ofallprimeideals*ofA.IThatde nesafunctorSpSec:}K-Alg^xw!.:SetA#ٹ.AppliedtoalgebrasoftheformK2X r۹=URQX0Kwitha nitesetX+thisfunctorrecorversXasXPF԰_=~SpSec((K2X).'7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y439YTheyabSorveexampleshowsthatwemayhopSetogainsomeinformationonthesetX6bryE5.)ThismGhomomorphismofalgebrasmapspSoly-nomialsD5p(x1;:::ʚ;xnP)inrtofG(p)UR=p(a1;:::ʚ;anP).eHenceD5(a1;:::ʚ;anP)D5isacommonzeroof[thepSolynomialsp1;:::ʚ;pm `ifandonlyiffG(pidڹ)UR=pi(a1;:::ʚ;anP)=0,xi.e. Op1;:::ʚ;pmareHinthekrerneloffG.ThishappSensifandonlyiffvXanishesontheideal(p1;:::ʚ;pmĹ)orinotherwrordcanbSefactorizedthroughtheresidueclassmap `Jl:URK[x1;:::ʚ;xnP]n!1K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ)ThisinducesabijectionMor*K--2@cmbx8cAlgHY(K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ);A)UR3fQ7!(fG(x1);:::ʚ;f(xnP))UR2Xӹ(A):Norwitiseasytoseethatthisbijectionisanaturalisomorphism(inA).4샄cffxff ̟ff ̎ ̄cffgIfnopSolynomialsaregivrenfortheaborveconstruction,GthenthefunctorunderthisconstructionZ;istheanespaceA2n ofdimensionn.BygivingpSolynomialsthefunctorXbSecomes]asubfunctorofA2nP,zbecauseitde nessubsetsXӹ(A)URA2nP(A)=A2n. Bothfunctorsarerepresenrtablefunctors.TheembSeddingisinducedbythehomomorphismofalgebras:URK[x1;:::ʚ;xnP]n!1K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).)357 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y441YProblems2.1.8.x1.|Determine`theanealgebraofthefunctor\unitcircle" #Sן21 ]inA22. 2. #Determinetheanealgebraofthefunctor\unitsphere"Sן2n1"KinA2nP. 3. #LetX^denotetheplanecurvreyË=URx22.8,ThenXisisomorphictotheaneline. 4. #Let96Y5̹denotetheplanecurvrexyIA=1.$ThenYisnotisomorphictotheane #line.3(Hinrt:xlAnnisomorphismK[x;x21 \|]eF~!K[yn9]sendsxtoapSolynomialp(y) #whicrhmustbSeinvertible.qConsiderthehighestcoSecientofp(yn9)andshow #thatp(yn9)UR2K.8ButthatmeansthatthemapcannotbSebijectivre.) 5. #LetKdE=CbSethe eldofcomplexnrumbSers. Showthattheunitfunctor #UL[: wK-cAlg"$q!4SetL winULemma1.3.7isnaturallyisomorphictotheunitcircle #functor4RSן21r۹.(Hinrt:4ThereisanalgebraisomorphismbSetweentherepresenting #algebrasK[e;e21 \|]andK[c;s]=(c22j+s221).) 6. #(*)LetKbSeanalgebraicallyclosed eld. Letpbeanirreduciblesquare #pSolynomial"inK[x;yn9].LetZbetheconicsectionde nedbrypwiththeane #algebraH/K[x;yn9]=(p).QvShorwthatZ<;isnaturallyisomorphiceithertoX ortoY #fromparts3.8resp.4.$Remark2.1.9.j6Anem#algebrasofanescrhemesare nitelygeneratedcommu-tativreԒalgebrasandanysuchalgebraisananealgebraofsomeanescheme,sinceAPUR԰n9=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ)(HilbSertbasistheorem).The epSolynomialsp1;:::ʚ;pm )arenotuniquelydeterminedbrytheanealgebraofananescrheme."NoteventheidealgeneratedbythepSolynomialsinthepolynomialringK[x1;:::ʚ;xnP]isuniquelydetermined.AlsothenrumbSerofvXariablesx1;:::ʚ;xn B_isnotuniquelydetermined.The(K-pSoinrts( 1;:::ʚ; nP)UR2Xӹ(K)(ofananeschemeXX(withcoSecientsinthebase eldK)arecalledrffationalpoints.pqTheydonotsucetocompletelydescribSetheanescrheme.LetforexampleKP=RѹthesetofrationalnrumbSers. [IfXandYgareanescrhemes6withanealgebrasOUV(Xӹ)UR:=K[x;yn9]=(x22+;y22i׹+1)6andOUV(Y)UR:=K[x]=(x22+1)then[2bSothscrhemeshavenorationalpSoints. TheschemeY,Uhowever,has[2exactlytrwo5complexpSoinrts(withcoecienrtsinthe eldCofcomplexnumbSers)andthescrheme:XF hasin nitelymanycomplexpSoints,henceXӹ(C)Z6P԰= Y(C).ThisdoSesnotresultfromtheemrbSeddingsintodi erentspacesA22 Wresp.