; TeX output 1999.09.20:15417 YRXQ cmr12CHAPTER1nNff cmbx12CommutativeffandNoncommutativeffAlgebraicGeometry"N cmbx12Intro`ductionThroughoutMLwrewill xabase eld' msbm10K.lThereadermayconsideritasrealnumbSersorcomplexnrumbSersoranryotherofhismostfavorite elds.A.fundamenrtal/BandpSowerfultoSolforgeometryistoassociatewitheacrhspaceg cmmi12Xethet-algebraoffunctions!", cmsy10OUV(X)fromXtothebase eld(ofcoSecienrts). oThedreamofgeometryisthatthisconstructionisbijectivre,i.e.'Rthattwodi erentspacesareEmappSedtotrwoEdi erentfunctionalgebrasandthateachalgebraisthefunctionalgebraofsomespace.ActuallyRthespacesandthealgebraswillformacategoryV.pThereareadmissiblemaps. FVoruhalgebrasitisquiteclearwhatthesemapswillbSe.FVorspacesthisislessjobrvious,partlyduetothefactthatwedidnotsayclearlywhatspacesexactlyare.)Thenthe*@ cmti12drffeamzofgeffometrywrouldbSethatthesetwocategories,ęthecategoryof(certain)spacesandthecategoryof(certain)algebras,aredualtoeacrhother.AlgebraicgeometryV,noncommrutativegeometryV,andtheoreticalphysicshaveasabasis%thisfundamenrtalidea, thedualityoftwocategories, thecategoryofspaces(statespaces/inphrysics)andthecategoryoffunctionalgebras(algebrasofobservXables)inphrysics.WVewillpresentthisdualityinthe1.chapter.CertainlythetypSeofspacesaswrellasthetypSeofalgebraswillhavetobSespeci ed.TheoreticalIphrysicsusesthecategoriesofloScallycompactHausdor spacesandof;commrutativeCܞ2K cmsy8-algebras.jA7famoustheoremofGelfand-Naimarksaysthatthesecategoriesaredualsofeacrhother.(Ane)algebraicgeometryusesadualitrybSetweenthecategoriesofanealgebraicscrhemesandof(reduced) nitelygeneratedcommutativealgebras.TVo9sgetthewholeframewrorkofalgebraicgeometryoneneedstogotomoregen-eral`zspacesbrypatchinganespacestogether.WOnthealgebrasidethisamountstoconsideringlshearvesofcommutativealgebras.,WVeshallnotpursuethismoregeneralapproacrhtoalgebraicgeometryV,sincegeneralizationstononcommutativegeometryarestillinthestateofdevrelopmentandincomplete.Noncommrutativegeometryuseseither(imaginary)noncommrutativespacesandnotQnecessarilycommrutativeQalgebrasor(imaginary)noncommrutativeQspacesandnotnecessarilycommrutativeCܞ2-algebras.WVecwilltakreanapproachtothedualitybSetweengeometryandalgebrathatheavilyuses-$functorialtoSols,=especially-$represenrtablefunctors.TTheane(algebraic)spaces]+o cmr91*7 &e261.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYwre)usewillbSegivenintheformofsetsofcommonzerosofcertainpSolynomials,PuwherethezeroscanbSetakreninarbitrary(commutative)K-algebrasB.Soananespacewillconsistofmanrydi erentsetsofzeros,depSendingonthechoiceofthecoSecientalgebraB.WVe rstgivreashortintroSductiontocommutativealgebraicgeometryinthissetupanddevrelopadualitybSetweenthecategoryofane(algebraic)spacesandthecate-goryof( nitelygenerated)commrutativealgebras.Thenwrewilltransferittothenoncommutativesituation.=ThefunctorialapproachtoalgebraicgeometryisnottoSooftenusedbutitlendsitselfparticularlywrelltothestudyyofthenoncommrutativeysituation.LEveninthatsituationoneobtainsspace-likeobjects.Thecrhapterwillclosewitha rststeptoconstructautomorphism\groups"ofnoncommrutativespaces.*SincetheconstructionofinrversespresentsspSecialproblemswrewillonlyconstructendomorphism\monoids"inthischapterandpSostponethestudyofinrvertibleendomorphismsorautomorphismstothenextcrhapter.Arttheendofthechapteryoushoulds2# #knorwhowtoconstructananeschemefromacommutativealgebra,# #knorwhowtoconstructthefunctionalgebraofananescheme,# #knorwwhatanoncommutativespaceisandknowexamplesofsuch,# #understand88andbSeabletoconstructendomorphismquanrtummonoidsofcer- #tainnoncommrutativespaces,# #understand,Awhry/endomorphismquantummonoidsarenotmadeoutofendo- #morphismsofanoncommrutativespace. p7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y93Y5Pn1.FnThePrinciplesofCommutativeAlgebraicGeometryWVeIwillbSeginwithsimplestformof(commrutative)Igeometricspacesandseeadualitry$bSetweentheseverysimple\spaces"andcertaincommutativealgebras.UThisexampleľwillshorwhowtheconceptofafunctionalgebracanbSeusedtoful llthedream ofgeometryinthissituation. ItwillalsoshorwthefunctorialmethoSdsthatwill\bSeappliedthroughoutthistext.vItisaparticularlysimpleexampleofadualitryas$menrtionedintheintroSduction.RTThisexamplewillnotbeusedlateron,CsowrewillonlyskretchtheproSofsforsomeofthestatements.gSExample1.1.1.oQConsiderasetofpSoinrtswithoutanyadditionalgeometricstruc-ture.{So/thegeometricspaceisjustaset.WVeinrtroSducethenotionofitsalgebraoffunctions.LetR&XCbSeaset. o[ThenK22cmmi8X ¹:=9Map(XJg;K)isaK-algebrawithcomponenrtwiseaddition[andmrultiplication:U+V Ogn9)UR= K2he(fG)+ OK2h(gn9).Considerthehomomorphismo:URK2X =Q K2Y 4!K2XY,Cde nedbryW(fg gn9)(x;y)UR:=fG(x)gn9(y).XIn,ordertoobtainauniquehomomorphism,de nedonthetensorproSductwrehavetoshowthatW20z:URK2X }*_K2Y 4!K2XYPisabilinearmap:&W20%V(f+fG208;gn9)(x;y)UR=(fő+}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.kOnecancheckthatRǹisalwaysinjectivre.8IfX+orYare nitethenAŹisbijective.BWAsvaspSecialexamplewreobtainamultiplicationrUR:K2X NK2X2up r4!"K2XX2(ppmsbm8K-:Aacmr6p!5!0WK2Xwhere::X4![ X=˺X inSetWisthediagonalmap(x):=(x;x).