; TeX output 1994.03.09:1637no]K+XQ cmr121ǐ"@v<"VG cmbx10EndomorphismBialgebrasofDiagramsKandofNon-CommrutativeAlgebrasandSpaces5BODOPVAREIGIShDepartmenrtofMathematics,UniversityofMunich~Municrh,Germany}dBialgebrasRandHopfalgebrasharveRavrerycomplicatedstructure. ItisnoteasytoconstructSexplicitexamplesofsucrhandcheckallthenecessarypropSerties.ThisgetsevenmoreߤcomplicatedifwrehavetoverifythatsomethinglikeacomoSdulealgebraoverabialgebraisgivren.BialgebrasandcomoSdulealgebras,horwever,ariseinavrerynaturalwayinnon-commu-tativregeometryandinrepresentationtheoryV.QWewanttostudysomegeneralprinciplesonhorwtoconstructsuchbialgebrasandcomoSdulealgebras.TheazleadingideathroughoutthispapSeristhenotionofanactionascanbeenseenmostclearlyintheexampleofvrectorspaces.8Givenavectorspace,g cmmi12V:wecanassoSciatewithititsendomorphismb.@ cmti12algebrffaEnd!(Vp)that,pinturn,de nesanactionEnd!(Vp)-!", cmsy10V G!URV.!sTherebisalsoFlthegenerallineargrffoupGL(Vp)thatde nesanactionGL(Vp)"V M!V.L+InFlthecaseoftheendomorphismalgebrawreareinthepleasantsituationthatEnd~(Vp)isavectorspaceitselfsothatwrecanwritetheactionalsoasEndX(Vp) V G!URV.0TheactionofGL'4(V)onVcanalsobSedescribedusingthetensorproductbryexpandingthegroupGLF(Vp)tothegroupalgebraKܞ(GL(Vp))toobtainK(GL(Vp)) V G!URV.WVearegoingto ndanaloguesofEnd(Vp)orKܞ(GL(V))actingonnon-commrutativegeometric"spacesoroncertaindiagrams.OThiswillleadtobialgebras,Hopfalgebras,andcomoSdulealgebras.Therearetrwowell-knownproScedurestoobtainbialgebrasfromendomorphismsofcer-tainBobjects.AHInthe rstsectionwrewillconstructendomorphismspacesinthecategoryofnon-commrutativespaces.8TheseendomorphismspacesaredescribSedthroughbialgebras.*no]K2ǐ[o]Inthesecondsectionwre nd(co-)endomorphismcoalgebrasofcertaindiagramsofvectorspaces,Lxgraded8vrectorspaces,di erenrtialgradedvectorspaces,Lxorothers.#UnderadditionalconditionstheyagainwillturnouttobSebialgebras.Theobjectsconstructedinthe rstsectionwillprimarilybSealgebras,whereasinthesec-ondDsectiontheobjectscoSend n(!n9)willharveDthenaturalstructureofacoalgebra.Nevrerthelesswrewillshowinthethirdsectionthattheconstructionsofbialgebrasfromnon-commutativespacesandofbialgebrasfromdiagramsofvrectorspaces,.remoteastheymayseem,.arecloselyrelated,in.factthatthecaseofanendomorphismspaceofanon-commrutative.spaceisaspSecialcaseofacoendomorphismbialgebraofacertaindiagram.4Someotherconstructionsofe0endomorphismspacesfromtheliteraturewillalsobSesubsumedunderthemoregeneralconstructionwDofcoSendomorphismbialgebrasofdiagrams.޴WVealsowill ndsucrhbialgebrascoactingonLiealgebras.ThisTindicatesthatasuitablesettingofnon-commrutativeTgeometrymighrtbSeobtainedbryhpconsidering(monoidal)diagramsofvectorspaces(whichcanbSeconsideredaspartiallyde nedalgebras)asageneralizationofanenon-commrutativespaces.HlSotheproblemof ndingnon-commrutative(non-ane)schemesmightbSeresolvedinthisdirection.In}ythelastsectionwreshowthatsimilarresultsholdforHopfalgebrasactingonnon-commrutativespacesresp.7ondiagrams.TheunivrersalHopfalgebracoactingonanalgebraiskusuallyobtainedastheHopfenrvelopSekoftheunivrersalbialgebraactingonthisalgebra.WVeshorwthatthisHopfalgebracanalsobSeobtainedasthe(co-)endomorphismbialgebraofaspSeci cdiagramconstructedfromthegivrenalgebra.ThroughoutthispapSerKFshalldenoteanarbitrary eldand standsfor 2cmmi8K;.(V1EndomorphismsofNon-CommrutativeGeometricSpacesb#In Xthissectionwrediscusssomesimplebackgroundonnon-commutativespacesandtheirendomorphisms."ʫ7"Vff cmbx101.1%@Anenon-comm=utativespaces@InyalgebraicgeometryananealgebraicspaceisgivrenasasubsetofKܞ2n consistingofallpSoinrtsIsatisfyingcertainpolynomialequalities.zAtrypicalexampleistheunitcirclein0N cmbx12R2t : cmbx92givrenby[CircF(R)UR=f(x;yn9)2R |{Ycmr82jx 2j+y 2=1g:ActuallyBoneisinrterestedincircleswitharbitraryradiusroveranycommutativeR-algebraB.8Theyarede nedinasimilarwrayby}CircK(B)UR=f(x;yn9)2B 2[ jx 2j+yn9 2rS 2h=UR0g:If forexamplerS22 !gis1insteadof1,therearenopSoinrtswithcoecienrtsinthe eldofrealnrumbSersbutlotsofpoinrtswithcoecienrtsinthe eldofcomplexnumbSers.FVurthermorewthisde nesafunctorCirc :cR-AlgTc 'ZN!Sets erfromwthecategoryofallcommrutativeR-algebrastothecategoryofsets,sinceanalgebrahomomorphismfQ:URBX E!B2K cmsy80pinducesnamapofthecorrespSondingunitcirclesCirc (fG): :CircĦ(B) c!Circ(B20i?).2ThecoSordinatesforthepoinrtswhichweconsideraretakenfromalgebraB,thecffoordinatedomain, qno]K3ǐ[o]andKCirc(B)KisthesetofallpSoinrtsofthespace,.^thatisthegivenmanifoldorgeometricspace.OnehasthissetforallcrhoicesofcoSordinatedomainsB.IngeneralafunctorX%:URKܞ-AlgTc=%!Sets40withX(B)UR:=f(b̸1;:::ʜ;b̹nP)2B nCVjp̸1(b̸1;:::ʜ;b̹nP)UR=:::uD=p̹rb(b̸1;:::ʜ;b̹nP)=0gQp(1)iscalledan/}h! cmsl12anealgebraicspace.8Thisfunctorisrepresenrtedbythealgebra`AUR=Kܞ[x̸1;:::ʜ;x̹nP]=(p̸1(x̸1;:::ʜ;x̹nP);:::;p̹rb(x̸1;:::;x̹nP));henceX(B)UR=Kܞ-AlgTc0(A;B).8SoforthecirclewrehavewUCirc(B)PUR԰n:=R-AlgTc (R[x;yn9]=(x 2j+y 2rS 2);B):TheYVonedaLemmashorwsthatthisrepresentingalgebraAisuniquelydetermined(uptoisomorphism)brythefunctorX.NItisusuallyconsideredasthe"functionalgebra"ofthegeometric,spaceunderconsideration.aIndeedthereisamapAX(Kܞ)UR !URK,FE(a;fG)7!f(a),wherefQ2URX(Kܞ)P԰n:=K-AlgTc0(A;K).Allofthishasnothingtodowiththecommrutativityofthealgebrasunderconsideration.Henceionecanusenon-commrutativeiKܞ-algebrasAandBaswrell. Certainquestionsinphrysicsinfactrequiresuchalgebras.8InsteadofrepresentingalgebrasbdAUR=Kܞ[x̸1;:::ʜ;x̹nP]=(p̸1(x̸1;:::ʜ;x̹nP);:::;p̹rb(x̸1;:::;x̹nP))wretakenowrepresentingalgebras`AUR=Kܞhx̸1;:::ʜ;x̹nPi=(p̸1(x̸1;:::ʜ;x̹nP);:::;p̹rb(x̸1;:::;x̹nP))whereIKܞhx̸1;:::ʜ;x̹nPiisthealgebraofpSolynomialsinnon-commrutingvXariablesor,equiva-lenrtlyV,thetensoralgebraofthe nite-dimensionalvectorspacewithbasisx̸1;:::ʜ;x̹nP.DEFINITION1.1af&(B-pSoinrtsofquantumspaces)A3yfunctorX=UKܞ-Alg(A;)iscffalled3yananenon-commrutativespace3yoraquanrtumspaceand35theelementsofX(B)UR=Kܞ-Alg(A;B)35theB-pSoinrtsofX.Asuin(1)theB-pSoinrtsofaquantumspaceareelementsofB2nCV,sothatX(B)isasubsetofB2nCV.WVellb&knorwnexamplesofquantumspacesarethequantumplaneswithfunctionalgebras[5]k|8OUV(A 2j0ڍq \|)UR=Kܞhx;yn9i=(xyq 1 ʵyx);qË2URKFnf0gand[6]"!;JOUV(Aߍ2j0g0 \|)UR=Kܞhx;yn9i=(xyyx+y 2.=):Asin(commrutative)algebraicgeometryonecanconsiderthealgebraAYj=OUV(X)( m=Kܞhx̸1;:::ʜ;x̹nPi=(p̸1(x̸1;:::ʜ;x̹nP);:::;p̹rb(x̸1;:::;x̹nP))(whicrhrepresentsthequantumspaceX)asthefunction4algebrffaofXconsistingofthe(natural)functionsfromX(B)tothecoSordinatealgebraU1(B)whereU :ٚKܞ-AlgQ"!Setsistheunderlyingfunctor.:SotheelemenrtsofAare1certainfunctionsfromX(B)toB.WVedenotethesetofallnaturaltransformationsorfunctionsfromX{toUbryMap g(X;U1)no]K4ǐ[o]LEMMA1.2F{(thefunctionalgebraofaspace)LffetpX0:]Kܞ-Alg!m!Sets,^beparepresentablefunctorwithrepresentingalgebraA.+Thenthereis35anisomorphismAPUR԰n:=Map (X;U1)inducinganaturffaltransformationAX(B)UR !URB.|ProSof:+TheunderlyingfunctorU.:Kܞ-Alg# !SetsisrepresenrtedbythealgebraKܞ[x]=Kܞhxi.8SobrytheYVonedaLemmaweget}V)9Map g(X;U1)PUR԰n:=Map(Kܞ-Alg(A;);Kܞ-Alg(Kܞ[x];))PUR԰n:=K-Alg(K[x];A)PUR԰n:=A:ffǟffffffffǎ&CORrOLLARVY1.3cz(theunivrersalpropSertyofthefunctionalgebra)LffetAXaandAbeasinLemma1.2.tIf'o:CIX(B) =!BܔisAanaturaltransformationthentherffe35existsauniquemap~'::URC1 I!A35suchthatthefollowingdiagramcommutesU1L/pAX(B)/pzB:bfd$0DO line10D-x0/p>{m'tҰD@tܰ@t氐@t@H@HRx0T CFX(B)/pږ*FfeD?औb~{'71ProSof:8ThisfollorwsfromtheYVonedalemma.ffǟffffffffǎ&IfXEnandY1arequanrtumspaces,7thenwewillcallanaturaltransformationfQ:URX% l~!Ysimplyamap[ofquantumspffaces. SothequanrtumspacesformacategoryQwhichisanrtiequivXalentdtothecategoryof nitelygeneratedKܞ-algebrasK-Alg. ThesetofmapsfromX{toY>willbSedenotedbryMap g(X;Y).If*\X/isaquanrtumspace,:IA=OUV(X)*\itsfunctionalgebra,andICAatrwo-sided*\idealof01A,AthenAˮ !!ˮA=I!isanepimorphismwhicrhinducesamonomorphism:X̹IT!XfortheassoSciatedquanrtumspaces.OInparticulartheB-poinrtsX̹IM(B)v=Kܞ-Alg(A=I;B)canbSeidenrti edwithacertainsubsetofB-pSointsinX(B).BConverselyeverysubspaceYXݜ(i.e.7tY(B)BX(B)/functoriallyforallB)isinducedbryanepimorphismOUV(X)B ]!BO(Y).(Observre,however,thatnotevreryepimorphisminthecategoryKܞ-Algt`issurjective,e.g.Kܞ[x]UR !URK(x)isanepimorphismbutnotasurjection.)"F1.2%@Thecomm=utativepartofaquantumspace@TheE7quanrtumplaneA22j0RAqde nesafunctorfromthecategoryofalgebrastothecategoryzofysets.WVecallitsrestrictiontocommrutativeyalgebrasthecommrutativeypart(A22j0RAq \|)commof*thequanrtumplane.IngeneraltherestrictionofaquantumspaceXotocommutativealgebrasiscalledthecffommutative [partofthequanrtumspaceandisdenotedbyXcomm.$.The*commrutativepartofaquantumspacerepresentedbyanalgebraAisalwaysananealgebraic(commrutative)scheme,?[sinceitisrepresentedbythealgebraA=[A;A],?[where[A;A]denotesthetrwo-sidedidealgeneratedbryallcommutators[a;b]]x=abQbafora;b]x2A. pInparticular,the!commrutativepartXcomm#EofaquantumspaceXisindeedasubspaceofX.FVoracommrutativealgebraBthespacesXcomm%EandX}harvethesameB-pSoinrts:Xcomm.$(B)UR=X(B).,fno]K5ǐ[+FVor`thequanrtumplaneA22j0RAqandcommutative eldsBBfthesetofB-pSointsconsistsexactlyzofHthetrwoHcoSordinateaxesinB22 since(A22j0RAq \|)comm#visrepresenrtedbyKܞ[x;yn9]=(xyXqyx)P԰i=Kܞ[x;yn9]=(xy)forqË6=UR1."ʫ1.3%@Comm=utingpuoints@InalgebraicgeometryanrytwopSoints(b̸1;:::ʜ;b̹m)and(b20RA1;:::ʜ;b20RAnP)withcoSecienrtsinthesamecoSordinatealgebraB8harvethepropertrythattheircoordinatesmrutuallycommuteunderthemrultiplication/b̹idb20RAj=Zb20RAjf b̹i ofB5sinceBiscommrutative.vThis/doSesnotholdanrylongerfornon-commrutativealgebrasBandarbitraryquanrtumspacesX{andY.DEFINITION1.4af&(commrutingpSoints)If?Ak6=OUV(X)andA209o=OUV(Y)andifp:A !B<2X(B)andp209o:A20 !B<2Y(B)?arffetwopffoints withcoordinatesinB,wesaythattheyarecommrutinghpSoints ifforallaUR2A;a20#2A20we<havep(a)p209(a20)?=p209(a20)p(a),~Ei.e.<theimagesofthealgebrffahomomorphismspandp20cffommute35elementwise.Obrviouslyritissucienttorequirethisjustforasetofalgebrageneratorsa̹ilofAandta20RAjڛofA209.֛InparticularifAisoftheformA@=Kܞhx̸1;:::ʜ;x̹mi=IfandtA20BisoftheformA20$D=V Kܞhx̸1;:::ʜ;x̹nPi=J4TthentheB-pSoinrtsaregivenby(b̸1;:::ʜ;b̹m)resp.(b20RA1;:::ʜ;b20RAnP)withcoSordinatesinB,andthetrwopointscommutei b̹idb20RAj\=URb20RAjf b̹iOforalliandj.ThesetofcommrutingpSoints.+4(X?Y)(B)UR=X(B)?Y(B)UR:=f(p;qn9)2X(B)Y(B)jpand|qXcommrute33gis asubfunctorofXY sincecommrutativity oftrwo elementsinBispreservedbyalgebrahomomorphisms=f+:B~ c!B20i?.2WVecallittheorthoffgonalproductofthequanrtumspacesXandY.LEMMA1.5F{(theorthogonalproSduct)IfnX0andYkarffequantumspaceswithfunctionalgebrasA=OUV(X)nandA20@=O(Y)thentheorthoffgonal2productX?Y/isaquantumspacewith(representing)functionalgebraOUV(X?Y)66=A A20#=UROUV(X)#W OUV(Y).ProSof:"no]K6ǐ[o]ThislemmashorwsthatthesetofcommutingpSointsk((b̸1;:::ʜ;b̹m);(b 0ڍ1;:::ʜ;b 0ڍnP))UR=(b̸1;:::ʜ;b̹m;b 0ڍ1;:::ʜ;b 0ڍnP)with>+b̹idb20RAjbJXb20RAjf b̹i,=UR0formsagainaquanrtumspace.aItisnoweasytoshowthatthecategoryQofF"quanrtumspacesisamonoidalcategorywiththeorthogonalproSductX?YB(inthesenseof[3]).