Representation theory of finite dimensional algebras Anton Cox

28 Pages·2008·0.31 MB·English

Representationtheoryoffinitedimensionalalgebras
AntonCox
NotesfortheLondonTaughtCourseCentre
Autumn2008
CentreforMathematicalScience
CityUniversity
NorthamptonSquare
LondonEC1V0HBEngland
1
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Recommendedreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter1. Algebrasandmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1. Associativealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3. Quivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4. Representationsofquivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter2. Semisimplicityandsomebasicstructuretheorems . . . . . . . . . . . . . . . . 17
2.1. Simplemodulesandsemisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2. Schur’slemmaandtheArtin-Wedderburntheorem . . . . . . . . . . . . . . . . . . 19
2.3. TheJacobsonradical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4. TheKrull-Schmidttheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter3. Projectiveandinjectivemodules . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1. Projectiveandinjectivemodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2. Idempotentsanddirectsumdecompositions . . . . . . . . . . . . . . . . . . . . . 31
3.3. Simpleandprojectivemodulesforboundquiveralgebras . . . . . . . . . . . . . . 34
3.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter4. RepresentationtypeandGabriel’stheorem . . . . . . . . . . . . . . . . . . . . 37
4.1. Representationtype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2. Representationtypeofquiveralgebras . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3. DimensionvectorsandCartanmatrices . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4. Reflectionfunctors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Chapter5. Furtherdirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1. Ringtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2. Almostsplitsequencesandthegeometryofrepresentations . . . . . . . . . . . . . 47
5.3. Localrepresentationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4. Representationsofotheralgebraicobjects . . . . . . . . . . . . . . . . . . . . . . 49
5.5. QuantumgroupsandtheRingel-Hallalgebra . . . . . . . . . . . . . . . . . . . . . 50
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3
6 INTRODUCTION
Innon-semisimplesettingsprojectiveandinjectivemodulesplayakeyrole. Inthiscourse
wewillonlybeabletotouchonthebasicdefinitions,andwillsaynothingofthevitalrolethey
playincohomology. Thisisinpartbecausewewillnothavethetimetodevelopthenecessary
backgroundincategorytheorywhichisanimportantpartofmoderndayalgebra.
Chapter3givesthebasicdefinitionsofprojectiveandinjectives,beforegoingontoastudy
Introduction oftheroleofidempotentsinrepresentationtheory.Usingtheseiswhatallowsustoreducetothe
studyofquivers,althoughwewillonlygiveanoutlineofthereductionmethodhere. Wethen
showhowsimple,projective,andinjectivemodulescanbeeasilyconstructedforquivers.
Thiscoursewillprovideabasicintroductiontotherepresentationtheoryofalgebras,concen- Chapter4introducesthenotionofrepresentationtype. Thisisameasureofhowharditis
tratingmainlyonthefinitedimensionalcase. Representationtheoryisconcernedwiththestudy tofullyunderstandtherepresentationtheoryofanalgebra. ThefundamentaltheoremofDrozd
ofhowvariousalgebraicobjectsactonvectorspaces, inamannerwhichrespectstheoriginal saysthateveryalgebrafallsintooneofthreetypes;thefirsttwobeing(inprinciple)completely
algebraicstructure.Finitedimensionalalgebras,whileofinterestintheirownright,providea(rel- understandable,whilethethirdisprovablyimpossibletofullyunderstand.
atively)elementarysettinginwhichtodevelopsomeofthebasiclanguage,whilestillexhibiting
Wewillsketchhowtheclassificationbytypecanbecarriedoutintwospecialcases: group
mostofthekeyfeaturesthatcanarise.
algebras, and forrepresentationsofquiverswithoutrelations. Thelattercase willallowusto
Itiscommonforafirstcourseinrepresentationtheorytoconcentrateonthecharactertheoryof introducesomemoreideasfromtherepresentationtheoryofquiverswhichareusedintheproof.
finitegroupsoverthecomplexnumbers.Thishasanumberofadvantages,notleastthatcharacters
FinallyinChapter5wewillindicatesomefurthertopicswhichthereadermaywishtoinves-
aremucheasiertoconstructthanthecorrespondingrepresentations.However,thetheoryisrather
tigate.
unrepresentativeincertaincrucialrespects.