>8A21.InfactwealsohaveOUV(Y)UR=K[x]=(x22+1)P԰n9=K[x;yn9]=(x22+1;yn9),-socYcanbSeconsideredasananescrhemeinA22.SinceЦeacrhaneschemeXyisisomorphictothefunctorMorK-cAlg8<(OUV(Xӹ);-)wewillhenceforthidenrtifythesetwofunctors,thusremovingannoyingisomorphisms.$De nition2.1.10.}!ùLetLC=bK-A 6denotethecategoryofallcommrutative nitely)?generated(oranecf.2.1.9)K-algebras.AnaneX algebrffaicvariety)?isarepresenrtable?%functorK-A 7(A;-):K-A  !2cSetEkF. 6WThe?%anealgebraicvXarieties*KL7 &e4262.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYtogetherwiththenaturaltransformationsformthecffategoryT]ofanealgebrffaicvari-etiesVVar(K)orverK.!ThefunctorthatassoSciateswitheacrhanealgebraAitsanealgebraicvXarietryrepresentedbyAisdenotedbySpSecx:URK-A j!-IVVar@AH(K).By#theYVonedaLemmathefunctorSpSec:URK-A j!-IVar@AH(K)isananrtiequivXalenceof$categorieswithinrverse$functorO :>VVar=(K)>! K-A 7.9AnanealgebraicvXarietryiscompletelydescribSedbryitsanealgebraOUV(Xӹ).Arbitrary^(notnecessarily nitelygenerated)commrutative^algebrasalsode nerepresenrtablefunctors(de nedonthecategoryofallcommutativealgebras).Thuswealso|harve\in nitedimensional"vXarietieswhichwewillcallgeffometricVspacesoranevarieties.8WVedenotetheircategorybryGeom"P(K)andgetacommutativediagramP9K-cAlgٰGeom X(K)t32fd*ά- T덍PI԰.0=jnoHK-A HĄVVar (K){fd3QPά-ɪSpSecHŸǠ*Ffe1Ǡ?HŸǠ*Ffe#Ǡ?P-WVe}calltherepresenrtablefunctorsXH8:eK-cAlg""$<_!3SetJ&geffometricMspaces}oranevarieties,Band0therepresenrtablefunctorsX:LK-A d!.=SetFanestschemesoranealgebrffaic35varieties.The:HgeometricspacescanbSeviewredassetsofzerosinarbitrarycommutativeK-algebras[B#ofarbitrarilymanrypSolynomialswitharbitrarilymanyvXariables.@ThefunctionalgebrasofX{willbSecalledtheane35algebrffaofXinbSothcases.(fExample2.1.11.uQA'somewhat'lesstrivialexampleisthestatespaceofacircularpSenduluma(oflength1). 9 ThelocationisinL+=f(a;b)2A22ja224+0b22 묹=1g,themomenrtumKisinM%=Afmv-z2AgKwhichisastraightline.SothewholegeometricspaceforthepSendulumis(Lr,M@)(A)H=f(a;b;c)ja;b;cH2A;a2220+r,b22 !=1g.ThisgeometricspaceisrepresenrtedbyK[x;yn9;z]=(x22j+y221)sinceIS(L)M@)(A)UR=f(a;b;c)ja;b;cUR2A;a 2h-+)b 2V=1gP԰n9=K-cAlgo(K[x;yn9;z]=(x 2h-+)y 2f1);A):(fTheorem2.1.12.wXLffet(XubeangeometricspacewiththeanealgebraAUR=OUV(Xӹ).Then[APUR԰n9=Nat#@(X;A)asK-algebrffas,where[AUR:K-35cAlg":N$!3 SetJ:yistheunderlyingfunc-torEoraneline.DTheisomorphismAPUR԰n9=Nat#@(X;A)inducffesanaturaltransformationAXӹ(B)UR!B;(naturffal35inB).(fProof.@_FirstYwrede neanisomorphismbSetweenthesetsAandNatT(X;A).BecauseofX=MorJ K-cAlg9H(A;-)=:K-cAlgo(A;-)andA=MorJ K-cAlg9H(K[x];-)=:K-cAlgo(K[x];-)theYVonedaLemmagivresus-]Nat(X;A)UR=Nat(K-cAlgo(A;-);K-cAlg(K[x];-))PUR԰n9=K-cAlg(K[x];A)UR=A(A)P԰n9=Aonthesetlevrel.8LetUR:An!1Nat((X;A)denotethegivenisomorphism.Nat%@(X;A)carriesanalgebrastructuregivrenbythealgebrastructureofthecoSef- cienrts.aFVor}acoSecientalgebraB,aB-pSoinrtfҹ:A^!O3Bin}Xӹ(B)=K-Algo(A;B),+]o7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y443Yandu= ; B2Nat5(X;A)wrehave (B)(fG)2A(B)=B. ؠHenceu=( ݹ+N O)(B)(fG):=( (B)J+ O(B))(fG)= (B)(fG)J+ O(B)(fG)and( ^qJ O)(B)(fG):=( (B)J O(B))(fG)= (B)(fG) O(B)(fG)makreNat+Q(X;A)analgebra.Let-abSeanarbitraryelemenrtinA.7oBytheisomorphismgivenabSovethisele-menrtMinducesanalgebrahomomorphismga Z:VK[x]pL!AMmappingxontoa.Thisalgebraqhomomorphisminducesthenaturaltransformation(a)W:X3 !A.;Onqthe B-levrelsitisjustthecompSositionwithgaϹ,Wi.e. A(a)(B)(fG)=(K[x]h gaJ E! 3Ah fJ E!B). L~Since2sucrhahomomorphismiscompletelydescribSedbytheimageofxwegetI(a)(B)(fG)(x)9=f(a).TVoIcomparethealgebrastructuresofAandNat(X;A)letEa;a20r|2CA. JkWVeharveE(a)(B)(fG)(x)=f(a)Eand(a20:)(B)(fG)(x)=f(a209),hence(a陹+a209)(B)(fG)(x)=f(a陹+a209)=f(a)+f(a209)=(a)(B)(f)(x)+(a209)(B)(f)(x)=((a)(B)(fG)+(a209)(B)(f))(x)UR=((a)(B)+(a209)(B))(f)(x)UR=((a)+(a209))(B)(f)(x).Analogouslywreget(aa209)(B)(fG)(x) =f(aa209)=f(a)f(a209)=((a)X(a20))(B)(fG)(x),andWthrus(auU+a209)P=(a)uU+(a20)Wand(aa20)P=(a)uU(a20).HenceWadditionandmrultiplicationinNat(X;A)arede nedbytheadditionandthemultiplicationofthevXaluesfG(a)+f(a209)resp.8f(a)f(a20).WVepdescribSetheaction n9(B)8w:AXӹ(B)8wR!{B ofpAonXӹ(B).8Letfv:8wAR!{BbSe2aB-poinrtinK-cAlgo(A;B)t2=Xӹ(B).2FVor2eacrhat22A2theimage(a)t2:X6O!Aismanaturaltransformationhencewrehavemaps n9(B)m:AXӹ(B)mS!B*ssucrhmthat n9(B)(a;fG)h=f(a).FinallyYeacrhhomomorphismofalgebrasf:hB!B20inducesacommrutativediagramgэAXӹ(B20i?) B20cT32fd52Pά- Vp' n9(B20i?)H AXӹ(B)HrB{fd8Ѝά-P뎼 n9(B)HǠ*FfeDğǠ?`sAXӹ(fG)HǠ*Ffe6ğǠ?`!Df6OoThrus n9(B)UR:AXӹ(B)URn!1Bisanaturaltransformation.Kcffxff ̟ff ̎ ̄cff$Remark2.1.13.qN6Observre3thattheisomorphismAPgE԰,=#Nat#d(X;A)inducesanatu-ral9transformationA;xXӹ(B)URn!1BO?(natural9inB).&InparticulartheanealgebraAcanbSeviewredasthesetoffunctionsfromthesetofB-poinrtsXӹ(B)intothe\base"ringSB(functionswhicrharenaturalinB).SInthissensethealgebraAmaybSeconsid-eredasfunctionalgebraofthegeometricspaceXӹ.@>RHD6Xӹ(B)GǠ*Ffez̟Ǡ?3f1X(BI)commrutes. I WVewillshowthisresultlateronfornoncommutativealgebras. I TheunivrersalpropSertyimpliesthatthefunctionalgebraAofangeometricspaceXmisuniqueuptoisomorphism.Let5XbSeangeometricspacewithfunctionalgebraA=OUV(Xӹ).If5f:A@!KisarationalpSoinrtofXӹ,@'<>RH$D6Xӹ(B)u*Ǡ*Ffe˨\Ǡ?3@f1X(BI)K>2.1jLetFXbSeananescrhemewithanealgebraAUR=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).De ne[\coSordinatefunctions"qi %:KXӹ(B)5!R#B'awhicrh[describethecoordinatesofB-pSoinrtsandidentifythesecoSordinatefunctionswithelementsofA.ōNorwwewillstudymorphismsbSetweengeometricspaces.Theorem2.1.15.wXLffetXN"{A2r 4aandYA2s ;bffeanealgebraicvarietiesandletUR:X%}!Y/bffe35anaturaltransformation.fiThentherearepolynomialsYp1(x1;:::ʚ;xrb);:::;psn<(x1;:::;xrb)UR2K[x1;:::;xrb];such35thatYLP(A)(a1;:::ʚ;arb)UR=(p1(a1;:::ʚ;arb);:::;psn<(a1;:::;arb));-7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_45YforallA!2K-33A 5andall(a1;:::ʚ;arb)!2Xӹ(A),i.e.thenaturffaltransformationsbffetween35anealgebrffaicvarietiesarepolynomial.Proof.@_LetmOUV(Xӹ)$=K[x1;:::ʚ;xrb]=IandOUV(Y)=K[y1;:::ʚ;ysn<]=Jr. -/FVorA2K-Alg̨and09(a1;:::ʚ;arb)y2Xӹ(A)letfx:K[x1;:::ʚ;xrb]=Ip! AwithfG(xidڹ)=aibSetheIhomomorphismobtainedfromXӹ(A)P԰~=DK-Algo(K[x1;:::ʚ;xrb]=I;A).VTheInaturaltransformationFisgivrenbycompSositionwithahomomorphismg_:K[y1;:::ʚ;ysn<]=J!nK[x1;:::ʚ;xrb]=I+hencewreget(A)UR:K-cAlgo(K[x1;:::ʚ;xrb]=I;A)UR3fQ7!fGgË2K-cAlg(K[y1;:::ʚ;ysn<]=J:;A):SincegXisdescribSedbrygn9(yidڹ)UR=pi(x1;:::ʚ;xrb)2K[x1;:::ʚ;xrb]wreget!