Sequential structures in cluster algebras and representation theory

163 Pages·2017·1.11 MB·English

by  Florian Gellert

SEQUENTIAL STRUCTURES IN
CLUSTER ALGEBRAS AND
REPRESENTATION THEORY
Florian Gellert
BielefeldUniversity
Athesissubmittedforthedegreeof
DoctorofMathematics(Dr.math.)
BielefeldinApril2017

Abstract
Thisthesisdealswitharangeofquestionsinclusteralgebrasandtherepresentationtheory
ofquivers. Inparticular,weprovidesolutionstothefollowingproblems:
1. Doesaclusteralgebraadmitaquantisationandifitdoes,howuniqueisit?
2. Whatisthesmallestsimply-lacedquiverwithoutloopsand2-cycleswhoseprincipal
extensiondoesnotadmitamaximalgreensequence?
3. Consideringtheposetofquiverrepresentationsofcertainorientationsoftype A
n
diagramsinducedbyinclusion,whatisthewidthofsuchaposet?
Inparticular,foragivenclusteralgebraweconstructabasisofthosematriceswhichpro-
videaquantisation. Leadingtothesmallestsimply-lacedquiverasproposedabove,we
proveseveralcombinatoriallemmasforparticularquiverswithuptofourmutablevertices.
Furthermore,weintroduceanewkindofperiodicityintheorientedexchangegraphofprin-
cipallyextendedclusteralgebras. Thisperiodicitywestudyinmoredetailforaparticular
extendedDynkinquiveroftypeA˜n−1andshowthatityieldsaninfinitesequenceofcluster
tiltingobjectsinsidethepreinjectivecomponentoftheassociatedclustercategory.
Key words: (Quantum) Cluster Algebra, Cluster Category, (Maximal) Green Sequences,
RepresentationTheoryofQuivers
i

Acknowledgements
IwouldliketothankmysupervisorProf.HenningKrauseforhissupportandadviceduring
the work on my PhD project. Special thanks go to Dr. Philipp Lampe with whom I was
fortunatetodoresearchtogether,leadingtomanyinterestingconversationsandresults.
Besidescreatingapleasantenvironmentinitself,thanksgotoallmembersoftheBIREP
researchgroupforplentytalks,meetingsandsocialgatheringsthathaveaccompaniedme
throughoutmytimehere.
IamalsogratefultoBielefeldUniversityandtheBielefeldGraduateSchoolinTheoretical
Sciencesforfinancialsupport,aswellastothemanyorganisingpartiesofconferencesand
workshopsIwasluckytoattend.
Bielefeld,21April2017 FlorianGellert
Man braucht fürs Schreiben jede
Menge Zeit zum Verschwenden.
Ian McEwana
aZEITMAGAZIN NR.
27/2015, available at
http://www.zeit.de/zeit-magazin/2015/27/ian-mcewan-rettung
iii