Mostimportantoftheseisthatsuchrepresentationsarealwayssemisimple.Thismeansthatit
Recommendedreading
isenoughtoclassifythoserepresentationswhichhavenosub-representations,which(foragiven
group)isafinitenumber. Allotherrepresentationscanthenbeconstructedfromtheseviadirect Duetothelimitednumberoflecturesavailable,thelectureswillconsistofanoutlineofthe
sums. Ingeneralonecannothopetoconstructallrepresentationsofanalgebra. Infact,wewill maintheorytogetherwithsomeexamples. Thesenoteswillfillinmoreofthedetails,butwith
seethatsuchanaimisprovablyimpossibletoachieve(exceptincertainspecialcases).Insteadwe onlysketchesoftheproofsinplaces. EachChapterendswithabriefselectionofexercisesfor
willdevelopvarioustoolstoanalyserepresentationsingeneral. thereader. Forafarmorecomprehensivetreatmentofthismaterial(togetherwithmanymore
Inthiscoursewewillfocusnotongroupalgebras(althoughthesewillplayarole),butrather examplesandexercises)thereaderisrecommendedtolookat[ASS06,ChaptersI-III].
oncertainalgebrasassociatedtoquivers.Thesehavemanyadvantages(evenovergroupalgebras) Forsimplicity,[ASS06]onlyconsidersalgebrasoveralgebraicallyclosedfields.Anexcellent
intermsofeaseofcomputationandofconstructingexamples,butarerichenoughtogiveabetter (ifrapid)introductionwhichconsidersmoregeneralringscanbefoundin[Ben91,Chapters1and
flavour of general aspects of representation theory. Indeed, it will turn out that to understand 4]. Othernotesavailableonthewebwhichcoversimilarmaterialare[Bru03]and[Bar06]. Our
therepresentationtheoryofanyfinitedimensionalalgebraoveranalgebraicallyclosedfieldit expositiondrawsonallofthesesources,aswellasunpublishedlecturenotesofErdmann.
isenoughtounderstandtherepresentationtheoryassociatedtoquiversandquotientsofquivers.
Twobookswhichgointomoreadvancedtopicsthanthiscourse(andwhicharenotentirely
Thisincludesgroupalgebrasasaspecialcase(andoveranyalgebraicallyclosedfield,notjustthe
suitableforthebeginner)are[ARS94]and[GR97].
complexnumbers).
InChapter1wewillbeginwithvariousbasicdefinitionsandexamples. Firstwewilllook
atalgebrasandmodules,andthenatquiversandtheirrepresentations. Wewillthenseethatthe
quiversettinggivesrisetoexamplesinthealgebrasetting.
Chapter2coversthecoreclassicalrepresentationtheoryofalgebras.Webeginwithananalysis
oftherelationbetweensimplerepresentationsandrepresentationsingeneral,andthenconsiderfor
whichalgebraswecanreducetothestudyofsimplesalone.Suchalgebrasarecalledsemisimple,
andtheArtin-WedderburnTheoremwillgiveacompleteclassificationinthiscase.
Ifanalgebraisnotsemisimple,thentheJacobsonradicalofthealgebracanberegardedasa
measureofitsnon-semisimplicity.Wewilldevelopthebasicpropertiesofthis.TheKrull-Schmidt
Theoremthentellsusthatitisenoughtodeterminetheindecomposablemodules(aclasswhich
containsthesimplemodulesbutisingeneralmuchlarger).
5
8 1.ALGEBRASANDMODULES
wi=xai11xai22...xaittforsomet.Giventwoelements(cid:229) ni=1l iwiand(cid:229) mi=1l i′w′itheproductisdefinedto
betheelement
n m
(cid:229) (cid:229) l il i′wiw′j
CHAPTER 1 i=1j=1
wherewiwj denotestheelementobtainedfromwi andwj byconcatenation. Thisisaninfinite
Algebrasandmodules dimensionalassociativealgebrawithidentitygivenbythetrivialpolynomial1. Ifn>1thenthe
algebraisnon-commutative.
(c)GivenagroupG,wedenotebykGthegroupalgebraobtainedbyconsideringthevector
Inthiscoursewewillbeinterestedintherepresentationtheoryoffinitedimensionalalgebras spaceofformallinearcombinationsofgroupelements.Giventwoelements(cid:229) ni=1l igiand(cid:229) mi=1m ihi
definedoverafield.Webeginbyrecallingcertainbasicdefinitionsconcerningfields. withl i,m i∈kandgi,hi∈Gwedefinetheproducttobetheelement
coeffiDcEieFnINtsITinIOkNha1s.0a.1ro.oAtinfiekl.dAkfiiesldalhgaesbcrahiacraalclytercilsotiscedpiiffpeviesrtyhenosmn-aclolenssttapnotsiptiovleyninotmegiearlswuicthh (cid:229)n (cid:229)m l im jgihj.
i=1j=1
that
p Theidentityelementistheidentityelemente∈GregardedasanelementofkG.ThealgebrakGis
(cid:229)
1=0. finitedimensionalifandonlyifGisafinitegroup,andiscommutativeifandonlyifGisabelian.