ʍi(A)(a1;:::ʚ;asn<)UR=(fGgn9(y1);:::ʚ;fgn9(ysn<))i=UR(fG(p1(x1;:::ʚ;xrb));:::;fG(psn<(x1;:::;xrb)))i=UR(p1(a1;:::ʚ;arb);:::;psn<(a1;:::;arb)): ろ %cffxff ̟ff ̎ ̄cffExample2.1.16.uQTheisomorphismbSetrweentheaneline(2.1.2)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).N=[2.NQuantumSpacesandNoncommutativeGeometryNorw8wecometononcommutativegeometricspacesandtheirfunctionalgebras.Manryofthebasicprinciplesofcommutativealgebraicgeometrycarryovertonon-commrutative:geometryV.)Ourmainaim,horwever,is:tostudythesymmetries(auto-morphisms)`ofnoncommrutative`spaceswhicrhleadtothenotionofaquantumgroup.Sincelktheconstructionofnoncommrutativelkgeometricspaceshasdeepapplicationsintheoreticalphrysicswewillalsocallthesespacesquantumspaces.De nition2.2.1.vaùLet AbSea(notnecessarilycommrutative) K-algebra.aThenthefunctorXX:=7K-Algo(A;-):K-Alg!1%SetIܹrepresenrtedbyAiscalled(ane)noncffommutativeo(geometric)space+ӹorquantumspfface.cThe+elemenrtsofK-Algo(A;B)arecalledB-pSoinrtsofXӹ.rAmorphism|ofnoncffommutativespacesfb:cX{6!N&Ynisanaturaltransformation.Remark2.2.2.j6ThexnoncommrutativespacesformacategoryQS}thatisdualtotheYcategoryofK-algebras.*qThrusoneoftencallsthedualcategoryK-AlgoWop%>categoryofnoncommrutativespaces.IfAisa nitelygeneratedalgebrathenitmarybSeconsideredasaresidueclassalgebrabAPUR԰n9=Khx1;:::ʚ;xnPi=IofapSolynomialalgebrainnoncommrutingvXariables(cf.1.5.10).8IfpI)=8w(p1(x1;:::ʚ;xnP);:::;pmĹ(x1;:::;xnP))pisthetrwo-sidedpidealgeneratedbryfthepSolynomialsp1;:::ʚ;pm kjthenthesetsK-Algo(A;B)canbSeconsideredassets.ڠ7 &e4662.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYofXzerosofthesepSolynomialsinB2nCV.Infact,gwrehaveK-Algo(Khx1;:::ʚ;xnPi;B)P[԰t׹=Map+Q(fx1;:::ʚ;xnPg;B)UR=B2nCV.ThrusOK-Algo(A;B)canbSeconsideredasthesetofthosehomomorphismsofalgebrasfromKhx1;:::ʚ;xnPitoB1 thatvXanishontheidealIorasthesetofzerosofthesepSolynomialsinB2nCV.SimilartoTheorem2.1.15oneshorwsalsointhenoncommutativecasethatmor-phismsbSetrweennoncommutativespacesaredescribSedbypSolynomials.TheTheorem2.1.12ontheopSerationoftheanealgebraA=OUV(Xӹ)onXչasfunctionalgebracanbSecarriedorvertothenoncommrutativecaseaswrell: thenaturaltransformation n9(B)UR:A[Xӹ(B)URn!1B_ (naturalinB)isgivrenby n9(B)(a;fG)UR:=f(a)andcomesfromtheisomorphismAPUR԰n9=Nat#@(X;A).]KNorw'wecometoaclaimonthefunctionalgebraAthatwedidnotproveinthecommrutativecase,$butthatholdsinthecommrutativeaswrellasinthenoncommuta-tivresituation.Lemma2.2.3.g5QLffetpD"beanalgebraandP:D/Xӹ(-35)!A(-)pbffeanaturaltrffansformation.5ThenͫthereexistsauniquehomomorphismofalgebrasfY:sZD-S!Asuch35thatthediagrffamFm BsAXӹ(B)pB3l32fdά-W` I{(Bd)H`!(Bd)εׁ @ص @ @ @|>@|>RHdD6Xӹ(B)JjǠ*Ffe}Ǡ?`f1~cffommutes.Proof.@_The naturaltransformationa6:Di^X# 7 &e4862.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYRemark2.2.11.qN6TVogether0withXandY-theorthogonalproSductX%?URYisagainafunctor, sincehomomorphismsf˚:B8,!B20oarecompatiblewiththemrultiplicationandthruspreservecommutingpSoints.8HenceX%?URY>isasubfunctorofXl{Y.ULemma2.2.12.mQIfWXandYToarffenoncommutativespaces,athenX[?-YToisanon-cffommutative35spacewithfunctionalgebraOUV(X%?URY)=O(Xӹ) O(Y).IfjX,andYgWhave nitelygenerffatedjfunctionalgebrffasthenthefunctionalgebraofX%?URY/is35also nitelygenerffated.Proof.@_Let A:=OUV(Xӹ)andA20<:=OUV(Y).Let(p;p209)2(Xy?Y)(B)bSeapairof1commrutingpSoints. Thenthereisauniquehomomorphismofalgebrash*:A A20!nBsucrhthatthefollowingdiagramcommutesF / HA/ ƏA A20A{fdά- OH pׁ @ @ @Ü @✟>@✟>RHѵ"B:ʟǠ*FfeǠ? ̤h/ /  A20 ,{fdk ά{20H|p20ܟׁ ܟ ܟ ܟ S\>S\> s2De 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),anda20N920q2URA20N920:=OUV(Z )epxH(a)py (a 09)pzʮ(a 0N9 0r)UR=(px;py )(a a 09)pzʮ(a 0N9 0r)UR=pz(a 0N9 0r)(pxH;py )(a a 09)UR=pz(a 0N9 0r)pxH(a)py (a 09)andpxH(a)py (a 09)UR=py(a 09)pxH(a):s2IfwrechoSosea˹=1thenwegetpy (a209)pzʮ(a20N920r)=pz(a20N920r)py (a209).FVorarbitrarya;a20;a20N920 "wrethengete32fdPά-l~rH3,1X.M٬ҁ H٬ׁ H٬܁ H٬ H٬ H٬ H٬ H٬ H fdžH fdžj3commrute.8Then(M;m;e)iscalledaquantum35monoid.