; TeX output 1999.09.20:15487 YRXQ cmr12CHAPTER1RNff cmbx12CommutativeffandNoncommutativeffAlgebraicGeometry/^o cmr917*7 &e1820:1. %COMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY&VN cmbx123.7DVQuantumMonoidsandtheirActionsonQuantumSpacesWVeusetheorthogonalproSductinrtroducedintheprevioussectionas\product"to?de nethenotionofamonoid(somemarycallitanalgebraw.r.t.8theorthogonalproSduct).UPObservreIxthatonthegeometricleveltheorthogonalproSductconsistsonlyofcommrutingpSoints.1Sowheneverwede neamorphismonthegeometricsidewithdomain_0anorthogonalproSductofquanrtumspacesg cmmi12fQ:UR!", cmsy10X%?YQ4!ZS32fdPά-k~rH3,1X.!q% cmsy6M٬ҁ H٬ׁ H٬܁ H٬ H٬ H٬ H٬ H٬ H fdžH fdžjcommrute.8Then(M;m;e)iscalledaquantum35monoid.#7Prop`osition1.3.2.OLffetMbeanoncommutativespacewithfunctionalgebraHV.Then35H isabialgebrffaifandonlyifMisaquantummonoid.,- cmcsc10Proof.@_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):Thereadershouldcrhecktheotheridenrtities.7 &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+:ThereadershouldcrheckthepropSertiesofthecounit.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.9Theidentitymorphismof/67 &e2220:1. %COMMUT: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 $isthedeformedvrersionofM諍2 %themultiplicativemonoidofthe2i2-matricesofcommrutativealgebras.wu2.&LetAߍ2j0]Gqι=URKhx;yn9i=(xyH`tf[Tׁ @[T @[T @[T @ԟ>@ԟ>RyYQ?URXŸǠ*Ffe"Ǡ?`gI{?1X.X.E2. #LetYzAbSeaK-algebra.UAY]K-algebraM@(A)togetherwithahomomorphismof #algebras :A4!5M@(A)9 AiscalledanalgebrffaQcoactinguniversallyon A(or #simplyauniversalalgebrffaforA)ifforevreryK-algebraBandeveryhomomor- #phismbofK-algebrasfi:!A4!\BZ TAbthereexistsauniquehomomorphismof #algebrasgË:URM@(A)4!1BsucrhthatthefollowingdiagramcommutesK]ōH@AH~M@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.;cmmi6A׌BytheunivrersalpropSertiestheuniversalalgebraM@(A)forAandtheuniversalquanrtumspaceM(Xӹ)forX{areuniqueuptoisomorphism.Z7 &e2420:1. %COMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYProp`osition1.3.8.O1.s+LffetpKAbeaK-algebrawithuniversalalgebraM@(A)and #:.A! M@(A) A.ThenM(A)isabialgebrffaandAisanM(A)-cffomodule #algebrffa35bys2.2. #IfѺBlisabialgebrffaandiffQ:URA!Bm  AѺde nesthestructureofaB-comodule #algebrffa5onAthenthereisauniquehomomorphismgË:URM@(A)!Bof5bialgebras #such35thatthefollowingdiagrffamcommutesMЍH2PAHKM@(A) Aw{fdPЍά-H`f$ׁ @$ @$ @$ @>@>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!1MAacmr6(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.Ajj7 &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$1M(A)pR 1X.APM@(A) A+M@(A) M(A) Al:2fd9Nά-}C1M(A)pR x1M(A)pR 1X.AܔP P?_P攴P P?^PPʍPԍ?]P&D P&D q@M@(A) APUR԰n:=M(A) K AǠ@feKǠ?x1M(A)pR  1X.A"XandnrAM@(A) A~tA2fdfά-mpǠM@feq$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$Ǡ?x 1M(A)pR 1X.ArA,M@(A) APUR԰n:=K M(A) A:~t32fd5氍ά-mInnfactthesediagramsimplybrytheuniquenessoftheinducedhomomorphismsofalgebras*c( 1M"(A)~L)=(1M"(A)TX  ),:R(1M"(A) )=1M"(A)and*c (1M"(A)~L)=1M"(A)~L.8FinallyAisanM@(A)-comoSdulealgebrabrythede nitionofand.NorwXassumethatastructureofaB-comoSdulealgebraonAisgivenbyabialgebraBQandfQ:URA4!1Bа 5A.:ThenthereisauniquehomomorphismofalgebrasgË:M@(A)4!BsucrhthatthediagramM)VH@AH~M@(A) A{fdPЍά--pH`;UfWׁ @W @W @W @؝>@؝>RHUBE AŸǠ*FfeǠ?q-tgI{ 1X.Aw7 &e2620:1. %COMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYcommrutes.8ThenthefollowingdiagramI؍ѭTAѭM@(A) A`bDfd ά-h-nRѭM@(A) AѭAM@(A) M(A) ATTTԟ̇fd9Nά-ȣ*y 1X.Aԟ:2fd9Nά-c1M(A)pR έ*@oafdؤݐʬQnؤ;tQxؤQؤQlQlsέbn`@fen`? gI{ 1X.Aέ4n`@fehn`? gI{ g 1X.A@YxBE A@YJPBE B AM4fdXPά-(X.BX 1X.A432fdXPά- =1X.BX f$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#=Hom)ppmsbm8K%؊(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)xbSethetensoralgebraofthevrectorspaceH7 &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) alPzߟr6iv)=GFindanisomorphismBPX԰ @=B2op forthebialgebraBX=URKha;bi=(a22;ab+ba).=G(compareproblem1.22).~;7  ,- cmcsc10+@ cmti12)ppmsbm8( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line10