Massey products on cycles of projective lines and

arXiv:math/0612761v3[math.QA]10Jan2007MASSEYPRODUCTSONCYCLESOFPROJECTIVELINESANDTRIGONOMETRICSOLUTIONSOFTHEYANG-BAXTEREQUATIONSA.POLISHCHUKAbstract.Weshowthatanondegenerateunitarysolutionr(u,v)oftheassociativeYang-Baxterequation(AYBE)forMat(N,C)(see[7])withtheLaurentseriesatu=0oftheformr(u,v)=1⊗1u+r0(v)+...satisfiesthequantumYang-Baxterequation,providedtheprojectionofr0(v)toslN⊗slNhasaperiod.WeclassifyallsuchsolutionsoftheAYBEextendingtheworkofSchedler[8].WealsocharacterizesolutionscomingfromtripleMasseyproductsinthederivedcategoryofcoherentsheavesoncyclesofprojectivelines.IntroductionThispaperisconcernedwithsolutionsoftheassociativeYang-Baxterequation(AYBE)r12(−u′,v)r13(u+u′,v+v′)−r23(u+u′,v′)r12(u,v)+r13(u,v+v′)r23(u′,v′)=0,(0.1)wherer(u,v)isameromorphicfunctionoftwocomplexvariables(u,v)inaneighborhoodof(0,0)takingvaluesinA⊗A,whereA=Mat(N,C)isthematrixalgebra.Hereweusethenotationr12=r⊗1∈A⊗A⊗A,etc.Wewillrefertoasolutionof(0.1)asanassociativer-matrix.Thisequationwasintroducedintheaboveformin[7]inconnectionwithtripleMasseyproductsforsimplevectorbundlesonellipticcurvesandtheirdegenerations.Itisusuallycoupledwiththeunitarityconditionr21(−u,−v)=−r(u,v).(0.2)Notethattheconstantversionof(0.1)wasindependentlyintroducedin[1]inconnectionwiththenotionofinfinitesimalbialgebra(whereAcanbeanyassociativealgebra).TheAYBEiscloselyrelatedtotheclassicalYang-Baxterequation(CYBE)withspectralparameter[r12(v),r13(v+v′)]−[r23(v′),r12(v)]+[r13(v+v′),r23(v′)]=0(0.3)fortheLiealgebraslN(sor(v)takesvaluesinslN⊗slN)andalsowiththequantumYang-Baxterequation(QYBE)withspectralparameterR12(v)R13(v+v′)R23(v′)=R23(v′)R13(v+v′)R12(v),(0.4)whereR(v)takesvaluesinA⊗A.Intheseminalwork[3]BelavinandDrinfeldmadeathoroughstudyoftheCYBEforsimpleLiealgebras.Inparticular,theyshowedthatallnondegeneratesolutionsareequivalenttoeitherelliptic,trigonometric,orrationalsolutions,andgaveacompleteclassificationintheellipticandtrigonometriccases.InthepresentpaperweextendsomeoftheirresultsandtechniquestotheAYBE.Inaddition,weshowthatoftensolutionsoftheAYBEareautomaticallysolutionsoftheQYBE(forfixedu).WewillbemostlystudyingunitarysolutionsoftheAYBE(i.e.,solutionsof(0.1)and(0.2))thathavetheLaurentexpansionatu=0oftheformr(u,v)=1⊗1u+r0(v)+ur1(v)+...(0.5)ThisworkwaspartiallysupportedbytheNSFgrantDMS-0601034.1Itiseasytoseethatinthiscaser0(v)isasolutionoftheCYBE.Hence,denotingbypr:Mat(N,C)→slNtheprojectionalongC·1weobtainthatr0(v)=(pr⊗pr)r0(v)isasolutionoftheCYBEforslN.Weprovethatifr(u,v)isnondegenerate(i.e.,thetensorr(u,v)∈A⊗Aisnondegenerateforgeneric(u,v))thensoisr0.Thus,r0fallswithinBelavin-Drinfeldclassification.Furthermore,weshowthatifr0iseitherellipticortrigonometricthenr(u,v)isuniquelydeterminedbyr0uptocertainnaturaltransformations.Thenaturalquestionraisedin[7]iswhichsolutionsoftheCYBEforslNextendtounitarysolutionsoftheAYBEoftheform(0.5).In[7]weshowedthatthisisthecaseforallellipticsolutionsandgavesomeexampleswithtrigonometricsolutions.In[8]SchedlerstudiedfurtherthisquestionfortrigonometricsolutionsoftheCYBEoftheformr0(v)=r+evr211−ev,whererisaconstantsolutionoftheCYBE.HediscoveredthatnotalltrigonometricsolutionsoftheCYBEcanbeextendedtosolutionsoftheAYBE,andfoundanicecombinatorialstructurethatgovernsthesituation(calledassociativeBDtriples).InthispaperwecompletethepicturebygivingtheanswertotheabovequestionforarbitrarytrigonometricsolutionsoftheCYBE(seeTheorem0.1below).Wewillalsoprovethateverynondegenerateunitarysolutionr(u,v)oftheAYBEwiththeLaurentexpansionatu=0oftheform(0.5)satisfiestheQYBEwithspectralparameterforfixedu,providedr0(v)eitherhasaperiod(i.e.,itiseitherellipticortrigonometric)orhasnoinfinitesimalsymmetries(seeTheorem1.4).Thus,ourworkonextendingtrigonometricclassicalr-matrices(withspectralparameter)tosolutionsoftheAYBEleadstoexplicitformulasforthecorrespondingquantumr-matrices.TheconnectionwiththeQYBEwasnoticedbeforeforellipticsolutionsconstructedin[7](becausetheyaregivenessentiallybyBelavin’sellipticR-matrix)andalsoforthosetrigonometricsolutionsthatareconstructedin[8].AnimportantinputforourstudyoftrigonometricsolutionsoftheAYBEisthegeometricpicturewithMasseyproductsdevelopedin[7]thatinvolvesconsideringsimplevectorbundlesonellipticcurvesandtheirrationaldegenerations.Inloc.cit.weconstructedallellipticsolutionsinthiswayandsometrigonometricsolutionscomingfromsimplevectorbundlesontheunionoftwoprojectivelinesgluedattwopoints.Inthispaperweconsiderthecaseofbundlesonacycleofprojectivelinesofarbitrarylength.WecomputeexplicitlycorrespondingsolutionsoftheAYBE.Thenwenoticethatsimilarformulamakesenseinamoregeneralcontextandprovethisbyadirectcalculation.Thecompletenessoftheobtainedlistoftrigonometricsolutionsisthencheckedbycombiningtheargumentsof[8]withthoseof[3](modifiedappropriatelyforthecaseoftheAYBE).Itisinterestingthatcontrarytotheinitialexpectationexpressedin[7]notalltrigonometricsolutionsoftheAYBEcanbeobtainedfromthetripleMasseyproductsoncyclesofprojectivelines(seeTheorem5.3).Thismakesuswonderwhetherthereissomegeneralizationofourgeometricsetup.Anotherquestionthats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