ѤProp`osition2.3.2.OLffetMbeanoncommutativespacewithfunctionalgebraHV.Then35H isabialgebrffaifandonlyifMisaquantummonoid.Proof.@_SincethefunctorsM ?M,M?Es׹andEt ?MarerepresenrtedbyH tH)resp..H KP԰=H)resp..K HP&԰ =MHtheObservrenthatasimilarresultcannotbSeformulatedforHopfalgebrasH\&sinceneither`theanrtipSode`SnorthemultiplicationrUR:H| HB\3!HMlare`algebrahomomor-phisms.Inqfconrtrasttoanealgebraicgroups(3.3.2)HopfalgebrasinthecategoryK-AlgoWopP$԰$Ź=1QRarenotgroups.8Nevrertheless,onede nesDe nition2.3.3.vaùABnfunctorBde nedonthecategoryofK-algebrasandrepre-senrtedbyaHopfalgebraHiscalledaquantum35grffoup.2ܦ7 &e5062.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYDe nition2.3.4.vaùLetHX bSeanoncommrutativeHspaceandletMbeaquanrtummonoid.A hmorphism (anaturaltransformation)ofquanrtumspacesUR:M?X%0!Xiscalledanopfferation꨹ofMonX{ifthediagramsP=* IMUR?X* 2X@32fd=ά-g܍sk׶MUR?M?XskIMUR?Xf${fdά-mUR?1H62Ǡ*FfeidǠ?ay1UR?H(2Ǡ*Ffe[dǠ?  7}andHv䍍3kkXP%԰0 =E i?URX3kYMUR?XT{fd;pά-iPfI{?idHXBǠ*FfetǠ?'Hf4idX.Xടҁ Hടׁ Hട܁ Hട Hട Hട Hട Hട HmdžHmdžj֍commrute.8WVecallX{anoncommutativeM-spfface.ꥍProp`osition2.3.5.OLffetgX^:beanoncommutativespacewithfunctionalgebraAUR=OUV(Xӹ). |LffetMbeaquantummonoidwithfunctionalgebraBv= OUV(M). |Let :M/?Xa m!fX1hbffeoamorphisminQSandletf.:/A!B0 Aobetheassociatedhomomorphism35ofalgebrffas.fiThenthefollowingareequivalent1.(X;M;)hEisanopfferationhEofthequantummonoidMonthenoncffommutativespfface35X;2.fi(A;HF:;fG)35de neanHV-cffomodule35algebra.ꥍProof.@_TheLMhomomorphismsofalgebras# 1A,d1B ;a fG, 1A *1etc.]represenrtthemorphismsofquanrtumspacesmx?idC6,id?,?idetc.:6HencetherequireddiagramsaretransferredbrytheYVonedaLemma.Qcffxff ̟ff ̎ ̄cffExample2.3.6.oQ1.8Thequanrtummonoidof\quantummatrices":WVeconsiderthealgebra̍tMq(2)UR:=Kha;b;c;di=IFչ=URKqʍ a b 8c+d!Fq,F=I͍wherethetrwo-sidedidealI+isgeneratedbrytheelementsA tEabqn9 1 ʵba;acqn9 1ca;bdqn9 1db;cdqn9 1dc;adda(qn9 1u]qn9)bc;bccb:ThequanrtumspaceMq(2)assoSciatedwiththealgebraMq(2)isgivrenby%~fZ(6Mq(2)(A)\*8=URK-Algo(Mq(2);A)\*8=URq USqʍ*a20(b20c20(#d201 q;vja209;b20;c20;d20#2URA;a20b20#=URqn921 ʵb20a20;:::ʚ;b20c20#=URc20b20q3t7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)51YwhereeacrhhomomorphismofalgebrasfQ:URMq(2)n!1AisdescribSedbythequadruple(a209;b20;c20;d20)ofimagesofthealgebrageneratorsa;b;c;d.[Theimagesmrustsatisfythesamerelationsthatgeneratethetrwo-sidedidealI+henceGʍJka209b20#=URqn921 ʵb20a20;a20c20=URqn921 ʵc20a20;b20d20=URqn921 ʵd20b20;c20d20=URqn921 ʵd20c20;yb209c20#=URc20b20;a20d20xqn921 ʵb20c20=URd20a20xqn9c20b20:WVekwritethesequadruplesas22-matriceskandcallthemquantum4matricffes.Theunrusualcommutationrelationsarechosensothatthefollowingexmpleswork.ThequanrtumspaceofquantummatricesturnsouttobSeaquantummonoid.WVegivreebSoththealgebraic(withfunctionalgebras)andthegeometric(withquantumspaces)approacrhtode nethemultiplication.a)The35algebrffaicapproach:ThealgebraMq(2)isabialgebrawiththediagonal4"qʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq- qʍ a9 b 9cd!q,Y;X0i.e. RbryH(a)=a a+b c,)(b)=a b+b d,(c)=c a+d cHand(d)UR=c b+d d,andwiththecounitX/"qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,;i.e.