#The,assoSciativitryoftheorthogonalproductarisesfromtheassociativitryofthetensorproSductofthefunctionalgebras.Theprecedinglemmashedssomelighrtonthereason,#Swhywehaverestrictedourcon-siderationstocommrutingpSoints.?Thereisageneralcredothatthefunctionalgebraofa"non-commrutative"spaceshouldbSegradedandharvepolynomialgrorwththatissomekindofpRaProincarse-Birkho -WittpRtheoremshouldhold.ButthefreeproSductofalgebras(whicrhwrouldcorrespSondtotheproductofthequanrtumspaces)growsexpSonentially(withthede-gree).HSomekindofcommrutationrelationamongtheelementsofthefunctionalgebraisrequiredX andthisisgivrenbylettingtheelementsofthetwofunctionalgebrasAandB&inthe0PorthogonalproSductofthequanrtumspacescommute.ThisisdonebythetensorproSduct."1.4%@Theendomorphismsofaquan=tumspace@In6thecategoryofquanrtumspaceswewantto ndananalogueoftheendomorphismalgebraEnd (Vp)ofavrectorspaceV0andofitsactiononV.9)ActuallywreconsiderasomewhatmoregeneralsituationofanactionH(X;Y)?X% l~!URYwhicrhresemblestheactionHoma(V;W)' Va !Wfor,Pvrectorspaces.ThetensorproSduct ofvectorspaceswillbSereplacedbytheorthogonalproSductofquanrtumspaces?.эDEFINITION1.6af&(thehomofquanrtumspaces)Lffet*XandY'~beaquantumspaces.PA*(universal)quantumspaceH(X;Y)togetherwithamapUR:H(X;Y)?X% l~!Y,suchthatforeveryquantumspffaceZxandeverymap h:URZ ?X% l~!Ytherffe35isauniquemap :URZI^ !H(X;Y)35suchthatthediagramW^\mp H(X;Y)?XmppY̟fd@D-K20mpܧ%} lD@l@l@l@X3@X3Rڍ3Z ?Xmpئڟ5*Ffe 5D?Ѝ O?1cffommutes,isH>calledahomomorphismspace.E̺X ?:=VVH(X;X)iscffalledanendomorphismspace.ApartfromthemapH(X;Y)?X| !Y,-whicrhwewillregardasamultiplicationofH(X;Y)onX{(withvXaluesinY),thisconstructionleadstofurthermrultiplications.LEMMA1.7F{(themrultiplicationofhoms)IfitherffeexisthomomorphismspacesH(X;Y),H(Y;Z ),andiH(X;Z )iforthequantumspffacesX,Y,andXZ ,thentherffeisamultiplicationmUR:H(Y;Z )?H(X;Y)UR !URH(X;Z )XwithrffespecttotheorthoffgonalproductstructureinQ.,Furthermorethisproductisassociativeandunitaryif35theneffcessary35homomorphismspffaces35exist.ProSof:8ConsiderthediagramPܠno]K7ǐS=u`Aw(H(Y;Z )?H(X;Y))?Xu`,wH(Y;Z )?(H(X;Y)?X)ԟƨfdD-Ý͍Px԰`=u`u`XH(Y;Z )?Y,ԟƨfd(D-P601?u`,]r@*Ffe]@D??Ƽm?1u`,wr@*Ffewˤ@D?;|~$,=(H(X;Z )?X,rpZ_҄fdꠍD-x*whicrh;8inducesauniquehomomorphismms:H(Y;Z )?H(X;Y)s 3!sH(X;Z ),O\the;8mrultipli-cation.wffǟffffffffǎ&CORrOLLARVY1.8cz(theendomorphismspaceisamonoid)ThewendomorphismspffacewE̺X `ofaquantumspffacewXJisamonoid(analgebrffa)inthecategoryQw9w.'r.t.theorthoffgonalproduct.'ThespaceH(X;Y)isaleftE̺Yϼ-spaceandarightE̺X-space(an35E̺Yϼ;E̺X-bimoffdule)inQ.ProSof:*rThemrultiplicationisgiveninLemma1.7./BTVogettheunitweconsidertheonepSointquanrtumspaceJ7v(B)UR=fu̹BN>gwithfunctionalgebraKFandthediagramXUigE̺X?X sX씌fdD-Kv0mpݯP<԰$=|D@|@|@|@œ3@œ3RKCJ7v?Xmp5*FfeC5D?kד!URE̹B OUV(H(X;Y))A ; OUV(H(X;Y))v!UROUV(H(X;Y))E_ E̹A:(+In"particularE̹A ]andE̹B `arebialgebras,AandAandOUV(H(X;Y))IarecomoSdulealgebrasorverE̹A,`andBandOUV(H(X;Y))I,7arecomoSdulealgebrasorverE̹BN>.FVurthermorethemapH(X;Y)?X% l~!URY>inducesanalgebrahomomorphismBX E!OUV(H(X;Y))E_ A.8WVrite=a(A;B)UR:=OUV(H(X;Y))BU: pno]K9ǐ[o]ThenwrehaveMap g(Z ;H(X;Y):)PUR԰n:=Map(Z ?X;Y+)h!andKܞ-Alg(a(A;B);Cܞ)PUR԰n:=K-Alg(B;CF A):and aunivrersalalgebrahomomorphismBX E!URa(A;B)c A.VHere wehaveusedthenotationofTVamrbara[15 ]fortheuniversalalgebra.FVromDe nition1.6andLemma1.7wregetthefollowingCORrOLLARVY1.10iv(thecoSendomorphismbialgebraofanalgebra)LffetAbea(non-commutative)algebra.LetE̹A beanalgebraandQ:+A x!E̹A  -Abeanalgebrffaxhomomorphism,ksuchthatforeveryalgebraBt~andalgebrahomomorphism h:URA !BE A35therffeisauniquealgebrahomomorphism :URE̹A 36!B;such35thatS/#A#柷E̹A AK"fdЍD-*S[0卒| ǥPD@ѥP@ۥP@P@d^n@d^nR1ɍBE A_Є*Ffe$ğ_D?<D T 1PBcffommutes. ThenE̹A representstheendomorphismspaceofXV!=NKܞ-Alg(A;-33),d"E̹A isabialgebrffa,35andAisanE̹A-comodulealgebra.Sincethecategoryofalgebrasisdualtothecategoryofquanrtumspace,wecallE̹A othecffoendomorphismbialgebrabxofAanditselemenrtscffoendomorphisms. {FVrombxPropSosition1.9weobtainimmediatelyCORrOLLARVY1.11iv(theunivrersalpropSertiesofthecoendomorphismbialgebra)The=cffoendomorphismbialgebraE̹A ofanalgebraAandthemap:hA !E̹A Q mAhavethefollowing35universalprffoperties:a)ForeveryalgebrffaB*andalgebrahomomorphism h:URA !Bڰ ?Athereisauniquealgebrahomomorphism35 :URE̹A 36!B;suchthatthediagrffamS/#A#柷E̹A AK"fdЍD-*S[0卒| ǥPD@ѥP@ۥP@P@d^n@d^nR1ɍBE A_Є*Ffe$ğ_D?<D T 1PBcffommutes.b).ForeverybialgebrffaBɒandalgebrahomomorphism :P:&A |!B dA.makingAintoacffomoduleFalgebrathereisauniquebialgebrahomomorphism <:SE̹A 1*!Bsuchthatthediagrffam#A#柷E̹A AK"fdЍD-*S[0卒| ǥPD@ѥP@ۥP@P@d^n@d^nR1ɍBE A_Є*Ffe$ğ_D?<D T 1PBcffommutes.^ffǟffffffffǎ ۠no]O10ǐ]o]2CoOendomorphismBialgebrasofDiagramsb#WVeJharveseenthat,similartocommutativealgebraicgeometryV,non-commutativespacesrepresenrtedwbynon-commutativealgebrasinducenon-commutativeendomorphismspaceswhicrharerepresentedbybialgebras.Norw(2wemovetoaseeminglyunrelatedsubjectwhichisstudiedinrepresentationtheoryandTVannakXadualitry.