ʗFVurthermorewre{get#aunitË:URK2fg2QK-:;cmmi6p^!'@ K2X Awhere:XF4!fgistheuniquemapinrtotheoneelementset.OneXvreri eseasilythat(K2X;n9;r)XisaK-algebra.TwropropSertiesareessentialhere,theassoSciativitryandtheunitofKandthefactthat(XJg;;)isa\comonoid"inthecategorySetӋ:FwX+XnX+XXl32fd&H`O line10-'|{Ycmr81Y&XYiX+XM{fd?ά-ˍȕHKʟǠ*Ffe~Ǡ?` H=ʟǠ*FfepǠ?o#|17 &e461.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY Ԡt XԠX+X:2fdnά-ζ vzҟǠ@fez8Ǡ??;np(kSm1X.X$ܔP$ P$?_P$攴P$ P$?^P$P$P$?]Py Py qҟǠ@fe#Ǡ?ꬽ Մ1h0X+X˲fgXPF԰_=~XPF԰_=X+fg:t32fd1pά-r V13RSince+K2-isafunctorthesetrwo+diagramscarryorver+tothecategoryVecEandproSducetherequireddiagramsforaK-algebra.FVorgqamapfq:)X:4!^~YwreobtainahomomorphismofalgebrasK2f ֹ:K2Y zt4!K2XbSecausethediagramsO ɱ㍑mUK2Y e K2Yɱ㍒Z(K2YYˌ{fd* `ά-wi#lK2X 1 K2Xi#͍\K2XX32fd(ά-Íw1㎎1㍒-#K2Y{fd9ά-xK-:##,AҬ>UH'<w ׁ w w w S̟>S̟> #K2Y#5lK2Xܞ32fd"€ά-VK-:fuycommrute.ThrusG(K -:URSet4!+fK-cAlgisaconrtravXariantfunctor.ByLthede nitionoftheset-theoretic(cartesian)proSductwreknowthatK2X ͠=!u cmex10Q UWXrK.0This`idenrtitydoSesnotonlyholdonthesetlevel,աitholdsalsoforthealgebraRJstructuresonK2X 1resp.8Q7XK.WVenorwconstructaninversefunctor謍}RSpSec":URK-cAlg!4!3`SetFI:FVorJ~eacrhpSointxq2X<thereJ~isamaximalidealmx eofQ՟X\Kde nedbymx AX:=qff@p2Map+Q(XJg;K)jfG(x)=0g.IfYIXJ̹isa nitesetthentheseareexactlyallmaximalidealsofQHVXeK.QTVoshorwthisweobservethefollowing.QThesurjectivehomomorphismpxk:Q UWXrKi4!Khaskrernelmx ?shencemxisamaximalideal.\IfmiQXmKisamaximalidealC{anda샹=( 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) 2mb1hencethei-thfactor0 :::´K:::0ofQ,UXIKisinm.2RSotheelemenrtsaUR2mmusthaveatleastonecommoncompSonent jX=qN0sincem6=K.-gButmorethanonesucrhacompSonentisimpSossiblesincewewrouldvgetzerodivisorsintheresidueclassalgebra..ThusmB=mx wherevx2Xgisthejӹ-thelemenrtsoftheset.(g7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y95YOnewcaneasilyshorwmorenamelythattheidealsmxarepreciselyallprimeidealsofMap(XJg;K).With)eacrhcommutativealgebraAwecanassoSciatethesetSpec(A)ofallprimeideals*ofA.IThatde nesafunctorSpSec:}K-Alg^4!.:SetA#ٹ.Appliedtoalgebrasoftheform~K2X Mx=/QFXKwitha nitesetXthisfunctorrecorvers~XasXP!r԰:Z=SpSec,z6(K2X).Thrusthedreamofgeometryissatis edinthisparticularexample.pThe*abSorveexampleshowsthatwemayhopSetogainsomeinformationonthespace(set)XkbryknowingitsalgebraoffunctionsK2X qandapplyingthefunctorSpSectoit.FVor nitesetsandcertainalgebrasthefunctorsK2-}EandSpSecactuallyde neacategorydualitryV.8Wearegoingtoexpandthisdualitrytolargercategories.WVe)shallcarrysomegeometricstructureinrtothesetsXandwillstudythecon-nectionbSetrweenthesegeometricspacesandtheiralgebrasoffunctions. UNFVorthispurpSosewrewilldescribesetsofpoinrtsbytheircoSordinates.ExamplesarethecircleortheparabSola.MoregenerallythegeometricspaceswrearegoingtoconsideraresoEcalledanescrhemesdescribSedbypSolynomialequations.=WVewillseethatsuchgeometricspacesarecompletelydescribSedbrytheiralgebrasoffunctions.HeretheYVonedaLemmawillplaryacentralr^ ole.WVetwill,Zhorwever,taketadi erenrtapproachtofunctionsalgebrasandgeometricspaces,0than"onedoSesinalgebraicgeometryV.We"usethefunctorialapproacrh,whichlends Nitselftoaneasieraccesstotheprinciplesofnoncommrutative NgeometryV.Wewill/ede negeometricspacesascertainfunctorsfromthecategoryofcommrutativealgebras.Xtothecategoryofsets.Thesesetswillharve.Xastronggeometricalmeaning.TheN&functorswillassoSciatewitheacrhalgebraAthesetofpoinrtsofa\geometricvXarietry",wherethepSointshavecoSordinatesinthealgebraA.pDe nition1.1.2.vaùTheYfunctorAUR=A21V:K-cAlg!4!3`SetI ܹ(theYunderlyingfunctor)thatassoSciateswitheacrhcommutativeK-algebraAitsspace(set)ofpSoints(elements)A꨹iscalledtheane35line.Lemma1.1.3.g5QThe35functor\aneline"isarffepresentable35functor.,- cmcsc10Proof.@_ByάLemma2.3.5therepresenrtingobjectisK[x].7Observethatitisuniqueuptoisomorphism.Ecffxff ̟ff ̎ ̄cffDe nition1.1.4.vaùThefunctorA22 j:FfK-cAlg#4!7BSetO:\thatassoSciateswitheacrhcommrutativealgebraAthespace(set)ofpSoinrts(elements)oftheplaneA22iscalledtheane35plane.Lemma1.1.5.g5QThe35functor\aneplane"isarffepresentable35functor.Proof.@_SimilarDtoLemma2.3.9therepresenrtingobjectisK[x1;x2].ThisDalgebraisuniqueuptoisomorphism.Ņcffxff ̟ff ̎ ̄cffLetp1(x1;:::ʚ;xnP);:::;pmĹ(x1;:::;xnP)UR2K[x1;:::;xnP]bSeafamilyofpolynomials.WVeMwranttoconsiderthe(geometric)vXarietyofzerosofthesepSolynomials.atObserve87 &e661.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYthatrKmarynotcontainsucientlymanyzerosforthesepSolynomials.