Bases of representations of type A affine Lie algebras via q

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

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aBASESOFREPRESENTATIONSOFTYPEAAFFINELIEALGEBRASVIAQUIVERVARIETIESANDSTATISTICALMECHANICSIGORB.FRENKELANDALISTAIRSAVAGEAbstract.Werelatetwoapparentlydi erentbasesintherepresentationsofa neLiealgebrasoftypeA:onearisingfromstatisticalmechanics,theotherfromgaugetheory.Weshowthatthetwoaregovernedbythesamecombina-toricsthatalsorespectstheweightspacedecompositionoftherepresentations.Inparticular,weareabletogiveanalternativeandmuchsimplergeometricproofofthemainresultof[DJKMO2]ontheconstructionofbasesofa neLiealgebrarepresentations.Atthesametime,wegiveasimpleparametrizationoftheirreduciblecomponentsofNakajimaquivervarietiesassociatedtoin niteandcyclicquivers.Wealsode nenewvarietieswhoseirreduciblecomponentsareinone-to-onecorrespondencewithbasesofthehighestweightrepresenta-tionsofgl n+1.IntroductionAremarkablerelationbetweenrepresentationtheoryofa neLiealgebrasandmodelsofstatisticalmechanicsbasedontheYang-Baxterequationhasbeendis-coveredandintensivelystudiedbyE.Date,M.Jimbo,A.Kuniba,T.MiwaandM.Okado(see[DJKMO1,DJKMO2]andreferencestherein).Oneoftheimportant ndingsoftheaboveauthorsisthattheone-dimensionalcon gurationsumsforthesemodelsgiverisetocharactersofintegrablehighestweightrepresentationsofa neLiealgebras.Thisrelationyieldscertainexplicitbasesintherepresenta-tionsthatadmitpurecombinatorialdescriptionsandimplyvariousidentitiesforthecharacters.Anotherastonishingrelationbetweenrepresentationtheoryofa neLiealgebrasandmodulispacesofsolutionsofself-dualYang-Millsequationshasbeenaccom-plishedbyH.Nakajima[N1,N3],whoobservedaprofoundlinkbetweenhisearlierworkwithP.KronheimerandtheresultsofG.Lusztig[L1,L2].AttheheartofbothworksthatprecededtheNakajimadiscoveryarequivervarietiesassociatedwithextendedDynkindiagrams.Nakajimaintroducedaspecialclassofquivervarietiesassociatedwithintegrablehighestweightrepresentationsofa neLieal-gebrasandobtainedageometricdescriptionoftheaction.Healsode nedcertain

Lagrangiansubvarietieswhoseirreduciblecomponentsyieldageometricbasisofthea neLiealgebrarepresentations.

Thecentralgoalofthepresentpaperistorelatethetwoapparentlydi erentbasesintherepresentationsofa neLiealgebrasoftypeA:onearisingfromstatis-ticalmechanics,theotherfromgaugetheory.Weshowthatthetwoaregovernedbythesamecombinatoricsthatalsorespectstheweightspacedecompositionoftherepresentations.Thisidenti cationallowsonetogiveanaturalconceptualframe-worktotheintricatestructureofstatisticalmechanicalmodelsandalsotomakeexplicitcalculationsinaseeminglyintractablegeometricsetting.Inparticular,weareabletogiveanalternativeandmuchsimplergeometricproofofthemainresult

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

2IGORB.FRENKELANDALISTAIRSAVAGE

of[DJKMO2]ontheconstructionofabasisofa neLiealgebrarepresentations.Atthesametime,wegiveasimpleparametrizationoftheirreduciblecomponentsofNakajimaquivervarietiesassociatedtoin niteandcyclicquivers.

Thecomparisonofthetwoverydi erenttheoriesbringssomesurprisesandsuggestsinterestingnewdirections.Inparticular,theYoungdiagramsthatareroutinelyusedinrepresentationtheoryoftypeALiealgebrasacquireanexplicitgeometricmeaning:Theypicturepreciselyrepresentationsofthecorrespondingquiverssatisfyingastabilityconditionforlevel1(seeFigure2inthetext).Ontheotherhand,thealgebraicconstructionsof[DJKMO2]involvesubstantiallythe n+1,whicharenotdirectlycoveredbyNaka-highestweightrepresentationsofgl

jima’stheory.Wede nenewvarietiesbyrelaxingthenilpotencyconditioninthede nitionofNakajima’squivervarietiesandshowthattheirreduciblecomponentsofthesenewvarietiesareinone-to-onecorrespondencewithbasesofthehighest n+1.Wealsomentionsomeinterestingproblemsthatweightrepresentationsofgl

ariseasaresultofthecomparisionofgeometricandalgebraicconstructions.

Westronglybelievethatthemainresultsofthecurrentpaperre ectaverygeneralprinciplethatassertstheprofoundgeometricorgaugetheoreticoriginofvariousalgebraicandcombinatorialstructuresofintegrablemodelsinstatisticalmechanics.Therelationofbothsubjectstotherepresentationtheoryofa neLiealgebrasisanecessaryprerequisiteofthisprinciple.Howeverweexpectmuchmore;namelythatvariousspeci cconstructionsappearinginintegrablemodelsofstatisticalmechanicsthatincludetensorproducts,fusionproducts,branchingrules,Bethe’sansatzandtheYang-Baxterequationitselfre ectcertaingeometricfactsaboutNakajimavarieties,Malkin-Nakajimatensorproductvarieties,variousLagrangiansubvarietiesandcorrespondinggaugetheoriesoncommutativeand,possibly,noncommutativespaces.Thepresentpaperisasmallbutindicativesteptowardthisvastprogram.

Thepaperisorganizedasfollows.InSection1werecallthede nitionofLusztig’squivervarietiesandcharacterizationsoftheirreduciblecomponentsintypesA∞(1)andAn.WealsointroduceaversionofLusztig’squivervarietiesfortheLiealge- bragln+1.Section2containsthede nitionofNakajima’squivervarietiesandtheLiealgebraactiononasuitablespaceofconstructiblefunctionsonthesevarietiesisgiveninSection3.InSection4wegiveanenumerationoftheirreduciblecompo-nentsofthequivervarietiesforlevel1intermsofYoungdiagrams.WealsoidentifythegeometricactionofthetypeA∞LiealgebrainthebasisenumeratedbyYoungdiagrams.InSection5weextendtheenumerationoftheirreduciblecomponentsofthequivervarietiestoarbitrarylevelandweestablishamatchwiththeind …… 此处隐藏：5760字，全部文档内容请下载后查看。喜欢就下载吧 ……