ڨTheprincipalquestionhereiswhetheragroup,amonoid,oranalgebraare%completelydetermined,LifalltheirrepresenrtationsormoSdulesare"known".WVecertainlyharvedtospSecifywhattheterm"knorwn"shouldmeaninthiscontext.tCertainreasons,D mainlythefundamenrtaltheoremonthestructureofcoalgebrasandcomoSdules,makeiteasiertoconsidercomoSdulesratherthanmodulesasrepresenrtations.The\purpSoseofthissectionistoshorwthatforeachdiagramof nitedimensionalvectorspaces>nthereisanassoSciatedcoalgebrawhicrhbeharves>nlikethedualofanendomorphismalgebra.InhdparticularwreassoSciatewiththetrivialdiagramthatconsistsofjustone nitedimensional=vrectorspaceVacoalgebraCthatisthedualoftheendomorphismalgebraofthisJ'vrectorspace:CP{԰c=!End'.n(Vp)2P԰=V V2\t.W\IfJ'thediagramofvectorspaceshasadditionalpropSertiesLthentheassociatedcoalgebrawillharveLadditionalproperties.^ThisconstructionrendersbialgebrasandcomoSdulealgebrasassociatedwithcertaindiagrams.IfޫwrestartwithacoalgebraCIthenwecanconstructthediagramofallits nitedimensional comoSdulesandcomodulehomomorphisms,Qandalsothediagramorcategoryof"allcomoSdules. NThenthecoalgebraCcanberecorvered"fromtheunderlyingfunctor!Ë:URC5omoSd-!C1 I!AasthecoalgebraassoSciatedwiththisdiagram.EacrhibialgebraB:oinducesatensorproSductinthecategoryofcomodulesC5omod-!BorverZB.ThisbialgebracanalsobSerecorveredZfromtheunderlyingfunctorasthebialgebraassoSciated֍withthegivrendiagram.2,AdditionalpropertiesofBqinduceadditionalfeaturesofC5omoSd-!BandconrverselyV."ʫ2.1%@Thebasecategory@AllpthestructuresconsideredinthispapSerarebuiltonunderlyingvrectorspacesoveragiven eld.CertaingeneralizationsofourconstructionsarestraighrtforwardwhencewrewilluseageneralcategoryAinsteadofthecategoryofvrectorspaces.WVeassumethatAisanabSeliancategoryandamonoidalcategorywithanassociativreunitaryDtensorproSduct ~:AA~ C!~ADasin[8].FWVealsoassumethatthecategoryAiscoScomplete([7]and[13 ]p.23)andthatcolimitscommrutewithtensorproducts.FinallywreassumethatthemonoidalcategoryAisquasisymmetricinthesenseof[13 ]orbrffaidedthatisthereisabifunctorialisomorphismË:URX+ Y G!YG X+sucrhthatthediagramTDw@S3X+ (YG Zܞ)w@ (YG Zܞ) Xªrfd*D-<qw@w@>3YG (ZF X)tªrfd* D<!р w@.u> *Ffeur$ D?h5i2 w@.` *Ffe`O$ԯD60e1 .S3(X+ Yp) Z. (YG X) Zafd*D-͍V  1..>3YG (X+ Zܞ)tafd* DE!р anddthecorrespSondingdiagramforn921/Lcommrute. iscalledasymmetryifinadditionn91cX&;Y'(=UR̹Yx;Xʶholds.WVegivreafewinterestingexamplesofsuchcategories.1):ThecategoryVec@ofallvrectorspacesoverKwiththeusualtensorproSductisasymmetric no]O11ǐ[o]monoidalcategoryandsatis estheconditionsforA.2)=GThecategoryC5omoSd-!H*ofcomoSdulesorver=GacoquasitriangularorbraidedbialgebraH*[13 ]withthediagonalactionofHonthetensorproSductofcomodulesorverK[andwithcolimitsinVecland8thecanonicalHV-comoSdulestructureontheseisaquasisymmetricmonoidalcategoryandsatis estheconditionsforAsincetensorproSductscommrutewithcolimits.3)ThecategoryofN-gradedvrectorspacesisisomorphictothecategoryC5omoSd-!HV,withNH{=%Kܞ[N]themonoidalgebraorverNthemonoidofinrtegers. dHencethecategorylofgradedvrectorspaces(withtheusualgradedtensorproSduct)isasymmetricmonoidalcategorywhicrhsatis estheconditionsforA.4) pThecategoryofcrhaincomplexesofvectorspacesKܞ-FC5omp]isisomorphictothecategoryC5omoSd-!Hof `comoSdulesorver `theHopfalgebraHx=OKܞhx;yn9;y21 ʵi=(xy.y+@yx;x22) `with(x)O=xp 1+yn921% xCand(yn9)F=yU py'[10 ,C11].DH1DisCacotriangularHopfalgebra([10]p.D373)andKܞ-FC5ompisasymmetricmonoidalcategorywiththetotaltensorproSductofcomplexeswhicrhsatis estheconditionsforA.5)HSupSer-symmetricspacesarede nedasthesymmetricmonoidalcategoryC5omod-!H6ofcomoSdulesorvertheHopfalgebraHB=URKܞ[Z=2Z]whichiscotriangular[1].WVe9|aregoingtoassumethroughoutthatthecategoryAissymmetric,M1sothatwrecanusewthefullstrengthofthecoherencetheoremsformonoidalcategories.ThrusmostofthetimewrewilldeleteallassoSciativityV,5unity,5andsymmetrymorphismsassumingthatourcategories,arestrictsymmetricmonoidalcategories. Mostofthegivrenresultsholdalsooverquasisymmetric =monoidalcategories,bcf.[4],but =itgetsquitetecrhnicalifonewantstocheckallthedetails.WVesarythatanobjectXofacategoryAhasadual(X2,ev)whereX2 Misanobjectandev:URX2\/ XF .!I+isamorphisminA,ifthereisamorphismdb:URIF!X+ X2 /sucrhthat$(XFK`y cmr10dbcI 1g  !X+ X2\/ XFո1 evg  <1!1X)UR=1̹X 1;XЍ(X2 ٸ1 dbg X!"UX2\/ X+ X2 ev 1g 5! X2)UR=1̹Xҟq% cmsy6]:ThecategoryAisrigidifevreryobjectofAhasadual.ThefullsubScategoryofobjectsinAharvingdualsinthesenseof[12 ]isdenotedbyA̸0.The categoryA̸0thenisarigidsymmetricmonoidalcategoryV.Observrethatavectorspacehasadualinthissensei itis nitedimensional."{ڍ2.2%@Cohomomorphismsofdiagrams@InthissubsectionthecategoryAdoSesnotharvetobesymmetricorquasisymmetric.WVeconsiderdiagrams(commrutativeornot)inA.QTheyaregivrenbytheobjectsatthevrerticesrandthemorphismsalongtheedges.Theverticesandthe(directed)edgesorarrowsalonede netheshapSeofthediagram,e.-[g.atriangleorasquare.-[Thisshapecanbemadeinrtoacategoryofitsownright[3,7 ],Zadiagrffam8scheme,andtheconcretediagramcanthenbSeconsideredasafunctor,sendingthevrerticestotheobjectsattheverticesandthearrowstothemorphismsofthediagram.8SothediagramscrhemeforcommutativetrianglesR}Ƒ=1Ƒ=2u 撄fd-`D-x% ³`jyT 7D@7@7@7@~|0@~|0R³`H}32@*Ffeܟ2@D?i\ Yhas8atotalofthreeobjectsf1;2;3g8and6morphismsf ; O; ;id ʢ̸1;id ʢ̸2;id ʢ̸3g,idenrtities8in- ߠno]O12ǐ[o]cluded.Let!