Bases of representations of type A affine Lie algebras via q(2)

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.InSection5weextendtheenumerationoftheirreduciblecomponentsofthequivervarietiestoarbitrarylevelandweestablishamatchwiththeindexingofbasesofthecorrespondingrepresentationscomingfromstatisticalmechanics.Finally,inSection6,wecomparetheweightstructureofthebasesresultingfromquivervarietiesandthepathrealizationsofstatisticalmechanicsandmakecertainoftheircompletecoincidence.

Theresearchofthe rstauthorwassupportedinpartbytheNationalScienceFoundation(NSF).TheresearchofthesecondauthorwassupportedinpartbytheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC).

1.Lusztig’sQuiverVarieties

Inthissection,wewillrecounttheexplicitdescriptiongivenin[L1]oftheirre-

(1)duciblecomponentsofLusztig’squivervarietyinthecaseoftypesA∞andAn.

Seethisreferencefordetails,includingproofs.

LetIbeasetofverticesoftheDynkingraphofaKac-MoodyLiealgebragandletHbethesetofpairsconsistingofanedgetogetherwithanorientationofit.Forh∈H,letin(h)(resp.out(h))betheincoming(resp.outgoing)vertexof

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