#Thuswearego-ing9toadmitzerosinextension eldsofKormoregenerallyinarbitrarycommrutativeK-algebras.In2zthefollorwingrathersimplebuildupofcommutativealgebraicgeometryV,nthereadershouldcarefullyvrerifyinwhichstatementsandproSofsthecommutativityisreally]needed.MostofthefollorwingwillbSeverballygeneralizedtonotnecessarilycommrutativealgebras.XDe nition1.1.6.vaùGivrenQ~asetofpSolynomialsfp1;:::ʚ;pmgURK[x1;:::ʚ;xnP].ThefunctorXthatassoSciateswitheacrhcommutativealgebraAthesetXӹ(A)ofzerosoftheNpSolynomials(pidڹ)inA2n uiscalledananealgebrffaicvarietyNorananescheme(inqA2nP)withde ningpSolynomialsp1;:::ʚ;pmĹ.RTheelemenrtsinXӹ(A)arecalledtheA-pffoints꨹ofXӹ.Theorem1.1.7.pThe aneschemeX˸inA2n5withde ningpffolynomialsp1;:::ʚ;pmis35arffepresentable35functorwithrffepresenting35algebrae{ OUV(Xӹ)UR:=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ);cffalled35theanealgebraofthefunctorX.Proof.@_First9wreshowthattheaneschemeXU :8K-cAlg%/4!9lSetRwiththede ningKpSolynomialsp1;:::ʚ;pm isafunctor.XLetfm:gnA4!iBQbeKahomomorphismofcommrutative#`algebras.TheinducedmapfG2n ,:A2n ^-4!-B2n fde nedbryapplicationoffontthecompSonenrtsrestrictstoXӹ(A)@!A2n astX(fG):X(A)4!X(B).ֻThistmapiswrell-de nedforlet(a1;:::ʚ;anP)UR2Xӹ(A)bSeazeroforallpolynomialsp1;:::ʚ;pm Jthenpidڹ(fG(a1);:::ʚ;f(anP))=fG(pi(a1;:::ʚ;anP))=fG(0)=0Lforallihencef2nO(a1;:::ʚ;anP)=(fG(a1);:::ʚ;f(anP))a 2B2n 4isazeroforallpSolynomials.MThrusXӹ(fG)a :X(A)4!ۥX(B)iswrell-de ned.8FVunctorialityofX{isclearnorw.Norw3weshowthatXisrepresentablebyOUV(Xӹ)UR=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).Ob-servrethat(p1;:::ʚ;pmĹ)denotesthe(two-sided)idealinK[x1;:::ʚ;xnP]generatedbythepSolynomialstp1;:::ʚ;pmĹ.WVeknorwthateachn-tupSel(a1;:::ʚ;anP)UR2A2nĹuniquelytdeter-minesanalgebrahomomorphismfN:wOK[x1;:::ʚ;xnP]4!+AbryfG(x1)wO=a1;:::ʚ;fG(xnP)=anP.F(TheDgpSolynomialringK[x1;:::ʚ;xn]inK-cAlg"ֹisfreeorverDgthesetfx1;:::ʚ;xng,orUaK[x1;:::ʚ;xnP]togetherwiththeemrbSedding :fx1;:::ʚ;xnPg4!/K[x1;:::ʚ;xnP]Uaisacounivrersal[[solutionoftheproblemgivenbytheunderlyingfunctorA':K-cAlg"4!Set-andJthesetfx1;:::ʚ;xnPgbP2SetK3.)OThisJhomomorphismofalgebrasmapspSolyno-mialsfp(x1;:::ʚ;xnP)inrtofG(p)(=p(a1;:::ʚ;anP).CHencef(a1;:::ʚ;anP)fisacommonzeroof[thepSolynomialsp1;:::ʚ;pm `ifandonlyiffG(pidڹ)UR=pi(a1;:::ʚ;anP)=0,xi.e. Op1;:::ʚ;pmareHinthekrerneloffG.ThishappSensifandonlyiffvXanishesontheideal(p1;:::ʚ;pmĹ)orinotherwrordcanbSefactorizedthroughtheresidueclassmape`Jl:URK[x1;:::ʚ;xnP]4!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):IF7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y97YNorwitiseasytoseethatthisbijectionisanaturalisomorphism(inA).4샄cffxff ̟ff ̎ ̄cffIfnopSolynomialsaregivrenfortheaborveconstruction,GthenthefunctorunderthisconstructionZ;istheanespaceA2n ofdimensionn.BygivingpSolynomialsthefunctorXbSecomes]asubfunctorofA2nP,zbecauseitde nessubsetsXӹ(A)URA2nP(A)=A2n. Bothfunctorsarerepresenrtablefunctors.TheembSeddingisinducedbythehomomorphismofalgebras:URK[x1;:::ʚ;xnP]4!1K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).Problem1.1.1.R1.(Determinebtheanealgebraofthefunctor\unitcircle" #Sן21 ]inA22.2. #Determinetheanealgebraofthefunctor\unitsphere"Sן2n1"KinA2nP.3. #LetX^denotetheplanecurvreyË=URx22.8,ThenXisisomorphictotheaneline.4. #Let96Y5̹denotetheplanecurvrexyIA=1.$ThenYisnotisomorphictotheane #line.3(Hinrt:xmAnnisomorphismK[x;x21 \|]eG4!K[yn9]sendsxtoapSolynomialp(y) #whicrhmustbSeinvertible.qConsiderthehighestcoSecientofp(yn9)andshow #thatp(yn9)UR2K.8ButthatmeansthatthemapcannotbSebijectivre.)5. #LetKdF=CbSethe eldofcomplexnrumbSers. Showthattheunitfunctor #UL\: xK-cAlg"4!4SetL zinULemma2.3.7isnaturallyisomorphictotheunitcircle #functor4RSן21r۹.(Hinrt:4ThereisanalgebraisomorphismbSetweentherepresenting #algebrasK[e;e21 \|]andK[c;s]=(c22j+s221).)6. #2(DLetKbSeanalgebraicallyclosed eld."Letpbeanirreduciblesquarepolyno- #mialsinK[x;yn9].&LetZbSetheconicsectionde nedbrypwiththeanealgebra #K[x;yn9]=(p).Shorw,thatZ isnaturallyisomorphiceithertoXeortoY)(from #parts3.8resp.4.Remark1.1.8.j6Anem#algebrasofanescrhemesare nitelygeneratedcommu-tativreԑalgebrasandanysuchalgebraisananealgebraofsomeanescheme,sinceAPUR԰n:=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ)(HilbSertbasistheorem).The fpSolynomialsp1;:::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.prTheydonotsucetocompletelydescribSetheanescrheme.LetforexampleKP=RѹthesetofrationalnrumbSers. \IfXandYgareanescrhemes6withanealgebrasOUV(Xӹ)UR:=K[x;yn9]=(x22+;y22i׹+1)6andOUV(Y)UR:=K[x]=(x22+1)then[3bSothscrhemeshavenorationalpSoints. TheschemeY,Uhowever,has[3exactlytrwo5complexpSoinrts(withcoecienrtsinthe eldCofcomplexnumbSers)andthescrheme:XF hasin nitelymanycomplexpSoints,henceXӹ(C)Z6P԰= Y(C).ThisdoSesnotresultfromtheemrbSeddingsintodi erentspacesA22 Wresp.>9A21.InfactwealsohaveOUV(Y)UR=K[x]=(x22+1)P԰n:=K[x;yn9]=(x22+1;yn9),-socYcanbSeconsideredasananescrhemeinA22.`$7 &e861.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYSinceЧeacrhaneschemeXzisisomorphictothefunctorMorK-cAlg8=(OUV(Xӹ);-)wewillhenceforthidenrtifythesetwofunctors,thusremovingannoyingisomorphisms.eDe nition1.1.9.vaùLetK-A denotethecategoryofallcommrutative nitelygenerated(oranecf.Ԣ1.1.8)K-algebras.AnanealgebrffaicvarietyisarepresenrtablefunctorK-A 7(A;-)UR:K-A j4!-ISet@,. \TheanealgebraicvXarietiestogetherwiththenatural[transformationsformthecffategory*0ofanealgebrffaicvarieties[VVar(K)orverK.]ThefunctorthatassoSciateswitheacrhanealgebraAitsanealgebraicvXarietyrepresenrtedbyAisdenotedbySpSecx:URK-A j4!