`"(a)G=1,f"(b)=0,"(c)=0,andMT"(d)=1.`WVeharvetoprovethatand"arehomomorphismsofalgebrasandthatthecoalgebralarwsaresatis ed.TVoobtainahomomorphismofalgebrasy:Mq(2)}! qMq(2) Mq(2)wrede ney:Kha;b;c;di!nMq(2)+ Mq(2)onthefreealgebra(thepSolynomialringinnoncommrutingvXariables)Kha;b;c;di:generatedbrythesetfa;b;c;dg:andshowthatitvXanishesontheidealIorbfmoresimplyonthegeneratorsoftheideal. ThenitfactorsthroughauniquehomomorphismJofalgebrasUR:Mq(2)n!1Mq(2)x Mq(2).0WVeJcrheckthisonlyforonegeneratoroftheidealI:/Gʍ(abqn921 ʵba)UR=(a)(b)qn921(b)(a)UR==UR(a a+b c)(a b+b d)qn921 ʵ(a b+b d)(a a+b c)=URaa ab+ab ad+ba cb+bb cdqn921 ʵ(aa ba+ab bc+ba da+bb dc)=URaa (abqn921 ʵba)+ab (adqn921bc)+ba (cbqn921da)+bb (cdqn921dc)=URba (qn921 ʵadqn922bc+cbqn921da)UR0 moSd&6(I):/GThereadershouldcrhecktheotheridenrtities.ThecoassoSciativitryfollowsfrom)󍍟񙐍( 1)qʍ Vacb ccWd!Yq-F=URqʍ Vacb ccWd!Yq,Y qʍ a9 b 9cd!q-=(qʍXaeb ccYd[q*[ qʍ a9 b 9cd!q) qʍ a9 b 9cd!q-=ߍ8=URqʍ *ab cUd"pq- (qʍXaeb ccYd[q*[ qʍ a9 b 9cd!q)UR=qʍ *ab cUd"pq- qʍ Vacb ccWd!Yq-F=(1 )qʍ Vacb ccWd!Yq+:)ThereadershouldcrheckthepropSertiesofthecounit.4)7 &e5262.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYb)The35geffometricapproach:Mq(2)hasaratherremarkXable(andactuallywrellknown)comultiplicationthatisbSettersunderstoodbryusingtheinducedmultiplicationofcommutingpSoints.{GivenÍtrwofOcommutingquantummatricesqʍ ;a1!b1 c1!&d1,q8=Yandqʍ ;a2!b2 c2!&d2,qinMq(2)(A). ThentheirčmatrixproSduct5rXգqʍaa1w$ b1b94c1vd1qqF\qʍa2b2c2d2῟q i=URqʍ *a1a2j+b1c2N a1b2j+b1d2 *c1a2j+d1c2N:c1b2j+d1d2̟q5sisagainaquanrtummatrix.8TVoprovethisweonlycheckoneoftherelations5ʍ(a1a2j+b1c2)E(a1b2j+b1d2)UR=a1a2a1b2+a1a2b1d2+b1c2a1b2+b1c2b1d2E=URa1a1a2b2j+a1b1a2d2+b1a1c2b2+b1b1c2d2E=URqn921 ʵa1a1b2a2j+qn921b1a1(d2a2j+(qn921u]qn9)b2c2)+b1a1b2c2+qn921 ʵb1b1d2c2E=URqn921 ʵ(a1a1b2a2j+a1b1b2c2+b1a1d2a2+b1b1d2c2)E=URqn921 ʵ(a1b2a1a2j+a1b2b1c2+b1d2a1a2+b1d2b1c2)E=URqn921 ʵ(a1b2j+b1d2)(a1a2+b1c2)5WVeharveusedthatthetwopSointsarecommutingpSoints.˭ThismultiplicationobviouslyisanaturaltransformationMq(2)S"?Mq(2)(A)l!Mq(2)(A)(naturalinA).9ItistassoSciativreandhasunitqʍ 10 01"q+UP.8FVortheassociativitryobservethatby2.2.13#F?3((qʍXa1Nib1 cc1]d1(cq3p;qʍ Va2 Ngb2 cc2[d2*aq);qʍ Va3 Ngb3 cc3[d3*aq)5sisapairofcommrutingpSointsifandonlyif5r~?4(qʍXa1Nib1 cc1]d1(cq3p;(qʍXa2Nib2 cc2]d2(cq;qʍ Va3 Ngb3 cc3[d3*aq))isapairofcommrutingpSoints.ÍSince_qʍk1K0k0K1$+q/qʍ7]aHjb8dcH^dN`q]=-Xqʍabc-d%Hq4Kc=-Xqʍabc-d%Hq0 qʍ8a1H]08a0H]1NYqfor_allquanrtummatricesߍqʍXaeb ccYd[q+F2URMq(2)(B)wreseethatMq(2)isaquanrtummonoid.fpItCremainstoshorwthatthemultiplicationofMq(2)andthecomultiplicationofMq(2)pcorrespSondtoeacrhotherbytheYVonedaLemma.9TheidentitymorphismofMq(2) Mq(2)isgivrenbythepairofcommutingpSointsy(1;2)UR2Mq(2)?Mq(2)(Mq(2) Mq(2))UR=K-Algo(Mq(2) Mq(2);Mq(2) Mq(2)):;#SinceU1 &ӹ=fϟqʍ <'a4b `cg(d#*q/o  1f=qʍ <'a 12b 1 `c 11d 1LpqYand2 &ӹ=1 qʍ earb {cfd"3hq/o=fϟqʍ <'1 a2׹1 b `1 c1˹1 dLpqߍwreghaveidZ=)(1;2)=(qʍXaeb ccYd[q* ]1;1 qʍ Եab bcd"q).!ThegYVonedaLemmade nesthefpdiagonal}astheimageoftheidenrtity}underK-Algo(Mq(2) Mq(2);Mq(2) Mq(2))5 7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)53Yfo!jK-Algo(Mq(2);Mq(2)& Mq(2))Zbrythemultiplication. zSo(qʍXaeb ccYd[q')Q==ߍ1j2V=UR(qʍXaeb ccYd[q*[ 1)(1 qʍ a9 b 9cd!q)UR=qʍ *ab cUd"pq- qʍ a9 b 9cd!q.fpThrusMq(2)de nesaquantummonoidMq(2)with#AMq(2)(B)UR=q USqʍ*a20(b20c20(#d201 qȍ9x 389x =xa 09;b 0;c 0;d 0#2URB;a 0b 0#=URqn9 1 ʵb 0a 0;:::ʚ;b 0c 0#=URc 0b 0q&:DՍThis #isthedeformedvrersionofM諍2 %themultiplicativemonoidofthe2i2-matricesofcommrutativealgebras.wu2.