Ë:URD !AbSeadiagraminA.-@ThecategoryDandthrusthediagram!6isalwaysassumedrtobSesmall.z>WVecallthediagram niteifthefunctor!:ziDϿ %!ArfactorsthroughA̸0thatisifallobjectsofthediagramharveduals.THEOREM2.1V (theexistenceofcohomofdiagrams)LffetXw(DUV;!n9)and(D;!n920whencebitcanbSewrittenasaquotienrtofacertaincoproduct. `ThetrwobmorphismsF̹CandPOF̹D arecompSosedofmorphismsoftheform1 !n920 D?⣤) ~'$'2NB 1cffommutes.荑Byj[12 ]2.1.12thecoSendomorphismsetcoend" (!n9)isacoalgebra.ThecomrultiplicationarisesfromthecommrutativediagramS D?) ~'$'/(X) 1cffommute.ProSof:'(X):!n9(X)  !D# !n9(X)!isinfactanaturaltransformation.pThentheexistenceanduniquenessofahomomorphismofvrectorspaces}~'=:coSend"3A(!n9) !Disobvious.FThefact,thatthisisahomomorphismofcoalgebrasfollorwsfromtheuniversalpropSertyofC=coSendz(!n9)bry `^!`^CF !\RfdCgD-ZzZʓ;\ xlJ/nD@x@@+ğ.@+ğ.R\ w9!! 1 nD@x@@ğ.@ğ.R\ 37@dfek$@?Z=ب1\ 3)@dfe]$@D?M$Qr~m'ʩ 1WÌCF !W0CF C !dRfd+s D-{F1  ,0@dfed$@D?F$Xr~t'ѩ 1 ,1"@dfe1V$@D?F$7~6'@e ~'  19!9D6 !f҄fdBD-ٛ{Q'3,ˍN'lD@l@l@+ğ@+ğR3,I<} 1 lD@l@l@ğ@ğR9$D6 !9D6 D !_҄fd*(pD- 'G1 '!no]O15ǐ[o]where@therighrtsideofthecubSecommutesbytheuniversalpropSertyV.:SimilarlyoneprovesthatΏ~'preservresthecounit.Affǟffffffffǎ&TVo~PdescribSethecoalgebrastructureofcoend!1(!n9)wredenotetheimageofx\ 2UR!(X) !n9(X)2in̖coSend (!)̖bryw}fe0 x .مLetfurthermoreCPZx̹i2 bX̹i1pdenotethedualbasisin!n9(X)bX !n9(X)2.Then:V!n9(X) !coSend" -(!) !(X)isinducedbrythemap!(X) !(X)2 l!VcoSend" -(!)andisgivrenbys2(x s)UR=CXw}feޗ x ̹i4PX x̹id:#TheconstructionofUR:coSend!(!n9) !coSend(!n9) coSend ^"(!)furnishes(\(w}fe0 x 0)UR=CXw}feޗ x ̹i4PX w}fes x̹i !:CORrOLLARVY2.7cz(diagramrestrictionsinducemorphismsforthecohoms)LffetD8andDUV20qbediagramschemesandlet!Ë:URDUV20x :!Aand!n920:URDUV20x!Abffe nitediagrams.LffetL F]:LDآ -!DUV20obeL afunctor.ThenF}inducesahomomorphism/~':Lcohom$(!n920 (D20 eG;!n920!n920!n920h@*Ffe>4@D?čE1~CM'M 1cffommutes.THEOREM2.13\(cohomisanalgebra)Lffet(DUV;!n9)and(D;!n920TF@HTP@RTZ@\Td@dHl@dHlRO:\wŸ@|z@fe\@?O:[Ÿ@|z@fe@?tA,vMŸ@|z@fev@?:Bn!n9(X+ Yp):cohomG^(!n920Ts@HTs@RTs@\Ts@dH@dHR9-DF1`2D5s-DF43where triangle1commrutesbyCorollary2.12,esquare2commutesbythede nitionofthemrultiplication!ginTheorem2.13,/square3commutessince'isanaturalmonoidaltransfor-mation,ֹandtriangle4commrutesbytheuniversalpropSertyofcohom%(!n920andofcommrutativityconditions.ThenwrewillcalculatetheassoSciatedcoendo-morphismJbialgebras.VSinceinamonoidalcategorytherearetrwoJcompSositions,athetensorproSductջandthecompositionofmorphisms,wreareconstructingafree(partiallyde ned)algebrafromtheobjects,morphismsandrelations.LetX̸1;:::ʜ;X̹n bSeagivrensetofobjects. MtThenthereisafreemonoidalcategoryC5[X̸1;:::ʜ;X̹nP]tgeneratedbrytheX̸1;:::ʜ;X̹n Tconstructedinananalogouswraytasthefreemonoidalcategoryononegeneratingobjectin[3].LetYX̸1;:::ʜ;X̹n ybSeagivrensetofobjectsandlet'̸1;:::ʜ;'̹m bSeadditionalmorphismsbSetrweentheobjectsofthefreemonoidalcategoryC5[X̸1;:::ʜ;X̹nP]generatedbryX̸1;:::ʜ;X̹nP.Fyno]O21ǐ[o]ThenNythereisafreemonoidalcategoryC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m]NygeneratedbryX̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m.If9theobjectsX̸1;:::ʜ;X̹n Oandmorphisms'̸1;:::ʜ;'̹m >aretakreninA,Mthentheyinduceauniquemonoidalfunctor!Ë:URC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m]UR !URA.WVe7indicatehorwthevXariousfreemonoidalcategoriescanbSeobtained.C5[X̸1;:::ʜ;X̹nP]isgeneratedasfollorws.8Thesetofobjectsisgivenby)(O̸1)QX̸1;:::ʜ;X̹n areobjects,>(O̸2)QI+isanobject,(O̸3)Qif:Y̸1andY̸2areobjectsthenY̸1= C9Y̸2isanobject(actuallythisobjectshouldbSewritten_as(Y̸1j Y̸2)toarvoidproblemswiththeexplicitassoSciativitryconditions),(O̸4)Qtheseareallobjects.Thesetofmorphismsisgivrenby(M̸1)!foreacrhobjectthereisanidentitymorphism,(M̸2)!forD/eacrhobjectYtherearemorphismss:Ik Y=(R̸2)theassoSciativitryandunitarycoherenceconditionformonoidalcategories,(R̸3)theconditionsthatand2x,and2, 7and 2 areinrversesofeacrhother.)ThefreemonoidalcategoryC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m]forgivrenobjectsX̸1;:::ʜ;X̹n >andmorphismsW'̸1;:::ʜ;'̹m \z(wherethe'̸1;:::ʜ;'̹m \zareadditionalnewmorphismsbSetrweenWobjectsof"C5[X̸1;:::ʜ;X̹nP])isobtainedbryaddingthefollowingtothelistofconditionsforgeneratingthesetofmorphisms(M̸6)!'̸1;:::ʜ;'̹m laremorphisms.Ifkthereareadditionalcommrutativitykrelationsr̸1;:::ʜ;r̹k forthemorphismsexpressedbryWthe'̹id,ttheycanbSeaddedtothede ningcongruencerelationstode nethefreemonoidalcategoryC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m;r̸1;:::ʜ;r̹k#]bry(R̸4)r̸1;:::ʜ;r̹k :areinthecongruencerelation.W]no]O22ǐ[o]THEOREM3.1V (theinrvXarianceofthecoSendomorphismbialgebra)LffetX̸1;:::ʜ;X̹n RbeobjectsinA̸0 andlet'̸1;:::ʜ;'̹m bemorphismsinA̸0 betweentensorprffoductsoftheobjeffctsX̸1;:::ʜ;X̹nP;I.5Letr̸1;:::ʜ;r̹kxberelationsbetweenthegivenmorphismsinrA̸0.& De neD:=URC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m]randDUV20x:=URC5[X̸1;:::ʜ;X̹nP;'̸1;:::ʜ;'̹m;r̸1;:::ʜ;r̹k#]and$3let!:Dh F!A$3and!n920P :DUV20u!Abffethecorrespondingunderlyingfunctors.9cThencoSendz(!n9)PUR԰n:=coSend-u(!20: xDb '!DUV20zgthatistheidenrtityontheobjectsandthatsendsmorphismstotheircongruenceclassessothat!Ë=UR!n920=and~9wherffeIoisthetwo-sidedidealgeneratedbythedi erencesoftheimagesofthe'̸1;:::ʜ;'̹m 7under35themaps%#L!n9(dom1('̹id)) ! 