-IVVar@AH(K),SpSec&(A)=K-- NA !(A;-).BytheYVonedaLemmathefunctorBSpSec:URK-A j4!-IVVar@AH(K)isananrtiequivXalence(orduality)ofcategorieswithinversefunctor=AO:URVVarQ(K)UR4!1K-A 7:AnanealgebraicvXarietryiscompletelydescribSedbyitsanealgebraOUV(Xӹ).וThusthedreamofgeometryisrealized.Arbitrary^(notnecessarily nitelygenerated)commrutative^algebrasalsode nerepresenrtablefunctors(de nedonthecategoryofallcommutativealgebras).Thuswealso|harve\in nitedimensional"vXarietieswhichwewillcallgeffometricUspacesoranevarieties.8WVedenotetheircategorybryGeom"P(K)andgetacommutativediagramPB9K-cAlgٰGeom X(K)t32fd*ά- T덍PH԰.0=jnoHK-A HĄVVar (K){fd3QPά-ɪSpSecHŸǠ*Ffe1Ǡ?HŸǠ*Ffe#Ǡ?VWVe}calltherepresenrtablefunctorsXH9:fK-cAlg""4!3SetJ(geffometricMspaces}oranevarieties,Band0therepresenrtablefunctorsX:LK-A d4!.=SetFanesuschemesoranealgebrffaic35varieties.8ThisisanotherrealizationofthedreamofgeometryV.The:HgeometricspacescanbSeviewredassetsofzerosinarbitrarycommutativeK-algebras[B$ofarbitrarilymanrypSolynomialswitharbitrarilymanyvXariables.AThefunctionalgebraofX{willbSecalledtheane35algebrffaofXinbSothcases.eExample1.1.10.uQA'somewhat'lesstrivialexampleisthestatespaceofacircularpSendulumb(oflength1). 9 ThelocationisinL+=f(a;b)2A22ja225+1b22 묹=1g,themomenrtumuisinM6=URfp2Aguwhichisastraightline.SothewholegeometricspaceforthepSendulumis(LM@)(A)X=f(a;b;p)ja;b;pX2A;a22k+b22=1g.>ThisgeometricspaceisrepresenrtedbyK[x;yn9;z]=(x22j+y221)since|(LraM@)(A)UR=f(a;b;p)ja;b;pUR2A;a 22e+rab 2V=1gP԰n:=K-cAlgo(K[x;yn9;z]=(x 22e+ray 21);A):eThe4#trwoantiequivXalencesofcategoriesabSovegiverisetothequestionforthefunc-tionyalgebra.9IfarepresenrtablefunctorX%=URK-cAlgo(A;-)isviewedasgeometricsets r7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y99Yofu8zerosofcertainpSolynomials,i.e.asspaceswithcoordinatesinarbitrarycommruta-tivre%algebrasB,4(plusfunctorialbSehavior),4thenitisnotclearwhytherepresentingalgebraȬAshouldbSeanrythinglikeanalgebraoffunctionsonthesegeometricsets.-ItisnotevrenclearwherethesefunctionsshouldassumetheirvXalues.OOnlyifwecanshorw`thatAcanbSeviewedasareasonablealgebraoffunctions,|KweshouldtalkabSoutaLBrealizationofthedreamofgeometryV.]ButthiswillbSedoneinthefollorwingtheo-rem.WVehgwillconsiderfunctionsasmaps(coSordinatefunctions)fromthegeometricsetDiXӹ(B)tothesetofcoSordinatesB,ZmapsthatarenaturalinB.F#SucrhcoSordinatefunctionsarejustnaturaltransformationsfromX{totheunderlyingfunctorA.Theorem1.1.11.wXLffetјXkbeageometricspacewiththeanealgebraAUR=OUV(Xӹ).Then[APUR԰n:=Nat#@(X;A)asK-algebrffas,where[AUR:K-35cAlg":N!3 SetJ:xistheunderlyingfunc-torDoraneline.DTheisomorphismAPUR԰n:=Nat#@(X;A)inducffesanaturaltransformationAXӹ(B)UR!B;(naturffal35inB).Proof.@_FirstYwrede neanisomorphismbSetweenthesetsAandNatT(X;A).BecauseofX=MorJ K-cAlg9H(A;-)=:K-cAlgo(A;-)andA=MorJ K-cAlg9H(K[x];-)=:K-cAlgo(K[x];-)theYVonedaLemmagivresus1]Nat(X;A)UR=Nat(K-cAlgo(A;-);K-cAlg(K[x];-))PUR԰n:=K-cAlg(K[x];A)UR=A(A)P԰n:=Aonthesetlevrel. Let?:A4!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,.saB-pSoinrtp:A4!|Bin Xӹ(B)=K-Algo(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)4Mand(  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]4!AMmappingxontoa.ThisalgebraӾhomomorphisminducesthenaturaltransformation(a):XW4!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(a0k+a209)(B)(p)(x)=p(a0k+a20)=p(a)0k+p(a20)=(a)(B)(p)(x)0k+(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{CdescribSetheaction n9(B)Kt:A Xӹ(B)Kt4!uBIof{CAonXӹ(B).LetpKt:A4!uBbSe3aB-poinrtinK-cAlgo(A;B)t3=Xӹ(B).2FVor3eacrhat32A3theimage(a)t3:X64!A ٠7 &e1061.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYisnanaturaltransformationhencewrehavemaps n9(B)m:AXӹ(B)m4!B*tsucrhnthat n9(B)(a;p)<=p(a).wMFinally"eacrhhomomorphismofalgebrasf;:(natural8inB).&InparticulartheanealgebraAcanbSeviewredasthesetoffunctionsfromthesetofB-poinrtsXӹ(B)intothe\base"ringRB(functionswhicrharenaturalinB).RInthissensethealgebraAmaybSeconsid-eredasfunctionalgebraofthegeometricspaceXӹ.@>RHD6Xӹ(B)GǠ*Ffez̟Ǡ?3f1X(BI)_commrutes. I WVewillshowthisresultlateronfornoncommutativealgebras. I TheunivrersalpropSertyimpliesthatthefunctionalgebraAofangeometricspaceXmisuniqueuptoisomorphism.Let-XbSeangeometricspacewithfunctionalgebraAǹ=OUV(Xӹ).oIf-p:A4!Kis arationalpSoinrtofXӹ,Ri.e.ahomomorphismofalgebras,thenImp(p)?=K henceKer(p)isamaximalidealofAofcoSdimension1._Conrverselylet/%n eufm10mbeamaximalideal&ofAofcoSdimension1thenthisde nesarationalpoinrtpH:A4!A=mP԰0=K.IfKisalgebraiclyclosedandmanarbitrarymaximalidealofA,mthenA=misa nitelygeneratedK-algebraanda eldextensionofK,henceitcoincideswithK.iThrusthecoSdimension|ofmis1.ThesetofmaximalidealsofAiscalledthemaximal͌spffectrumSpSecmĹ(A).Thisistheapproacrhofalgebraicgeometrytorecoverthegeometricspaceof(rational)pSoinrtsfromthefunctionalgebraA.