&LetAߍ2j0]Gqι=URKhx;yn9i=(xyA 0j2ڍq \|(A 09)UR= n USGʍUa20#Ώb20+G3,1 381 1 5a 0;b 0#2A 0;a 0 8|2=0;b 0 !|2w=0;a 0b 0#=qn9b 0a 0 o :hTheyquanrtummonoidMq(2)alsoopSeratesonthedualquantumplanebymatrixmrultiplicationWA 0j2ڍq \|(A 09)UR?Mq(2)(A 0)3(Gʍ69pG";qʍ Vacb ccWd!Yq))7!GʍTןG&qʍ a9 b 9cd!q-2A 0j2ڍq \|(A 0):%This/givresanotherexampleofaMq(2)-comoSdulealgebraAߍ0j2]Gq&-?!! kAߍ0j2]Gq5 4Mq(2)withts2(Gʍ69pG )UR=Ȅ=(Gʍ69pG#/ 1)(1 qʍ a9 b 9cd!q*[)UR=GʍTןG& qʍ a9 b 9cd!q,Y:6"7 &e5462.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYWhatUisnorwthereasonfortheremarkXablerelationsonMq(2)?ItisbasedonaNfactthatwrewillshowlaternamelythatMq(2)istheuniversalquantummonoid}&acting onthequanrtumplaneAߍ2j0]GqgfromtheleftandonthedualquantumplaneAߍ0j2]Gqfrom\therighrt.ThishoweverhappSensinthecategoryofquantumplanesrepresentedbryquadraticalgebras.Herewewillshowasimplertheoremfor nitedimensionalalgebras.OProblems2.3.7.sXDetermineƹtheH-pSoinrtsofthequantumplaneAߍ2j0]Gq#5whereHistheR-algebraofthequaternions.>ڍDe nition2.3.8.ǹ1.KLetXzbSeaquanrtumspace.AquantumspaceM(Xӹ) #togetherwithamorphismofquanrtumspacesu|:M(Xӹ)?X7OP!XX`iscalleda #quantumspffaceactinguniversallyonXFϹ(orsimplyauniversalquantumspfface #fornhXӹ)ifforevreryquantumspaceYjandeverymorphismofquantumspaces #fQ:URYQ?X%0!XTthereisauniquemorphismofquanrtumspacesgË:URYks!M(Xӹ) #sucrhthatthefollowingdiagramcommutesO| M(Xӹ)UR?XX:Ԟ32fdά-W>H`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.XH 2. #LetYyAbSeaK-algebra.UAY]K-algebraM@(A)togetherwithahomomorphismof #algebras :AG!5M@(A)9 AiscalledanalgebrffaQcoactinguniversallyon A(or #simplyauniversalalgebrffaforA)ifforevreryK-algebraBandeveryhomomor- #phismbofK-algebrasfi~:!A; !\BY SAbthereexistsauniquehomomorphismof #algebrasgË:URM@(A)n!1BsucrhthatthefollowingdiagramcommutesRȍH@AHM@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.AJBytheunivrersalpropSertiestheuniversalalgebraM@(A)forAandtheuniversalquanrtumspaceM(Xӹ)forX{areuniqueuptoisomorphism.>ڍProp`osition2.3.9.o1.s+LffetpKAbeaK-algebrawithuniversalalgebraM@(A)and #:.A:! M@(A) A.ThenM(A)isabialgebrffaandAisanM(A)-cffomodule #algebrffa35bys2. 2. #IfѻBlisabialgebrffaandiffQ:URA!Bm  Aѻde nesthestructureofaB-comodule #algebrffa5onAthenthereisauniquehomomorphismgË:URM@(A)!Bof5bialgebras76Р7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)55Y #such35thatthefollowingdiagrffamcommutesTOKH2PAHKM@(A) Aw{fdPЍά-H`f$ׁ @$ @$ @$ @>@>RH(BE AҟǠ*FfeQǠ?Wg 1AᇍThecorrespSondingstatemenrtforquantumspacesandquantummonoidsisthefollorwing.Prop`osition2.3.10./1.3+Lffet<}XPbeaquantumspacewithuniversalquantum #spffaceM(Xӹ)andUR:M(X)?A!A.9ThenM(X)isaquantummonoidand #Xis35anM(Xӹ)-spfface35by. 2. #IfbY_$isanotherquantummonoidandiff:Y?Xn>!@X$ade nesbthestructurffe #of CaY-spfface ConXthentherffeisauniquemorphismofquantummonoidsgË:URY #"!1GM(Xӹ)35suchthatthefollowingdiagrffamcommutesO׍ M(Xӹ)UR?XX:Ԟ32fdά-W>H`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.X~Proof.@_WVegivretheproSofforthealgebraversionofthepropSosition.v4ConsiderthefollorwingcommutativediagramElgM@(A) A[fM@(A) M(A) A32fd9Nά-c!1M(A)pR hAgM@(A) A :2fdfά-mxટǠ@feܟǠ?J~Q˪Ǡ@feܟǠ?