0=T(n CM ҁn91s!(X̹id) ! 0=!(X) !20VtotheobjectAj2A,andthemrultiplicationandtheunitinDLqtothemultiplicationresp.^8theunitofthealgebraA~3inA.Then!̹A \isamonoidalfunctor.IfAis nitedimensional,thenthediagram(DUV;!̹A)is nite.FVora nitedimensionalalgebraAu2Kܞ-AlgBwrecanconstructthecoSendomorphismbialgebraE̹A ʈofAasinsubsection1.5.>WVealsocanconsiderthecorrespSondingdiagram(DUV;!̹A)inAandconstructitscoSendomorphismbialgebracoend!"(!̹A).-VTHEOREM3.4V (cohomomorphismsofnon-commrutativespacesandofdiagrams)Lffet SX&andYbequantumspaceswithfunctionalgebrasAUR=OUV(X) SandBX=URO(Y).YsLffetAbe nite35dimensional.fiThenH(X;Y)existswithOUV(H(X;Y))F=URa(A;B)P԰n:=cohom1J(!̹A;!̹BN>).ProSof:hGivrenӸaquantumspaceZandamapofquantumspacesZ ?X% l~!URY.LetC1=OUV(Z ).ThenTthemapinducesanalgebrahomomorphismf:BX !C} &A. vWVeTconstructtheassoSciateddiagrams(DUV;!̹A)and(D;!̹BN>).WVe willshorwnowthatthereisabijectionbSetweenthealgebrahomomorphismsfA:BX E!URC0 Aandthemonoidalnaturaltransformations':!̹B !C0 !̹A. 8Givrenfweget'bryN+O'(X n 6K)UR:!̹BN>(X n)=B n5 y!Cܞ n A n|mn7 1|(!*nCF A n=URC !̹A(X n 6K);7whereZ^m̹n :wCܞ2 n46!C6isthen-foldmrultiplication.Thisisanaturaltransformationsincethediagrams.no]O24ǐ5=iBE BCF A A)ԟ2fd`D-'(X+ X)uCF C A A@f f`DX`X鄟*X鄟*zcDm 1+ D  ۄ ۄ:zҟꠠM@fezꠠ?uUmc}`ؗҟE feED?p}m m6ҟꠠM@fe6ꠠD?=ݍ;o1 mkCF AfꠠD 鄟鄟:\xD1 1DXX ۄeX ۄezrvBr&]CF Ad 2fdD-/p''(X)O΍andaZrzxBr햮CF Av 2fdPD-/p'(X)YбKYHCF K+LTfd`D-P!'(I)! Zꠠ*Ffe>ꠠD?u!šꠠ*Ffe̟ꠠD?=L1 ucommrute.8FVurthermorethediagramsScoB2 rt B2 scuCF C A2 r ق A2 s:ğςfde0D-]'(X2 r ]) '(X2 s 7)P0㍒fCܞ2 r Cܞ2 s A2 r ق A2 sZTDX԰X҄5X҄5zPЎZ p`pĄTĄT:PPҟUM@feU?Ў2ҟP feP?PPGrҟUM@feGU?޿7Cܞ2 (r;C !̹A).0xIfAis nitedimensional,}thentheleftsideisrepresenrtedbya(A;B)andtherightsidebycohom%5<(!̹A;!̹BN>)(2.19).=ffǟffffffffǎ&CORrOLLARVY3.5cz(isomorphiccoSendomorphismbialgebras)Therffe35isauniqueisomorphismE̹AP 36԰ L=coSend4Y(!̹A)ofbialgebrassuchthatthediagramTϰëcʕWAëc.E̹A AbfdЍD-x0/pЎ5,Ұ@5,ܰ@5,氐@5,@{@{Rx0/ptKcoSend'(!̹A) A ڟ*Ffe D?cffommutes.ProSof:IfthecoendomorphismbialgebraE̹A pexists,thenitsatis estheunivrersalpropertrygivreninCorollary1.10.WҟffǟffffffffǎCORrOLLARVY3.6cz(TVamrbara[15 ]Thm.1.1)Lffet35A;B;bealgebrasandletAbe nitedimensional.fiThen la(A;B)PUR԰n:=T(BE A )=(xy CXox y ߸(1)$ ߸(2) \|;(1)1̹A jx;yË2URB;=S2A ):ProSof:This9isanimmediateconsequenceofTheorem3.3andTheorem3.4. &.}tThusweobtainqthecategoryQAofquadraticalgebras.ThedualQQofthiscategoryisthecategoryofquadraticquanrtumspaces.(KWVewillsimplydenotequadraticalgebrasby(A;RJ)whereweassumeRnURA A.WVeMconsiderthefreemonoidalcategoryD=C5[XJg;Yp;]Mwhere:YNv!X fX?:inDUV.Thenjeacrhquadraticalgebra(A;RJ)inducesamonoidalfunctor!߸(A;R )yo:/ Db ٻ!Ajwith!n9(X)UR=A,!(Yp)=RJ,and!(UR:Y G!X+ X)=:Rn !A A.FVoranrytwoquadraticalgebras(A;RJ)and(B;S),6whereAandR"1are nitedimensional,wrecanconstructtheuniversalalgebracohom%N(!߸(A;R )Jc;!߸(Bd;Sr}) )satisfyingCorollary2.18.1WVeshorwthatthisisthesamealgebraasthequantumhomomorphismspacehom((A;RJ);(B;S))constructedbryManin[5]4.4.4GthathastheuniversalpropSertygivenin[5]4.4GTheorem5.Inparticularitisagainaquadraticalgebra.no]O26ǐ[o]THEOREM3.7V (cohomofquadraticalgebras)Lffet35(A;RJ)and(B;S)bffequadraticalgebraswith(A;RJ) nite.fiThen~cohom(!߸(A;R )Jc;!߸(Bd;Sr}) )PUR԰n:=(BE A ;S] RJ ?5);wherffe35RJ2? histheannihilatorofRLin(A A)2V=URA2j A2.bDProSof: ByM=Theorem3.3thealgebracohom%(!߸(A;R )Jc;!߸(Bd;Sr}) )isgeneratedbrythevectorspacesBX A2zandvSpm RJ2N.JItsatis estherelationsgeneratedbrythemorphism:Y! vo!X X,whicrhinducesrelationsthroughthediagramM_'ٍ`S] RJ2lyE S] A2j A2!/{ӟ21 -:VךDVVҚVV͚0C0C:*BE B A2j A2:!̹ 1V@DXV @XV@XV @XV@X}X}zGivren[anelementsl  2072URS A2, A2_we[getequivXalentelementssl (  20)j̹R HURs 20.Sincethemap1 2 Qissurjectivre,everyelementinS] RJ2 k4isequivXalenttoanelementinB XB A2 A24soE0thatwrecandispSoseofthegeneratingsetS RJ2~altogether.FVurthermoreelemenrtsoftheformsq  2072URB}w B A2u A2HareequivXalenrttozeroifsUR2S;yand  q 20inducesrthezeromaponRBthatisifitisinRJ2?5, sothatthesetofrelationsisinducedbryS] RJ2?5.9ffǟffffffffǎ&GivrenanalgebraC˟inA,wecallanalgebrahomomorphismf:\(A;RJ) !C< (B;S)quadrffatic,ifitsatis esfG(A)URCF Band(m̹C @ BE B)(f f)(RJ)URCF SthatisifRspGbROCF S| ƨfdπpD-Pfu`,t@*Ffet̟@D?u`,_v@*Ffe_̟@?9dA A9n~CF C BE Bi<_҄fd5D-ǁf f99BԧCF BE Bܟ_҄fd(D-퍒m 1commrutes.Theesetofallquadratichomomorphismsfrom(A;RJ)toCژ (B;S)isdenotedbryuKܞ-AlgWqN((A;RJ);C (B;S)).$ThenoneprorvesasinTheorem3.4thatKܞ-AlgWqN((A;RJ);C (B;S))P[԰t= Mor q % qf8(!߸(A;R )Jc;C C!߸(Bd;Sr}) ):{HencewregetthefollowinguniversalpropSertywhichisdi erenrtfromtheoneinManin[5]4.8Theorem5.bDTHEOREM3.8V (univrersalpropSertyofcohomforquadraticalgebras)LffetN(A;RJ)and(B;S)bffequadraticalgebrasandlet(A;RJ)be nite.*rThenthereisaquadraticalgebrffahomomorphismN%:(A;RJ) 0L!cohom'%(!߸(A;R )Jc;!߸(Bd;Sr}) ) (B;S)suchthatforeveryalgebrffaC]3andeveryquadraticalgebrahomomorphism':(A;RJ) 9!C (B;S)therffeisaunique35algebrffahomomorphism~'::URcohom$(!߸(A;R )Jc;!߸(Bd;Sr}) )UR !URCsuchthatthediagramFrp(A;RJ)p`cohom(!߸(A;R )Jc;!߸(Bd;Sr}) ) (B;S)\ƨfdLD-*;u`,;`f'iDDPiDXjPiDحPiDPiDXiPiD⭾PiDPiDXhPiD쭽PiDP iDXgPiDP<ğP<ğqu`,$:CF (B;S)?@*Ffe?@D?čF|K~Dd'N 1cffommutes.^ffǟffffffffǎߠno]O27ǐ[o]3.4%@Completequadraticquan=tumspaces@TheVmostinrterestingdiagramforconstructingcomoSdulealgebrasoverbialgebrasisde nedorverthefreemonoidalcategoryD=URC5[XJg;]with:X9 XF .!X X.