ǐWVewillnotfollowthisapproachsinceitdoSesnoteasilyextendtononcommrutativegeometryV.:Problem1.1.2.nRLetX{bSeananescrhemewithanealgebraAUR=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ): Π7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y411YDe ne[\coSordinatefunctions"qi &:LXӹ(B)4!R%B'awhicrh[describethecoordinatesofB-pSoinrtsandidentifythesecoSordinatefunctionswithelementsofA.NorwwewillstudymorphismsbSetweengeometricspaces.Theorem1.1.13.wXLffetXN"{A2r 4aandYA2s ;bffeanealgebraicvarietiesandletUR:X%!Y/bffe35anaturaltransformation.fiThentherearepolynomialseYp1(x1;:::ʚ;xrb);:::;psn<(x1;:::;xrb)UR2K[x1;:::;xrb];such35thatLP(A)(a1;:::ʚ;arb)UR=(p1(a1;:::ʚ;arb);:::;psn<(a1;:::;arb));s2forallA.2K-33A Wandall(a1;:::ʚ;arb).2Xӹ(A),Ji.e.themorphismsbffetweenanealgebrffaic35varietiesareofpolynomialtype.Proof.@_LetmOUV(Xӹ)$=K[x1;:::ʚ;xrb]=IandOUV(Y)=K[y1;:::ʚ;ysn<]=Jr. -/FVorA2K-Alg̨and09(a1;:::ʚ;arb)z2Xӹ(A)letfy:K[x1;:::ʚ;xrb]=Ip4! AwithfG(xidڹ)=aibSetheIhomomorphismobtainedfromXӹ(A)P԰=DK-Algo(K[x1;:::ʚ;xrb]=I;A).VTheInaturaltransformationpisgivrenbycompSositionwithahomomorphismgË:URK[y1;:::ʚ;ysn<]=Jq4!K[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 eʍi(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)):ȍ %cffxff ̟ff ̎ ̄cffAnanalogousstatemenrtholdsforgeometricspaces.Example1.1.14.uQTheisomorphismbSetrweentheaneline(1.1.2)andthepara-bSolaeisgivrenbytheisomorphismfn:&K[x;yn9]=(yl~x22)4!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 &e1261.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY=[2.NQuantumSpacesandNoncommutativeGeometryNorw8wecometononcommutativegeometricspacesandtheirfunctionalgebras.ManryǘofthebasicprinciplesofcommutativealgebraicgeometryasintroSducedin1.1carry-orvertononcommutativegeometryV.nOurmainaim,bNhowever,is-tostudythesymmetries0p(automorphisms)ofnoncommrutative0pspaceswhicrhleadtothenotionofaquanrtumgroup.Sinceljtheconstructionofnoncommrutativeljgeometricspaceshasdeepapplicationsintheoreticalphrysicswewillalsocallthesespacesquantumspaces. ͍De nition1.2.1.vaùLet AbSea(notnecessarilycommrutative) K-algebra.bThenthefunctorXY:=7K-Algo(A;-):K-Alg4!1%SetI߹represenrtedbyAiscalled(ane)noncffommutativeo(geometric)space+Թorquantumspfface.dThe+elemenrtsofK-Algo(A;B)arecalledB-pffointsofXӹ.Amorphismofnoncommutativespacesf:ʦXy4!pY“isanaturaltransformation.Thisde nitionimpliesimmediatelyCorollary1.2.2.sWThenoncffommutativespacesformacategoryQSthatisdualto35thecffategory35ofK-algebrffas.Remark1.2.3.j6ThrusoneoftencallsthedualcategoryK-AlgoWop%Xcategoryofnon-commrutativespaces.IfAisa nitelygeneratedalgebrathenitmarybSeconsideredasaresidueclassalgebraaAPUR԰n:=Khx1;:::ʚ;xnPi=IofapSolynomialalgebrainnoncommrutingvXariables(cf.A.6). ,IfilIX=(p1(x1;:::ʚ;xnP);:::;pmĹ(x1;:::;xnP))ilisthetrwo-sidedilidealgeneratedbryfthepSolynomialsp1;:::ʚ;pm kkthenthesetsK-Algo(A;B)canbSeconsideredassetsofYzerosofthesepSolynomialsinB2nCV.Infact,gwrehaveK-Algo(Khx1;:::ʚ;xnPi;B)P[԰tٹ=Map+Q(fx1;:::ʚ;xnPg;B)UR=B2nCV.ThrusOK-Algo(A;B)canbSeconsideredasthesetofthosehomomorphismsofalgebrasfromKhx1;:::ʚ;xnPitoB1 thatvXanishontheidealIorasthesetofzerosofthesepSolynomialsinB2nCV.SimilartoTheorem1.1.13oneshorwsalsointhenoncommutativecasethatmor-phismsbSetrweennoncommutativespacesaredescribSedbypSolynomials.TheTheorem1.1.11ontheopSerationoftheanealgebraA=OUV(Xӹ)onXֹasfunctionalgebracanbSecarriedorvertothenoncommrutativecaseaswrell: thenaturaltransformation n9(B)a:AZXӹ(B)a4!۩B(naturalinB)isgivrenby n9(B)(a;p)a:=p(a)andcomesfromtheisomorphismAPUR԰n:=Nat#@(X;A).Norw'wecometoaclaimonthefunctionalgebraAthatwedidnotproveinthecommrutativecase,#butthatholdsinthecommrutativeaswrellasinthenoncommuta-tivresituation. B7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_13YLemma1.2.4.g5QLffetDbeasetand$:DQXӹ(-35)!wA(-)bffeanaturaltransfor-mation.fiThen35therffeexistsauniquemapfQ:URD!dAsuchthatthediagram<AXӹ(B)HB̶32fdCά-W` I{(Bd)@"D6Xӹ(B)ǠfeǠ?b{f1@b\(Bd)Ǫ@|>RHdD6Xӹ(B)JjǠ*Ffe}Ǡ?`f1commrutes.EsDe nition1.2.5.vaùThenoncommrutativespaceAߍ2j0]GqG$withthefunctionalgebraݵ/OUV(A 2j0ڍq \|)UR:=Khx;yn9i=(xyq 1 ʵyx)withx:qË2URKnf0giscalledthe(deformeffd)quantumplane.Thenoncommrutativespace"!Aߍ0j2]GqG$withthefunctionalgebra|SOUV(A 0j2ڍq \|)UR:=Khs;n9i=( 2Tw;n9 2.=;+qn9s)iscalledthedual35(deformeffd)quantumplane.8WVeharve^q|A 2j0ڍq \|(A)UR=q USqʍ*xlyqȍ$U 38$U (Ux;yË2A;xyË=qn9 1 ʵyn9x2Kq۳and9GNA 0j2ڍq \|(A)UR=G UTGʍVs;") eGȍ.g 38.g 2gs;Ë2A;s 2ɹ=0;n9 2=0;sË=qn9G :Օ7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_15YDe nition1.2.6.vaùLet"0XbSeanoncommrutative"0spacewithfunctionalgebraAandletXc *bSetherestrictionofthefunctorX͹:%K-Algi4!1SetHwtothecategoryofcommrutativeJalgebras:Xc&:K-cAlg"4!4>SetG!.VThenwrecallXcx}thecffommutativepartofthenoncommrutativespaceXӹ.Lemma1.2.7.g5QThe4cffommutativepartXc ʭofanoncommutativespaceX^isanane35variety.Proof.@_TheunderlyingfunctorAi̹:K-cAlg";4!3K-AlghasaleftadjoinrtfunctorK-Alg3lA7!A=[A;A]2K-cAlg#where%[A;A]denotesthetrwo-sided%idealofAgeneratedbrytheelementsab%!ba.TInfactforeachhomomorphismofalgebrasfπ:AUR4!1B۹withCacommrutativeCalgebraBthereisafactorizationthroughA=[A;A]sincef2vXanishesontheelemenrtsabba.HenceDifAUR=OUV(Xӹ)DisthefunctionalgebraofX^thenA=[A;A]istherepresenrtingalgebraforXc.y.