Contents
1 Introduction 1
2 Preliminaries 7
2.1 SetTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 GraphTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 RepresentationTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Clusteralgebras 15
3.1 Definitionsandclassicalresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Greensequences 33
4.1 Definitionsandfundamentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Permissiblevertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Theorientedpentatope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Periodicitiesintheorientedexchangegraph . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Periodicitiesinclustertheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.2 Greenpermissibleperiods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.3 TheextendedDynkincaseA˜n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Antichainsinposetsofquiverrepresentations 73
5.1 Posetproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Linearorientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Simplezigzag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Alternatingorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 ConclusionandOutlook 81
Bibliography 85
v
Contents
Appendix 89
A Cyclicquiverswithmaximalgreensequences 91
B ExtendedexchangematricesforproofsinSection4.3 109
B.1 ExtendedexchangematricesfortheproofofLemma4.3.6 . . . . . . . . . . . . . . 109
B.1.1 ExtendedexchangematricesformutationsofQsource. . . . . . . . . . . . . 109
tri
B.1.2 ExtendedexchangematricesformutationsofQsink . . . . . . . . . . . . . . 114
tri
B.2 B-andC-matricesfortheproofofTheorem4.3.9 . . . . . . . . . . . . . . . . . . . . 116
B.3 B-andC-matricesfortheproofofTheorem4.3.11. . . . . . . . . . . . . . . . . . . . 124
C Code 135
C.1 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
C.2 Greenpermissibleperiods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
vi
1
Introduction
Intheearlyyearsofthismillennium,SergeyFominandAndreiZelevinskyintroducedand
studiedclusteralgebrasinaseriesoffourarticles[FZ02;FZ03;BFZ05;FZ07],oneofwhichis
co-authoredbyArkadyBerenstein. TheirmotivationwastogainnewinsightsintoLusztig’s
canonicalbasisofquantumgroupsandtotalpositivity. Sincetheintroductionofcluster
algebras,thesealgebraicstructureshavebecomeawide-rangedandintenseresearchtopic
withmanyconnectionstoseveralbranchesofmathematicssuchasrepresentationtheory,
geometry,topologyandeventotheoreticalphysics.
Clusteralgebrasarecommutativealgebraswhichareconstructedbygenerators—called
clustervariables—whicharegroupedintooverlappingsetsofthesamecardinalityn —
calledclusters—andrelationsinsideanambientfield. Whenevertwoclusterssharen−1
clustervariables,therelationbetweenthetwonon-identicalelementsofthesetwosetsis
encodedinaskew-symmetricn×n integermatrix—theso-calledexchangematrix—and
theoperationexchangingthesetwoclustervariablesiscalledmutation.
Generalisingthisconcept,ArkadyBerensteinandAndreiZelevinskydefinedquantumcluster
algebrasin[BZ05]. Theseareq-deformationswhichspecialisetoordinaryclusteralgebras
intheclassicallimitq =1. Thesegeneralised,non-commutativealgebrasplayanimportant
rôle in cluster theory: on the one hand, quantisations are essential when trying to link
clusteralgebrastoLusztig’scanonicalbases,seeforexample[Lus93;Lec04;Lam11;Lam14;
GLS13;HL10]. Ontheotherhand,GoodearlandYakimov[GY14]comparequantumcluster
algebrasandtheirso-calledupperbounds,whichareintersectionsofLaurentpolynomial
ringsgeneratedbyaninitialandonce-mutatedclusters. Itisshownthatforalargeclassof
1
Chapter1. Introduction
casesthesetwoalgebrascoincide,yieldinganapproximationofclusteralgebrasbytheir
respectiveupperboundsintheclassicallimit.
Unfortunately,noteveryclusteralgebraadmitsaquantisation.Ifitexists,thentheassociated
exchangematrixB˜ isnecessarilyoffullrankasshownin[BZ05]. Inthisthesisweconsider
thereversedirectionandproveinconjunctionwiththeresultof[BZ05]thattheexistenceof
quantisationsofaclusteralgebradependsonlyontherankofB˜.
Theorem[Cf.Theorem3.2.4]. AclusteralgebraAassociatedtoanexchangematrixB˜ admits
aquantisationifandonlyifB˜ hasfullrank.
DependingonthedimensionofB˜,aquantisationoftheassociatedclusteralgebraisnot
necessarilyunique. Thisambiguitywemakemoreprecisebyrelatingallpossiblequanti-
sationsviamatricesweexplicitlyconstructfromagivenB˜ usingparticularminors. Inthis
fashion,wereobtainthefollowingresultfrom[GSV03],wherethesubjectisconsideredin
thelanguageofcertainPoissonstructures.
Theorem[Cf.Corollary3.2.12]. LetB˜ =(cid:2)B(cid:3)beanm×n exchangematrixoffullrankandr
C
thenumberofconnectedcomponentsofthemutablepartoftheassociatedquiver. Thenthe
solutionspaceofmatricesΛsatisfyingthedefinitionofquantumclusteralgebrastoagiven
skew-symmetriserD ofB isavectorspaceovertherationalnumbersofdimension(cid:0)m−n(cid:1).
2
Inparticular,thesetofallquantisationsliesinarationalvectorspaceofdimensionr +(cid:0)m−n(cid:1).
2
Aspreviouslyremarked,clusteralgebrasalsohavestrongconnectionstotheoreticalphysics.
Onesuchintersectionisgivenbyquantumdilogarithms,forwhichBernhardKellershowed
in[Kel11]thatcertainmutationsequences—calledgreensequences—inaprincipally
extendedclusteralgebraleadtoidentitiesofsuchfunctions. Theredorgreencolouringof
mutableverticesinseedsofsuchclusteralgebrasisdeterminedbythesignofthecolumnsof
theassociatedC-matrix. Here,thesign-coherenceofc-vectors,asproveninfullgenerality
in[Gro+14],isanessentialingredientofthewell-definednessofthisnotion.
Thedefinitionofgreensequencesinturnledtoawealthofnewquestions. Inparticular,the
existenceofmaximalgreensequences—thoseendinginquiversinwhicheachmutable
vertex is coloured red — given a particular quiver is of high interest. In [BDP14] it was
proventhatallacyclicquiversadmitamaximalgreensequenceandthisresulthassince
beenextendedtoallfinite-typeclusteralgebrasexceptthosewhichareofmutationtype
(cid:88) ,cf. theworkofMatthewMillsin[Mil16]. Theknowncasesofcyclicquiverswhichdo
7
notadmitamaximalgreensequenceareratherlimited. Besidesthosequiversoftype(cid:88) ,
7
coveredbyAhmetSevenin[Sev14],itwasshownbyGregMullerin[Mul16]thatcyclicquivers
onthreeverticesforwhicheverysideofthe3-cycleisamulti-edgedonotadmitamaximal
greensequence.
2