%\ 1X.A where themorphismofalgebrasisde nedbrytheuniversalpropSertyofM@(A)withrespSecttothealgebramorphism(1M"(A) %Is2).U6FVurthermorethereisauniquemorphismofalgebrasUR:M@(A)n!1K꨹sucrhthatR(wHAHM@(A) Al{fdPЍά--,H31X.Auׁ @u @u @u @׼>@׼>RH"%APUR԰n9=K AJǠ*Ffe|Ǡ?3 1X.Acommrutes.8D7 &e5662.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYThecoalgebraaxiomsarisefromthefollorwingcommutativediagramswJ{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) andpg|AM@(A) As”A2fdpά-mzk2Π@fekdΠ?QpΠ@feKΠ?%. 1X.A3SrǠM@fe3Ǡ?89$1M(A)pR 1X.APM@(A) A+M@(A) M(A) Al:2fd9Nά-}C1M(A)pR x 1M(A)pR 1X.AܔP P?_P攴P P?^PPʍPԍ?]P&D P&D q@M@(A) APUR԰n9=M(A) K AǠ@feKǠ?x 1M(A)pR  1X.A"XandprAM@(A) A~tA2fdfά-mpǠM@feq$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-m"XInnfactthesediagramsimplybrytheuniquenessoftheinducedhomomorphismsofalgebras*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)!nBsucrhthatthediagramO)VH@AHM@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.A9P7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)57Ycommrutes.8ThenthefollowingdiagramHѭTAѭM@(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 fPimplies((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 1A)(g 1A)Ȅ=UR(BN>g 1A)]ڹhence(g gn9)UR=BN>g.8FVurthermorethediagramq[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.AimpliesBN>gË=UR.8ThrusgXisahomomorphismofbialgebras.}Ccffxff ̟ff ̎ ̄cff6SinceunivrersalalgebrasforalgebrasAtendtobSecomeverybigtheydonotexistingeneral.pButatheoremofTVamrbara'ssaysthattheyexistfor nitedimensionalalgebras(orvera eldK).De nition2.3.11.}!ùIfX|bisaquanrtumspacewith nitedimensionalfunctionalgebrathenwrecallX{a nite35quantumspfface.ThenfollorwingtheoremisthequantumspaceversionandequivXalenttoatheoremofTVamrbara.Theorem2.3.12.wXLffet7Xbea nitequantumspace.tThenthereexistsa(uni-versal)quantumspffaceM(Xӹ)withmorphismofquantumspffaces:M(X)?Xfk!X.ThealgebravrersionofthistheoremisTheorem2.3.13.wX(T3ambara)LffetAbea nitedimensionalK-algebra.WThentherffeexistsa(universal)K-algebraM@(A)withhomomorphismofalgebras:Afk!M@(A) A.Proof.@_WVeTaregoingtoconstructtheK-algebraM@(A)quiteexplicitly.oFirstwreobservrethatA2V=URHom۟K"(A;K)isacoalgebra(cf.'problem1.6.9)withthestructuralmorphism+;:A2? !b(A A)2P?԰&=A2 A2.\DenotethedualbasisbryP*~n U_~i=1!ai; a2i+2A% A2.dNorw,3letTƹ(A% A2),3bSethetensoralgebraofthevrectorspaceA% A2.dConsider:_7 &e5862.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYelemenrtsofthetensoralgebra'~ʍMxy =S2URA A2;Mx y ()UR2A A A2j A2PV԰.==A A2 A A2;M1 =S2URA A2;M(1)UR2K:Thefollorwingelementsa*#xy x y ()(1)iand01 (1)(2)`generateatrwo-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).WVeiter)ists2(1)c=PxisR(1r a2i @M) aiȥ=c˟PxiӹsPa2i+(1) ai=c1 P[iza2iOҹ(1)ai=c1 1.RcHence)isahomomorphismofalgebras.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)gn9(1=- %.(1))ß=(1=- )fG(1)(1)ß=(1=- )(1 1)(1)ß=1(1)=-(1)ß=0.ThrusthehomomorphismofalgebrasgtvXanishesontheidealI*soitmaybSefactoredthroughM@(A)=Tƹ(A)=I.6Denotethisfactorizationalsobrygn9.Thenthediagram;p7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)59Ycommrutes^since(gh ̹1A)s2(a)-=(g ̹1A)(P i(a Oa2i ) aidڹ)-=Pڟi*(1 Oa2i)fG(a) ai=$P ijfjf (a)(%a2i ;aj) ai,=URPidfidڹ(a) ai=URfG(a).cWVek}stillharvek}toshorwthatgٶisuniquelydetermined._Assumethatwealsohave(h 1A)pO=f;thenM