}If!Ë:D !Aisa nitediagramorverthisdiagramscrhemewith!n9(X)o\=Vdand!()=f[:VQ V a%!V Vp,thenXwrecande neaquadraticalgebraA߸(Vx;fǸ)o:=HT(Vp)=(Im(fG)).fLetusfurthermorede neBX=URB߸(Vx;fǸ)1y:=coSend!(!n9).8WVesarythatBisabialgebrffa35withRJ-matrix.LEMMA3.9F{(spacesforabialgebrawithRJ-matrix)A߸(Vx;fǸ)\is35aB߸(Vx;fǸ)'-cffomodule35algebra.ProSof:ۙCertainly<allvrectorspacesVp2 n=Kܞx0Kytrwodimensionalandf:Ve 0V> o!V 0Vgivrenbythematrix(withqn9;pUR6=0)#*Rn=fTURC0URBURBURBfiUR@&eS1&f0T0}0S0&f0Oqn921}0S0 Op21<7x1qn921 ʵp21}0S0&f0T0}1fTC1CCCfiA:)RThebialgebrageneratedbrythismatrixisgeneratedbytheelementsaUR=x6 s;bUR=x6 n9;cUR=y s;dUR=y n9,Pwheres;Xisthedualbasistox;yn9.8TherelationsareacUR=qn9 1 ʵca;bd=qn9 1db;ad1qn9 1cbUR=da1qn9bc;abUR=p 1 \|ba;cd=p 1 \|dc;ad1p 1 \|bcUR=da1pcb:FVromthisfollorwsqn9bcUR=pcb.Thisisthetwoparameterversionofaquantummatrixbialgebraconstructed in[6]Chap.ڷ4,.4.10.The matrixR9hastrwo eigenvXalues̸1q)=%1(ofmultiplicitythree)and̸2V=URqn921 ʵp21G$(ofmrultiplicityone)whicrhleadtoalgebrasJ̹nfVOEkC1OEkCOEkCfiOEkA(Gwith/hXgJ̹i,=fJURC0URBURBURBURBURBURBURBURBfiUR@_S̹i)J%1F%}.*]./1=.A1=.Bb.Fݟ.U.Z-.^}.U.Z-.^}.nu0l̹ifJ{3C1{3C{3C{3C{3C{3C{3C{3C{3Cfi{3A82Tharvingr\dim(V̹id)r\entries̹i6anddim(W̹id)entries1. Thentheinducedhomomorphismf8 :V XYVU!V V4withrespSecttoasuitablebasisthroughtheW̹iNandV̹isatis esR̹ilf=Im(f̹iid uL)./ffǟffffffffǎ*3.5%@Liealgebras@SimilartoourconsiderationsabSout nitequanrtumspaces,VletD=URC5[X;m]beafreemonoidalcategoryonanobjectXwithamrultiplicationmk:Xe X]f!X. BThenevery nitedimensionalLiealgebraginducesadiagram(DUV;!n9)with!(X)aI=gand!(m)theLiebracrket. EssentiallyrthesameargumenrtsasinLemma3.9showthatthebialgebracoSend!U(!n9)makresgacomoSduleLiealgebra(theLiemultiplicationisacomoSdulehomomorphism)anditsunivrersalenvelopingalgebraacomoSdulebialgebra.Again$coSend"(!n9)$hasaunivrersalpropSertywithrespSecttoitscoactionong0.TTVoshowthisletgѢandg020bSeLiealgebras. WVesarythatalinearmapfQ:URgR ګ!C g020ismultiplicffativeifthediagram0no]O30ǐS=ɱ0gڨ gɱCF C g020 g020ƨfd+s D-Pf fu`,R@*Ffe@D?[;]u`,@*Ffeğ@D?$kDm [;]hAghACF g0204_҄fdNqD-fݍcommrutes.tThenthesetofallmultiplicativemapsMultD(g0;C @g209)iscertainlyafunctorinCܞ.эTHEOREM3.15\(theunivrersalbialgebracoactingonaLiealgebra)Lffet?jgojandg020=beLiealgebrasandletg020=be nitedimensional. ThenMult`(g0;CM qSg209)?jisarffepresentable35functorwithrffepresenting35objectcohom%}(!g ;!g%ޟ0).In35pffarticularthemultiplicativemapg020S !URcohom$(!g ;!g%ޟ0) gc5is35universal.ProSof:wItsisanalogoustotheproofofTheorem3.4.BInparticularwregetanisomorphismMult (g0;CF g209)PUR԰n:=Mor q % qf8(!g ;CF !g%ޟ0). ffǟffffffffǎ&WVexcomputenorwaconcreteexampleofauniversalbialgebracoactingonaLiealgebra.EXAMPLE3.16ZcS(theunivrersalbialgebracoactingonathree-dimensionalLiealgebraofuppSertriangularmatrices)Letwgta=DaKܞfx;yn9;zgbSethethreedimensionalLiealgebrawithbasisx;yn9;zandLiebracrket[x;yn9]N=0and[x;z]N=[z;yn9]=z.Theng02 isaLiecoalgebrawithdualbasiss;n9;andcobracrketA(s)=(n9)=0and()=(bDn9) , (bn9).ByATheorem3.3theunivrersalC1>Cfi>AJ+=fXURC0URBfiUR@fdSx 6#x [x Wy 6ey [y lz3 6{z3 \ Wz3 fX|C1|Cfi|Aׅ: ThecomrultiplicationisdescribSedinthetextafterCorollary2.6andisgivenby(M@)=M M@.8Withsomestraighrtforwardcomputationsonegetstherelationsasݍ(ab)cUR=c(ab);33(de)fQ=URfG(de);(ab)fQ=URc(de);fG(ab)UR=(de)c;R (ab)iUR=i(ab)UR=0;33(de)iUR=i(de)UR=0;gË=0;h=0:EXAMPLE3.17ZcS(theunivrersalbialgebracoactingonslx(2))LetgH=VHKܞfx;yn9;zg=slx(2)bSethethreedimensionalLiealgebrawithbasisx;yn9;z andLiebracrket[x;yn9]UR=z,'[z;x]UR=xand[yn9;z]UR=yn9."*Theng02isaLiecoalgebrawithdualbasiss;n9;andcobracrket(s)W=? WP ,((n9)W= W? n9,(and()W=P W s.TheunivrersalbialgebracoSend"\[(!n9)forthediagraminducedbygagainisgeneratedbytheelemenrtsofthematrix $֍oM6=fXURC0URBfiUR@fdSa""b3cSd! e2f#g!@Th3ifX>C1>Cfi>AJ+=fXURC0URBfiUR@fdSx 6#x [x Wy 6ey [y lz3 6{z3 \ Wz3 fX|C1|Cfi|Aׅ:#$ՍThecomrultiplicationisgivenby(M@)UR=M M.8HereonegetstherelationsasݍabUR=ba;33ac=ca;bc=cb;de=ed;d fQ=fGd;efQ=fGe;gn9h=hg;33gi=ig;33hi=ih;B,aUR=aicgË=URiagn9c;33b=chbi=hcib;33c=bgah=gn9bha;<~4dUR=fGgdi=gn9fid;33e=eifGh=iehf;33fQ=dhegË=hdgn9e;>GgË=URcdafQ=dcfGa;33h=bfce=fGbec;33i=aedb=eabd:̠no]O31ǐ]o]4AutomorphismsandHopfAlgebrasb#In=KthislastshortsectionwrewanttoextendourtechniquesofusingdiagramsfordeterminingbialgebrastoHopfalgebras.HopfalgebrasariseasfunctionringsofanealgebraicgroupswhicrhactasgroupofautomorphismsonalgebraicvXarieties..Theprobleminnon-commrutativegeometryisthatthede nitionW\ofanautomorphismgroupisnotthatclear.~Soonede nesthecffoautomorphismHopfcalgebrffaNofaspaceXtobSetheHopfenrvelopeNofthecoendomorphismbialgebraofX.The\constructionoftheHopfenrvelopSe\HofabialgebraB^bwrasgiveninapapSerofTVakeuchi[14 ].Hopf6algebrasalsoariseascoSendomorphismalgebrasofrigiddiagrams[17 ].)eSowre rststudytsomepropSertiesofrigidmonoidalcategoriesandthenshorwthatcoautomorphismHopfalgebrascanalsobSeobtainedfromdiagrams.8OneofthemaintheoremsinthisconrtextisTHEOREM4.1V (coSendomorphismHopfalgebrasofrigiddiagrams)(a)zLffetHgjbeaHopfalgebra. ;ThenthecategoryofrightH-comodulesthatare nitedi-mensionalasveffctorspaces,Misrigid,andtheunderlyingfunctor!e:C5omoffd-xH3 9!VeffcDismonoidal35andprffeservesdualobjects(uptoisomorphism).