Ifthereisnosuchpthenthefieldissaidtoi=h1avecharacteristic0. Afieldisinfiniteifitcontains (d)ThesetMn(k)ofn×nmatriceswithentriesinkisafinitedimensionalalgebra,thema-
trixalgebra,withtheusualmatrixmultiplication,andidentityelementthematrixI. Ifn>1it
infinitelymanyelements.
isnon-commutative. Equivalently,letV beann-dimensionalk-vectorspace, andconsiderthe
endomorphismalgebra
Henceforthkwilldenotesomefield.
Endk(V)={f:V−→V|f isk-linear}.
Thisisanalgebrawithmultiplicationgivenbycompositionoffunctions.FixingabasisforV the
1.1. Associativealgebras elementsofEndk(V)canbewrittenintermsofmatriceswithrespecttothisbasis,andinthisway
wecanidentifyEndk(V)withMn(k).
DEFINITION1.1.1. Analgebraoverk,ork-algebraisak-vectorspaceAwithabilinearmap (e)IfAisanalgebrathensoisAop,theoppositealgebra,whichequalsAasavectorspace,
A×A −→ A butwithmultiplicationmap(x,y)7−→yx.
(x,y) 7−→ xy.
AsusualinAlgebra,wearenotjustinterestedinobjects(inthiscasealgebras),butalsoin
Wesaythatthealgebraisassociativeifforallx,y,z∈Awehave functionsbetweenthemwhichrespecttheunderlyingstructures.
x(yz)=(xy)z. DEFINITION1.1.3. Ahomomorphismbetweenk-algebrasAandBisalinearmapf :A−→B
suchthatf (1)=1andf (xy)=f (x)f (y)forallx,y∈A.Thisisanisomorphismpreciselywhen
AnalgebraAisunitalifthereexistsanelement1∈Asuchthat1x=x1=xforallx∈A.Suchan
thelinearmapisabijection.
elementiscalledtheidentityinA.(Notethatsuchanelementisnecessarilyunique.)Wesaythat
analgebraisfinitedimensionaliftheunderlyingvectorspaceisfinitedimensional.AnalgebraA DEFINITION1.1.4. GivenanalgebraA,asubalgebraofAisasubspaceSofAcontaining1,
iscommutativeifxy=yxforallx,y∈A. suchthatforallx,y∈Swehavexy∈S.Aleft(respectivelyright)idealinAisasubspaceIofA
suchthatforallx∈Ianda∈Awehaveax∈I(respectivelyxa∈I).IfIisaleftandarightideal
Itiscommontoabuseterminologyandtakealgebratomeananassociativeunitalalgebra,and thenwesaythatIisanidealinA.
wewillfollowthisconvention.Thereareseveralimportantclassesofnon-associativealgebras(for
EXAMPLE1.1.5. (a)IfHisasubgroupofagroupG,thenkHisasubalgebraofkG.
exampleLiealgebras)butweshallnotconsiderthemhere. Thusallalgebrasweconsiderwill
beassociativeandunital. (b)GiventwoalgebrasAandB,andahomomorphismf :A−→B,thesetim(f )isasubal-
gebraofB,whileker(f )isanidealinA.
EXAMPLE1.1.2. (a)Letk[x1,...,xn]denotethevectorspaceofpolynomialsinthe(commut-
ing)variablesx1,...,xn. Thisisaninfinitedimensionalcommutativealgebrawithmultiplication Idempotentsplayacrucialroleintheanalysisofalgebras.
givenbytheusualmultiplicationofpolynomials,andidentitygivenbythetrivialpolynomial1. DEFINITION1.1.6. Anelemente∈Aisanidempotentife2=e.Twoidempotentse1ande2in
(b)Letkhx1,...,xnidenotethevectorspaceofpolynomialsinthenon-commutingvariables Aareorthogonalif
x1,...,xn. Ageneralelementisoftheform(cid:229) ni=1l iwi forsomenwhereforeachi, l i∈k and e1e2=e2e1=0.
7
1.2.MODULES 9 10 1.ALGEBRASANDMODULES
Anidempotenteiscalledprimitiveifitcannotbewrittenintheforme=e1+e2wheree1ande2 (undertherelationm+N=m′+Nifandonlyifm−m′∈N)hasanA-modulestructuregivenby
arenon-zeroorthogonalidempotents.Anidempotenteiscentralifea=aeforalla∈A. a(m+N)=am+N,andiscalledthequotientofMbyN.
