; TeX output 1999.09.20:1546 7 YRXQ cmr12CHAPTER1RNff cmbx12CommutativeffandNoncommutativeffAlgebraicGeometry/^o cmr911 *7 &e1220:1. %COMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY=[N cmbx122.NQuantumSpacesandNoncommutativeGeometryNorw8wecometononcommutativegeometricspacesandtheirfunctionalgebras.ManryǘofthebasicprinciplesofcommutativealgebraicgeometryasintroSducedin1.1carry-orvertononcommutativegeometryV.nOurmainaim,bNhowever,is-tostudythesymmetries0p(automorphisms)ofnoncommrutative0pspaceswhicrhleadtothenotionofaquanrtumgroup.Sinceljtheconstructionofnoncommrutativeljgeometricspaceshasdeepapplicationsintheoreticalphrysicswewillalsocallthesespacesquantumspaces. ͍De nition1.2.1.vaùLet g cmmi12AbSea(notnecessarilycommrutative) ) msbm10K-algebra.bThenthefunctor !", cmsy10XY:=7K-Algo(A;-):K-Alg4!1%SetI߹represenrtedbyAiscalled,@ cmti12(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-AlgoW2cmmi8op%Xcategoryofnon-commrutativespaces.IfAisa nitelygeneratedalgebrathenitmarybSeconsideredasaresidueclassalgebraaAPUR԰n:=Khx|{Ycmr81;:::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. 7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_13YLemma1.2.4.g5QLffetDbeasetand$:DQXӹ(-35)!wA(-)bffeanaturaltransfor-mation.fiThen35therffeexistsauniquemapfQ:URD!dAsuchthatthediagram<AXӹ(B)HB̶32fdCO line10-W` I{(Bd)@"D6Xӹ(B)ǠfeǠ?b{f!K cmsy81@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#u cmex10 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 :]7 &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.Problem1.2.2.nRLetbS3 fbSethesymmetricgroupandA=:=K[S3]bbethegroupalgebra;onS3.yDescribSethepoinrtsofXӹ(B)z =K-Algo(A;B);asasubspaceofA22(B).WhatisXc.y(B)andwhatistheanealgebraofXc?TVo$understandhorwHopfalgebras tintothecontextofnoncommutativespaceswrehavetobSetterunderstandthetensorproductinK-Algo.De nition1.2.9.vaùLet A>=OUV(Xӹ)andA20 Ϲ=OUV(Y)bSethefunctionalgebrasofthe-TnoncommrutativespacesX'resp. Y.TwroB-pSointspz:A4!BZinXӹ(B)andp20r :A204!/nBJ4in.Y(B)arecalledcffommutingpoints.ifwrehaveforalla2A.andalla20#2URA20Ap(a)p 09(a 0)UR=p 09(a 0)p(a);Íi.e.8iftheimagesofthetrwohomomorphismspandp20commrute.Remark1.2.10.qN6TVoshorwthatthepSointspandp20=commute,[itissucienttocrheckGmthattheimagesofthealgebrageneratorsp(x1);:::ʚ;p(xmĹ)commrutewiththeimages|5ofthealgebrageneratorsp209(y1);:::ʚ;p20(ynP)|5underthemrultiplication. Thismeansthatwrehavebidb 0ڍj\=URb 0ڍjf bi|ofortheB-pSoinrts(b1;:::ʚ;bmĹ)UR2Xӹ(B)and(b20RA1;:::ʚ;b20RAnP)UR2Y(B).,?7 &e1620:1. %COMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYDe nition1.2.11.}!ùThefunctorKP(X%?URY)(B):=f(p;p 09)2Xӹ(B)Y(B)jp;p 0commrute5e3giscalledtheorthoffgonal35product꨹ofthenoncommrutativespacesX{andY.YDRemark1.2.12.qN6TVogether0withXandY-theorthogonalproSductX%?URYisagainafunctor, sincehomomorphismsf˚:B4!B20oarecompatiblewiththemrultiplicationandthruspreservecommutingpSoints.8HenceX%?URY>isasubfunctorofXl{Y.Lemma1.2.13.mQIfWXandYTparffenoncommutativespaces,athenX[?-YTpisanon-cffommutative35spacewithfunctionalgebraOUV(X%?URY)=O(Xӹ) O(Y).IfjX,andYgWhave nitelygenerffatedjfunctionalgebrffasthenthefunctionalgebraofX%?URY/is35also nitelygenerffated.Proof.@_Let'iA:=OUV(Xӹ)andA20:=OUV(Y).$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):=B7 &eLů2.pQUANTUM!SP:A9CESANDNONCOMMUTATIVEGEOMETRY@p_17YIfwrechoSosea̹=1thenwegetpy (a209)pzʮ(a20N920r)=pz(a20N920r)py (a209).FVorarbitrarya;a20;a20N920 "wrethengete