^ {cffxff ̟ff ̎ ̄cffRemark1.2.8.j6FVorZanrycommutativealgebra(ofcoSecients)B`thespacesXand_Xc harvethesameB-pSoints:!Xӹ(B)8=Xc.y(B). The_trwospacesdi eronlyfornoncommrutativehalgebrasofcoSecienrts. Inparticularforcommutative eldsBanaswualgebrasofcoSecienrtsthequantumplaneAߍ2j0]GqThasonlyB-pSointsonthetwoaxessincethefunctionalgebraKhx;yn9i=(xyq21 ʵyx;xyyx)PUR԰n:=Kܞ[x;y]=(xy)de nesonlyB-pSoinrts(b1;b2)whereatleastoneofthecoSecienrtsiszero.Problem1.2.4.nRLetbS3 fbSethesymmetricgroupandA=:=K[S3]bbethegroupalgebra;onS3.yDescribSethepoinrtsofXӹ(B)z =K-Algo(A;B);asasubspaceofA22(B).WhatisXc.y(B)andwhatistheanealgebraofXc?TVo$understandhorwHopfalgebras tintothecontextofnoncommutativespaceswrehavetobSetterunderstandthetensorproductinK-Algo.De nition1.2.9.vaùLet A>=OUV(Xӹ)andA20 Ϲ=OUV(Y)bSethefunctionalgebrasofthe-TnoncommrutativespacesX'resp. Y.TwroB-pSointspz:A4!BZinXӹ(B)andp20r :A204!/nBJ4in.Y(B)arecalledcffommutingpoints.ifwrehaveforalla2A.andalla20#2URA20Ap(a)p 09(a 0)UR=p 09(a 0)p(a);Íi.e.8iftheimagesofthetrwohomomorphismspandp20commrute.Remark1.2.10.qN6TVoshorwthatthepSointspandp20=commute,[itissucienttocrheckGmthattheimagesofthealgebrageneratorsp(x1);:::ʚ;p(xmĹ)commrutewiththeimages|5ofthealgebrageneratorsp209(y1);:::ʚ;p20(ynP)|5underthemrultiplication. Thismeansthatwrehavebidb 0ڍj\=URb 0ڍjf bi|ofortheB-pSoinrts(b1;:::ʚ;bmĹ)UR2Xӹ(B)and(b20RA1;:::ʚ;b20RAnP)UR2Y(B).a7 &e1661.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYDe nition1.2.11.}!ùThefunctorKP(X%?URY)(B):=f(p;p 09)2Xӹ(B)Y(B)jp;p 0commrute5e3giscalledtheorthoffgonal35product꨹ofthenoncommrutativespacesX{andY.YDRemark1.2.12.qN6TVogether0withXandY-theorthogonalproSductX%?URYisagainafunctor, sincehomomorphismsf˚:B4!B20oarecompatiblewiththemrultiplicationandthruspreservecommutingpSoints.8HenceX%?URY>isasubfunctorofXl{Y.Lemma1.2.13.mQIfWXandYTparffenoncommutativespaces,athenX[?-YTpisanon-cffommutative35spacewithfunctionalgebraOUV(X%?URY)=O(Xӹ) O(Y).IfjX,andYgWhave nitelygenerffatedjfunctionalgebrffasthenthefunctionalgebraofX%?URY/is35also nitelygenerffated.Proof.@_Let'iA:=OUV(Xӹ)andA20:=OUV(Y).$Let(p;p209)2(X~?Y)(B)bSeapairofcommrutingpSoints.ThenthereisauniquehomomorphismofalgebrashUR:Au A20#4!jBsucrhthatthefollowingdiagramcommutesG]X/ HA/ ƏA A20A{fdά- OH pׁ @ @ @Ü @✟>@✟>RHѵ"B:ʟǠ*FfeǠ? ̤h/ /  A20 ,{fdk ά{20H|p20ܟׁ ܟ ܟ ܟ S\>S\> jDe neh(a a209)W:=p(a)p20(a20)andcrheckthenecessarypropSerties.Observrethatforanq=arbitraryhomomorphismofalgebrash:d:AI A204!\B Ctheq=imagesofelemenrtsoftheFformal 1Fand1l a20commruteFsincetheseelementsalreadycommuteinAl A209.Thruswehave8 (X%?URY)(B)P԰n:=K-Algo(A A 09;B):IfthealgebraAisgeneratedbrytheelementsa1;:::ʚ;am XandthealgebraA20͹isgeneratedbrytheelementsa20RA1;:::ʚ;a20RAn thenthealgebraAs A20isgeneratedbytheelemenrtsai 1and1 a20RAjf .ױcffxff ̟ff ̎ ̄cffYDProp`osition1.2.14.The5Corthoffgonalproductofnoncommutativespacesisasso-ciative,35i.e.fifornoncffommutativespacesX,Y,andZ'Awehave(X%?URY)?ZPI^԰bF=X?(YQ?Z ):Proof.@_LettB\zbSeacoecienrtalgebraandletpx 2vXӹ(B),7'py s2Y(B),7'andpzo2Z (B)>bSepoinrtssuchthat((pxH;py );pzʮ)>isapairofcommutingpSointsin((X?Y)UR?Z )(B).5Inparticular(pxH;py )isalsoapairofcommrutingpSoints.5ThuswehaveforallaUR2A:=OUV(Xӹ),a20#2A20:=OUV(Y),anda20N920q2URA20N920:=OUV(Z )pxH(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)and8pxH(a)py (a 09)UR=py(a 09)pxH(a):d7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_17YIfwrechoSosea̹=1thenwegetpy (a209)pzʮ(a20N920r)=pz(a20N920r)py (a209).FVorarbitrarya;a20;a20N920 "wrethengete32fdPά-k~rH3,1X.M٬ҁ H٬ׁ H٬܁ H٬ H٬ H٬ H٬ H٬ H fdžH fdžjcommrute.8Then(M;m;e)iscalledaquantum35monoid.#7Prop`osition1.3.2.OLffetMbeanoncommutativespacewithfunctionalgebraHV.Then35H isabialgebrffaifandonlyifMisaquantummonoid.Proof.@_SincethefunctorsM ?M,M?EsعandEt ?MarerepresenrtedbyH tH)resp..H KP԰=H)resp..K HP'԰=OHthei.e. RbryH(a)=a a+b c,*(b)=a b+b d,(c)=c a+d cHand(d)UR=c b+d d,andwiththecounit="qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,;i.e.`"(a)G=1,f"(b)=0,"(c)=0,andMU"(d)=1.`WVeharvetoprovethatand"arehomomorphismsofalgebrasandthatthecoalgebralarwsaresatis ed.TVoobtainahomomorphism[ofalgebrasUR:Mq(2)4!1Mq(2) Mq(2)[wrede neUR:Kha;b;c;diUR4!Mq(2) Mq(2)onthefreealgebra(thepSolynomialringinnoncommrutingvXariables)Kha;b;c;diݹgeneratedbrythesetfa;b;c;dgݹandshowthatitvXanishesontheidealIormoresimplyonthegeneratorsoftheideal. ThenitfactorsthroughauniquehomomorphismIofalgebrasUR:Mq(2)4!1Mq(2)x Mq(2).0WVeIcrheckthisonlyforonegeneratoroftheidealI:1"ʍ(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):1"Thereadershouldcrhecktheotheridenrtities.'B7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)21YThecoassoSciativitryfollowsfrom*G񙐍( 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+:*GThereadershouldcrheckthepropSertiesofthecounit.b)The35geffometricapproach:Mq(2)hasaratherremarkXable(andactuallywrellknown)comultiplicationthatisbSettersunderstoodbryusingtheinducedmultiplicationofcommutingpSoints.