CLUSTER ALGEBRAS AS HALL ALGEBRAS OF QUIVER REPRESENTATIONS

Description

Niveau: Supérieur, Doctorat, Bac+8
ar X iv :m at h. RT /0 41 01 87 v 2 2 6 A pr 2 00 5 CLUSTER ALGEBRAS AS HALL ALGEBRAS OF QUIVER REPRESENTATIONS PHILIPPE CALDERO AND FREDERIC CHAPOTON Abstract. Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables. 1. Introduction Cluster algebras were introduced in [FZ02] by S. Fomin and A. Zelevinsky in connection with the theory of dual canonical bases and total positivity. Coordinate rings of many varieties from Lie group theory – semisimple Lie groups, homogeneous spaces, generalized Grassmannian, double Bruhat cells, Schubert varieties – have a structure of cluster algebra, at least conjecturally, see [BFZ05, Sco03]. One of the goals of the theory is to provide a general framework for the study of canonical bases of these coordinate rings and their q-deformations. A (coefficient-free) cluster algebra A of rank n is a subalgebra of the field Q(u1, . . . , un). It is defined from a distinguished set of generators, called cluster variables, constructed by an induction process from a antisymmetrizable matrix B, see Section 2.1.

• called ext- configurations


• positive roots


• let ?i


• cluster algebras


• free indecomposable


• indecomposable modules


• finite cluster


• almost positive

Sujets

• Reineke
• Keller
• Root system

Informations

• Publié par: mijec
• Nombre de lectures: 63
• Langue: English

Extrait

CLUSTERALGEBRASASHALLALGEBRASOFQUIVER REPRESENTATIONS PHILIPPECALDEROANDFRE ´ DE ´ RICCHAPOTON Abstract. Recentarticleshaveshowntheconnectionbetweenrepresentation theoryofquiversandthetheoryofclusteralgebras.Inthisarticle,weprove thatsomeclusteralgebrasoftypeADEcanberecoveredfromthedataofthe correspondingquiverrepresentationcategory.Thisalsoprovidessomeexplicit formulasforclustervariables. 1. Introduction Clusteralgebraswereintroducedin[FZ02]byS.FominandA.Zelevinskyin connectionwiththetheoryofdualcanonicalbasesandtotalpositivity.Coordinate ringsofmanyvarietiesfromLiegrouptheory–semisimpleLiegroups,homogeneous spaces,generalizedGrassmannian,doubleBruhatcells,Schubertvarieties–have astructureofclusteralgebra,atleastconjecturally,see[BFZ05,Sco03].Oneof thegoalsofthetheoryistoprovideageneralframeworkforthestudyofcanonical basesofthesecoordinateringsandtheirq-deformations. A(coefficient-free)clusteralgebraAofranknisasubalgebraofthefield Q (u 1 ,...,u n ).Itisdefinedfromadistinguishedsetofgenerators,calledcluster variables,constructedbyaninductionprocessfromaantisymmetrizablematrix B,seeSection2.1.TheLaurentphenomenonassertsthatAisasubalgebraof Q [u 1 ±1 ,...,u n ±1 ].Thereexistsanotionofcompatibilitybetweentwoclustervari- ables;maximalsubsetsofpairwisecompatibleclustervariablesarecalledclusters. Allclustershavethesamecardinality,whichistherankoftheclusteralgebra. Aclusteralgebraisoffinitetypeifthenumberofclustervariablesisfinite.The classificationofclusteralgebrasoffinitetype[FZ03a]isafundamentalstepinthe theory.Themainresultisthattheseclusteralgebrascomefromanantisymmetrized Cartanmatrixoffinitetype,seeSection2.2.Moreover,inthiscasethecluster variablesareincorrespondencewiththesetofalmostpositiverootsΦ ≥−1 ,i.e. positiverootsoropposedsimpleroots,oftherootsystem. TheGabrieltheoremassertsthatthesetofindecomposablerepresentationsofa quiverQofDynkintypeisinbijectionwiththesetΦ + ofpositiveroots.Thecluster categoryCwasconstructedin[BMR + ,CCS04]asanextensionofthecategory mod k (Q)offinitedimensionalkQ-modules,suchthatthesetofindecomposable objectsofCisinbijectionwithΦ ≥−1 .ThecategoryCisnotabelianingeneral,but itisatriangulatedcategory,[Kel].In[BMR + ],thiscategoryisstudiedindepth. Theauthorsgiveacorrespondencebetweenclustervariablesandindecomposable objectsofC.Theyprovethatthecompatibilityoftwoclustervariablescorrespond tothevanishingoftheExtgroups;hence,clusterscorrespondtoso-calledext- configurations.Theyprovethatthereexistmanyanalogiesbetweenfinitecluster Date:April27,2005. 1