(b)Lffet(DUV;!n9)bea nitemonoidaldiagramandletD1berigid.[Thenthecoendomorphismbialgebrffa35coSend!(!n9)35isaHopfalgebra.ProSof:8[12 ]and[16].hffǟffffffffǎ&LEMMA4.2F{(therigidizationofamonoidalcategory)Lffet'4D|beasmallmonoidalcategory.BfThenthereexistsaunique(left-)rigidization(uptoisomorphism),i.e. Da}I(left-)rigidsmallmonoidalcffategory}IDUV2 andamonoidalfunctor:D % _~!DUV2 | suchfthatforevery(left-)rigidsmallmonoidalcffategoryfEyandmonoidalfunctoro:URD !EEtherffe35isauniquemonoidalfunctorUR:DUV2j!Esuch35thatTDƐٍźDƐٍ#DUV2Ҿ̟ªrfd'0D-<w@.h5(Ҿ̟ԯD@ܾ̟ޯ@̟诠@̟@L@LRw@HE z *Ffe= D?= ,cffommutes.ProSof: oTheconstructionfollorwsessentiallythesamewayastheconstructionofafreemonoidalcategoryorveragivren( nite)setofobjectsandmorphismsinsection3.1.#2ffǟffffffffǎ&CORrOLLARVY4.3cz(therigidizationofadiagram)EachR nitemonoidaldiagrffam!:|D 9+!ARhasauniquerigidization!n92 :|DUV2 /!ARsuchthat35!n92.=UR=!n9.ProSof:9AK niteUmonoidaldiagramisbryde nitionamonoidaldiagramintherigidcategoryA̸0.x ffǟffffffffǎ  no]O32ǐ[o]PRrOPOSITION4.4lԲ(extendingmonoidaltransformationstotherigidization)Givena nitemonoidaldiagrffam(DUV;!n9)andits(left-)rigidization(D2Z;!n92.=).GGivena(left-)HopfzalgebrffaKmandamonoidalnaturaltransformationf$:!JR !K ^!n9. ~7Thenzthereisauniqueextensiong+:[!n92 @!K} !n92.=, amonoidalnaturffaltransformation, suchthat(gn9UR:!2.= !KF !2.=)=(fQ:!Ë !KF !).ѓProSof:&:FVoraUXĔ2DletX2 2DUV2 vbSeitsdual. LetM:=!n9(X),M@2 :=!2.=(X2)aUandev:M@2 !  M S!}Ku bSektheevXaluation.B*MOisaKܞ-comodule.B*WVede neacomodulestructureonGM@2.+SinceM'+is nite-dimensional(ormoregenerallyhasaleftdualinA)thereisacanonicalisomorphismM@2 KP1԰J=ܙHom)V"(M;Kܞ).8Thens2(m2)UR2M@2 KFisde nedbrygavCXqrm ڍ(0) \|(m) m ڍ(1):=URCXm (m߸(0)) S(m߸(1)):ItistediousbutstraighrtforwardtocrheckthatthisisarighrtcomoSdulestructureonM@2.WVecrheckthecoassoSciativitry:.궟CX$\}m ڍ(0)(0)8(m) S(m ڍ(0)(1)) S(m ڍ(1) \|)=URCXm ڍ(0) \|(m߸(0)) m߸(1)$ S(m ڍ(1) \|)=URCXm (m߸(0)(0)8) m߸(1)$ m߸(0)(1)=URCXm (m߸(0) \|) W(m߸(1))=URCXm ڍ(0) \|(m) W(S(m ڍ(1)))=URCXm ڍ(0) \|(m) S(m ڍ(1)) S(m ڍ(2)):ThentheevXaluationev:URM@2 M6 !kQsatis esga4 CXDm ڍ(0) \|(m߸(0)) m ڍ(1) \|m߸(1)=URCXm ڍ(0)(m߸(0)) S(m߸(1) \|)m߸(2)=URm (m) 1̹KhenceitisacomoSdulehomomorphism.WVeneedauniqueKܞ-comoSdulestructureonM@2 sucrhthatevbecomesacomodulehomomorphism.Letc M@2 charvesuchacomoSdulestructurewiths2(m2)"8=CPqm2(0)Y m2(1)andletmUR2M+withs2(m)=CPm߸(0)$ m߸(1)G$thenwregetDcCXh*m ڍ(0) \|(m߸(0)) m ڍ(1) \|m߸(1)=URm (m) 1̹KgafromthefactthatevisacomoSdulehomomorphism.8Hencewreget궟CX$\}m ڍ(0) \|(m) m ڍ(1)=URCXm ڍ(0) \|(m߸(0)) m ڍ(1) \|"(m߸(1))=URCXm ڍ(0) \|(m߸(0)) m ڍ(1) \|m߸(1)S(m߸(2))=URCXm (m߸(0) \|) S(m߸(1)):ButfQthisispreciselytheinducedcomoSdule-structureonM@2 g9brytheantipSodefQS(ofKBgivenabSorve.FVoriterateddualsandtensorproSductsofobjectstheKܞ-comodulestructurearisesfromiterating1theproScessgivrenaborve1resp.fromusingthemrultiplicationofKotogivethetensorproSductacomodulestructure.3ffǟffffffffǎ&THEOREM4.5V (theHopfenrvelopSeofacoendomorphismbialgebra)Givenamonoidaldiagrffam(DUV;!n9)anditsrigidization(D2Z;!n92.=).\LffetB:=coSend!o(!n9),abial-gebrffa,r+and.Ho:=coSend$\(!n92.=),aHopfalgebrffa. SThenthereisabialgebrahomomorphismq:!8B>!H-bsuch@ thatforeveryHopfalgebrffaKandeverybialgebrahomomorphismfQ:URBX E!Ktherffe35isauniquebialgebrahomomorphismgË:URHB !Ksuch35that!bno]O33ǐMƐٍXBƐٍHҾ̟ªrfd)0D-<w@.هfҾ̟ԯD@ܾ̟ޯ@̟诠@̟@L@LRw@HK z *Ffe= D?= ,gcffommutes.BProSof:Themonoidalnaturaltransformation!n92!URH !n92inducesauniquehomomorphismË:URBX E!HsucrhthatMFzO!Ë=UR!n92.=z'BE !T"fdPD-0<Ϣ$P@٢$P@$P@$P@褟^n@褟^nR1ɍ#H !n92.=ҟ_Є*Ffe!_D?mӄ 1The.homomorphismfĹ:|B m!K :induces.amonoidalnaturaltransformation(f7 81):!Ë !URB }!Ë!K }!n9,DwhicrhmaybSeextendedto!n92!URK }!n92.=.HencethereisauniquegË:URHB !KFsucrhthatO󚍍zø!n92zH !n92Œ"fd D-0<;P@;P@;P@;P@<^n@<^nR1ɍKF !n92j_Є*Ffe_D?Ímg 1commrutes.8Thenthefollowingdiagramscommuteuú!SBE !fd+@D-1"|`H !p4DZZ4ZZ`)Z`)~|`9KF !T>DJ辟JѓhJ>ǴJ輟ѴJf۴J>J躟JJ^|`T $g`feL ?FM 1T,@$g`feL@D?sÍg 1|`,@Y?fe@D?Tf 1andOvƐٍXBƐٍHҾ̟ªrfd)0D-<w@.هfҾ̟ԯD@ܾ̟ޯ@̟诠@̟@L@LRw@HK5: z *Ffe= D?= ,gffǟffffffffǎ&WVeclosewithafewobservXationsandapplicationsofthisresult.EvrerybialgebraB!isthecoSendomorphismbialgebraofthediagramofall nite-dimen-sional-B-comoSdulesbry[10 ].SotheconstructionwithdiagramsgivenabSovegivestheHopfenrvelopSeofanarbitrarybialgebra."0no]O34ǐ[o]Ifa nite-dimensionalnon-commrutativealgebraAisgivrenthentherigidizationofthe nitemonoidaldiagram!̹A _:DF ,!AwithDF:=C5[X;m;u]generatedbryAasin3.2hasacoSendomorphismbialgebraH@whicrhistheHopfenvelopSeofthecoendomorphismbialgebraofA.8Asimilarremarkholdsforquadraticalgebras.In9fthecategoryofquanrtumspacestheendomorphismquantumspaceEL)ofaquantumspaceXfcanbSerestrictedtothe"automorphism"quanrtumspaceAbyusingthehomomor-phismfromtherepresenrtingbialgebraofEtoitsHopfenvelopSe.OThusAactsalsoonX.ThetexistenceandtheconstructionintherelevXanrtcasescanbSeobtainedfromassociatedrigidmonoidaldiagramsasintheabSorvetheorem.(VReferencesb#[1]' MiriamCohen,fSaraWVestreicrh:CFromSupSersymmetrytoQuantumCommutativityV.' 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Math.72,1990.H;nG#DO line10Cu cmex10<"VG cmbx107"Vff cmbx100N cmbx12/}h! cmsl12.@ cmti12-!", cmsy10,g cmmi12+XQ cmr12K`y cmr10t : cmbx9K cmsy82cmmi8 |{Ycmr8q% cmsy6;cmmi6Aacmr6KZ