|GivenÍtrwofOcommutingquantummatricesqʍ ;a1!b1 c1!&d1,q8=Yandqʍ ;a2!b2 c2!&d2,qinMq(2)(A). ThentheirčmatrixproSduct்Xգqʍaa1w$ b1b94c1vd1qqF\qʍa2b2c2d2῟q i=URqʍ *a1a2j+b1c2N a1b2j+b1d2 *c1a2j+d1c2N:c1b2j+d1d2̟q఍isagainaquanrtummatrix.8TVoprovethisweonlycheckoneoftherelations7z@ʍ(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)7z@WVeharveusedthatthetwopSointsarecommutingpSoints.˭ThismultiplicationobviouslyisanaturaltransformationMq(2)S"?Mq(2)(A)4!Mq(2)(A)(naturalinA).9ItistassoSciativreandhasunitqʍ 10 01"q+UP.8FVortheassociativitryobservethatby1.2.14$ˍ?3((qʍXa1Nib1 cc1]d1(cq3p;qʍ Va2 Ngb2 cc2[d2*aq);qʍ Va3 Ngb3 cc3[d3*aq)఍isapairofcommrutingpSointsifandonlyif்~?4(qʍXa1Nib1 cc1]d1(cq3p;(qʍXa2Nib2 cc2]d2(cq;qʍ Va3 Ngb3 cc3[d3*aq))isapairofcommrutingpSoints.ÍSince`qʍk1K0k0K1$+q/qʍ7^aHkb8dcH_dNaq]=-Yqʍabc-d%Hq4Ke=-Yqʍabc-d%Hq0 qʍ8b1H^08b0H^1NZqfor`allquanrtummatricesߍqʍXaeb ccYd[q+F2URMq(2)(B)wreseethatMq(2)isaquanrtummonoid.fpItCremainstoshorwthatthemultiplicationofMq(2)andthecomultiplicationofMq(2)pcorrespSondtoeacrhotherbytheYVonedaLemma.9Theidentitymorphismof87 &e2261.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYMq(2) Mq(2)isgivrenbythepairofcommutingpSointsf(1;2)UR2Mq(2)?Mq(2)(Mq(2) Mq(2))UR=K-Algo(Mq(2) Mq(2);Mq(2) Mq(2)):bڍSinceU1 &Թ=fПqʍ <(a5b acg)d#+q/o 1f=qʍ <(a 12b 1 ac 11d 1LqqYand2 &Թ=1 qʍ fasb {cgd"3iq/o=fПqʍ <(1 a2ع1 b a1 c1̹1 dLqqߍwreghaveidZ=)(1;2)=(qʍXaeb ccYd[q* ^1;1 qʍ Զab bcd"q).!ThegYVonedaLemmade nesthefpdiagonal8>astheimageoftheidenrtity8>underK-Algo(Mq(2)>= Mq(2);Mq(2) Mq(2))UR4!tK-Algo(Mq(2);Mq(2) Mq(2))brythemultiplication.2So(qʍXaeb ccYd[q')UR==1D2V=ߍ(qʍXaeb ccYd[q*[ 1)(1 qʍ a9 b 9cd!q)UR=qʍ *ab cUd"pq- qʍ a9 b 9cd!q.ThrusMq(2)de nesaquantummonoidMq(2)withbڍ#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&:ԧThis $isthedeformedvrersionofM諍2 %themultiplicativemonoidofthe2i2-matricesofcommrutativealgebras.wu2.&LetAߍ2j0]Gqι=URKhx;yn9i=(xyH`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.X.E2. #LetYzAbSeaK-algebra.UAY]K-algebraM@(A)togetherwithahomomorphismof #algebras :A4!5M@(A)9 AiscalledanalgebrffaQcoactinguniversallyon A(or #simplyauniversalalgebrffaforA)ifforevreryK-algebraBandeveryhomomor- #phismbofK-algebrasfi:!A4!\BZ TAbthereexistsauniquehomomorphismof #algebrasgË:URM@(A)4!1BsucrhthatthefollowingdiagramcommutesK]ōH@AH~M@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.A׌BytheunivrersalpropSertiestheuniversalalgebraM@(A)forAandtheuniversalquanrtumspaceM(Xӹ)forX{areuniqueuptoisomorphism.d07 &e2461.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYProp`osition1.3.8.O1.s+LffetpKAbeaK-algebrawithuniversalalgebraM@(A)and #:.A! M@(A) A.ThenM(A)isabialgebrffaandAisanM(A)-cffomodule #algebrffa35bys2.2. #IfѺBlisabialgebrffaandiffQ:URA!Bm  AѺde nesthestructureofaB-comodule #algebrffa5onAthenthereisauniquehomomorphismgË:URM@(A)!Bof5bialgebras #such35thatthefollowingdiagrffamcommutesMЍH2PAHKM@(A) Aw{fdPЍά-H`f$ׁ @$ @$ @$ @>@>RH(BE AҟǠ*FfeQǠ?Wg 1AThecorrespSondingstatemenrtforquantumspacesandquantummonoidsisthefollorwing.Prop`osition1.3.9.O1.s+Lffet}XPbeaquantumspacewithuniversalquantum #spffaceM(Xӹ)andUR:M(X)?A!A.9ThenM(X)isaquantummonoidand #Xis35anM(Xӹ)-spfface35by.2. #IfbY_$isanotherquantummonoidandiff:Y?Xn!@X$ade nesbthestructurffe #ofTaY-spffaceTonX'thentherffeisauniquemorphismofquantummonoidsg: #YQ! M(Xӹ)35suchthatthefollowingdiagrffamcommutesHk\ M(Xӹ)UR?XX:Ԟ32fdά-W>H`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.XProof.@_WVegivretheproSofforthealgebraversionofthepropSosition.v5Considerthefollorwingcommutativediagram?lfM@(A) A[fM@(A) M(A) A32fd9Nά-c!1M(A)pR hAfM@(A) A :2fdfά-mxટǠ@feܟǠ?J~Q˪Ǡ@feܟǠ?%\ 1X.A |where themorphismofalgebrasisde nedbrytheuniversalpropSertyofM@(A)withrespSecttothealgebramorphism(1M"(A) %Is2).U7FVurthermorethereisauniquemorphismofalgebrasUR:M@(A)4!1K꨹sucrhthatKxHAHM@(A) Al{fdPЍά--,H31X.Auׁ @u @u @u @׼>@׼>RH"%APUR԰n:=K AJǠ*Ffe|Ǡ?3 1X.Asޠ7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)25Ycommrutes.ThecoalgebraaxiomsarisefromthefollorwingcommutativediagramsuJ{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) andng|AM@(A) As”A2fdpά-mzk2Π@fekdΠ?QpΠ@feKΠ?%. 1X.A3SrǠM@fe3Ǡ?89$1M(A)pR 1X.APM@(A) A+M@(A) M(A) Al:2fd9Nά-}C1M(A)pR x1M(A)pR 1X.AܔP P?_P攴P P?^PPʍPԍ?]P&D P&D q@M@(A) APUR԰n:=M(A) K AǠ@feKǠ?x1M(A)pR  1X.A"XandrAM@(A) A~tA2fdfά-mpǠM@feq$Ǡ?ӍbB1X.A|2Π@fe|dΠ?QΠ@fe$Π?% 1X.AyDnM@(A) A.M@(A) M(A) A$:2fd-@ά-}Cl 1M(A)pR |2Ǡ@fe|dǠ?kS 1X.AǠ@fe$Ǡ?x 1M(A)pR 1X.ArA,M@(A) APUR԰n:=K M(A) A:~t32fd5氍ά-m"XInnfactthesediagramsimplybrytheuniquenessoftheinducedhomomorphismsofalgebras*c( 1M"(A)~L)=(1M"(A)TX  ),:R(1M"(A) )=1M"(A)and*c (1M"(A)~L)=1M"(A)~L.8FinallyAisanM@(A)-comoSdulealgebrabrythede nitionofand.NorwXassumethatastructureofaB-comoSdulealgebraonAisgivenbyabialgebraBQandfQ:URA4!1Bа 5A.:ThenthereisauniquehomomorphismofalgebrasgË:M@(A)4!BsucrhthatthediagramM)VH@AH~M@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.AC7 &e2661.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYcommrutes.8ThenthefollowingdiagramI؍ѭTAѭM@(A) A`bDfd ά-h-nRѭM@(A) AѭAM@(A) M(A) ATTTԟ̇fd9Nά-ȣ*y 1X.Aԟ:2fd9Nά-c1M(A)pR έ*@oafdؤݐʬQnؤ;tQxؤQؤQlQlsέbn`@fen`? gI{ 1X.Aέ4n`@fehn`? gI{ g 1X.A@YxBE A@YJPBE B AM4fdXPά-(X.BX 1X.A432fdXPά- =1X.BX f$implies((gh R/gn9) 1A)kT="(g g 1A)( 1A)kT="(gh g 1A)(1M"(A){ s2)kT=(gP K(g 1A)s2)S=j(1B 0 (g 1A)s2)(g 1A)S=j(1B 0 fG)f(i=(B 0 1A)f(i=(B 0 1A)(g 1A)Ȅ=UR(BN>g 1A)]ڹhence(g gn9)UR=BN>g.8FVurthermorethediagramsgË=UR.8ThrusgXisahomomorphismofbialgebras.}Ccffxff ̟ff ̎ ̄cffcSinceunivrersalalgebrasforalgebrasAtendtobSecomeverybigtheydonotexistingeneral.pButatheoremofTVamrbara'ssaysthattheyexistfor nitedimensionalalgebras(orvera eldK).De nition1.3.10.}!ùIfX|bisaquanrtumspacewith nitedimensionalfunctionalgebrathenwrecallX{a nite35quantumspfface.ThemfollorwingtheoremisthequantumspaceversionandequivXalenttoatheoremofTVamrbara.Theorem1.3.11.wXLffetXbea nitequantumspace.QThenthereexistsa(univer-sal)35quantumspfface35M(Xӹ)withmorphismofquantumspffaces35UR:M(X)?X%!X.ThealgebravrersionofthistheoremisTheorem1.3.12.wX޹(TVamrbara)'LffetAbea nitedimensionalK-algebra. >ThentherffeOexistsa(universal)K-algebraM@(A)withhomomorphismofalgebrasz{:IA!M@(A) A.Proof.@_WVe`aregoingtoconstructtheK-algebraM@(A)quiteexplicitly. FirstwreTobservethatA2 f#=HomK%؊(A;K)isacoalgebra(cf. problemA.6.8)withthestructuralImorphism:A2 j4!/A(A A)2P j԰ Ĺ=k A2Y A2. VDenoteIthedualbasisbryP* n U_ i=1ai }a2i'2URA A2.6{NorwxletTƹ(A A2)xbSethetensoralgebraofthevrectorspace7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)27YA A2.8Considerelemenrtsofthetensoralgebra'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)agenerateatrwo-sidedidealIFURTƹ(A A2).8Norwwede neYM@(A)UR:=Tƹ(A A )=Iand\thecoSoperation\:OA3a4!H+P*"n U_"i=114.(a@ ùa2i ) ai|)2OTƹ(A A2)=I A.Thisisawrell-de nedlinearmap.TVoshorwthatthismapisahomomorphismofalgebraswe rstdescribSethemul-tiplication(IofAbryaidaj =qPk@4 2kRAijJak#.ThenthecomultiplicationofA2 Misgivenbyn(%a2k N)=Plijj 2kRAijpgJa2i .ȹ Ea2jsince((%a2k);al+  EamĹ)=(%a2k;al!amĹ)=Plr 2rylKm (%a2k;arb)=E 2kylKm2=lPSijb5 2kRAijJ(%a2i ;al!ȹ)(%a2j ;amĹ)l=(P ij 2kRAijpga2i[ Ma2j;al amĹ).eNorw%*write1l=PQ 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 Fa2i 2) aidڹ)(P* n U_ jv=13(b Fa2j 3ι) ajf )UR=PijJ(a b Fa2ia Fa2j) aidaj\=EP ijvk! 2kRAijJ(a1 b Wba2i Wba2j ) akx=URPk#(a b (%a2k N)) ak=URPk#(ab Wba2k r) ak=URs2(ab).FVurthermorelwrehaves2(1)UR=Pidٹ(1p a2i 9K) ai,=URPiZda2i(1) ai=UR1 PTixa2iHй(1)ai=UR1 1.Hence]ڹisahomomorphismofalgebras.NorwrwehavetoshowthatthereisauniquegCforeachfG.yFirstofallfQ:URA4!1Bw tqAinducesCuniquelydeterminedlinearmapsfi,:URA4!1B@IwithCfG(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(Aɝ A2)4!^B. )IApplied tothegeneratorsoftheidealwregetgn9(ab2 a2ka b (%a2k N))3,=(12 a2kŹ)Plqfl!ȹ(ab) alPzߟrgn9(1=. %/(1))à=(1=. )fG(1)(1)à=(1=. )(1 1)(1)à=1(1)=.(1)à=0.ThrusthehomomorphismofalgebrasgtvXanishesontheidealI*soitmaybSefactoredthroughM@(A)=Tƹ(A)=I.6Denotethisfactorizationalsobrygn9.Thenthediagram7 &e2861.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYcommrutes^since(gh ̹1A)s2(a)-=(g ̹1A)(P i(a Oa2i ) aidڹ)-=Pڟi*(1 Oa2i)fG(a) ai=$P ijfjf (a)(%a2i ;aj) ai,=URPidfidڹ(a) ai=URfG(a).cWVek}stillharvek}toshorwthatgٶisuniquelydetermined._Assumethatwealsohave(h 1A)pP=f6iv)=GFindanisomorphismBPX԰ @=B2op forthebialgebraBX=URKha;bi=(a22;ab+ba).=G(compareproblem1.72).̠7 &e6I3.pQUANTUM!MONOIDSANDTHEIRA9CTIONSONQUANTUMSP:ACES)29YSketchofsolution:1/2.ߏThe޶bialgebrahastheformBX=URKha;bi=(a22;ab]+ba)޶with(a)=a] 1+b a,(b)UR=b b꨹and"(a)UR=0,"(b)=1.8Thecoactioniss2(x)=a 1+b x.3/4.L=AFrhasthebasis1;x;x22.TheFrdualcoalgebrahasthedualbasise;;2 vwith(e)UR=e e,()UR= e+e ҩand(2)UR=2j e+ +e 2.TheunivrersalbialgebraBX=URTƹ(A& A2)=Isatis ess2(x)=x e 1+x  x+xnM 2.Q x22V=URa 1+b x+c x22./ThrusitisgeneratedbytheelementsaUR=xnM e,bUR=x ҩandcUR=x 2.8Themrultiplicationtableandtherelationsarisefrom3cʍ>1 eUR=1;>1 =S=UR1 2V=0;>x22j eUR=(x e)(x e);>x22j =S=UR(x )(x e)+(x e)(x );>x22j 2V=UR(x 2)(x e)+(x )(x )+(x e)(x 2);Hʍ-C0UR=x23j e=(x22 e)(x e);-C0UR=x23j =S=(x22 )(x e)+(x22 e)(x );-C0UR=x23j 2V=(x22 2)(x e)+(x22 )(x )+(x22j e)(x 2)$G}WVeusetheabbreviationfu;vn9gUR:=u22v+uvu+vu22andharve&8ʍa23V=UR0;fa;bgUR=0;fa;cg+fb;agUR=0:Thecondition(1 s2)Ȅ=UR( 1)]ڹimplies;8ʍx(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:*0ThrusethecoSendomorphismbialgebrahasthealgebrageneratorsa D fwitheaW2f1;x;yn9;xygܹando2 nfe;s;;Sg. |Thegeneratorsoftherelations(ofI)aregivrenbrytheequations1.1and1.2. ATheyimplythat1 9 e퍹istheunitelement,nGthatF7 &e3061.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY1 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 neeʍZY{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;eandv\gets2(x)UR=a 1+b x+c y+W+d xy䕹andv\(yn9)UR=e 1+f x+g+W y+h xyn9.HenceBisgeneratedbrya;:::ʚ;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.ԅ;7  4F C cmbxti10/%n eufm10-2@cmbx8,- cmcsc10+o cmr9*@ cmti12(ppmsbm8' msbm10!u cmex10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10