1.2. Modules EXAMPLE 1.2.6. (a)ThealgebraAisa(leftorright)A-module,withrespecttotheusual
multiplicationmaponA.IfIisaleftidealofAthenIisasubmoduleoftheleftmoduleA.
Representationtheoryisconcernedwiththestudyofthewayinwhichcertainalgebraicobjects (b)IfA=kthenA-modulesarejustk-vectorspaces.
(inourcase,algebras)actonvectorspaces. Therearetwowaystoexpressthisconcept;interms
ofrepresentationsor(inmoremodernlanguage)intermsofmodules. tran(scfo)rImfAat=ionks[xa1,i.:.M.,x−n→]thMen(awnhAer-emaodiudleescisriabeks-vtehcetoarctsipoancoefMxi)t.ogetherwithcommutinglinear
DEFINITION1.2.1. GivenanalgebraAoverk,arepresentationofAisanalgebrahomomor- (d)EveryA-moduleMhasMandtheemptyvectorspace0assubmodules.
phism
f :A−→Endk(M) LEMMA 1.2.7(IsomorphismTheorem). If M andN areA-modulesandf :M−→N isa
forsomevectorspaceM. AleftA-moduleisak-vectorspaceMtogetherwithabilinearmap homomorphismofA-modulesthen
A×M−→M,whichwewilldenoteby(a,m)7−→am,suchthatforallm∈Mandx,y∈Awehave
im(f )∼=M/ker(f )
1m=mand(xy)m=x(ym).Similarly,arightA-moduleisak-vectorspaceMandabilinearmap
f :M×A−→Msuchthatm1=mandm(xy)=(mx)yforallm∈Mandx,y∈A.Wewilladopt asA-modules.
theconventionthatallmodulesareleftmodulesunlessstatedotherwise.
DEFINITION 1.2.2. AnA-moduleisfinitedimensionalifitisfinitedimensionalasavector PROOF. Copytheproofforlinearmapsbetweenvectorspaces,notingthattheadditionalstruc-
space. AnA-moduleM isgeneratedbyaset{m1:i∈I}(whereI issomeindexset)ifevery tureofamoduleispreserved. (cid:3)
elementmofMcanbewrittenintheform
(cid:229) DEFINITION 1.2.8. IfanA-moduleM hassubmodulesLandN suchthatM=L⊕N asa
m= aimi vectorspacethenwesaythatMisthedirectsumofLandN.AmoduleMisindecomposableifit
i∈I isnotthedirectsumoftwonon-zerosubmodules(andisdecomposableotherwise).AmoduleMis
forsomeai∈A.WesaythatMisfinitelygeneratedifitisgeneratedbyafinitesetofelements.If simple(orirreducible)ifMhasnosubmodulesexceptMand0.
AisafinitedimensionalalgebrathenMisfinitelygeneratedifandonlyifMisfinitedimensional.
LEMMA1.2.3. (a)Thereisanaturalequivalencebetweenleft(respectivelyright)A-modules Forvectorspaces,thenotionsofindecomposabilityandirreducibilitycoincide.However,this
andright(respectivelyleft)Aop-modules. isnotthecaseformodulesingeneral.
(b)ThereisanaturalequivalencebetweenrepresentationsofAandleftA-modules.
EXAMPLE1.2.9. LetC2denotethecyclicgroupwithelements{1,g},andconsiderthetwo-
PROOF. Wegivethecorrespondenceineachcase;detailsarelefttothereader. Givenaleft dimensionalkC2-moduleMwithbasis{m1,m2}wheregm1=m2andgm2=m1.IfM=N1⊕N2
moduleMforAwithbilinearmapf :A×M−→M,definearightAop-modulestructureonM withN1andN2non-zerotheneachNiisthespanofavectoroftheforml 1m1+l 2m2forsome
viathemapf ′:M×A−→Mgivenbyf ′(m,x)=f (x,m). Itiseasytoverifythatf isanAop- l 1,l 2∈k. Applyinggwededucethatl 1=±l 2,andhenceNimustbethespanofm1−m2or
homomorphism. m1+m2.ButN1=N2ifkhascharacteristic2,whichcontradictsourassumption.ThusMisnever
irreducible,butisindecomposableifandonlyifthecharacteristicofkis2. Wewillseethatthis
Givenarepresentationf :A−→Endk(M)ofAwedefineanA-modulestructureonM by examplegeneralisestoarbitrarygroupalgebraswhenweconsiderMaschke’sTheorem.
setting
am=f (a)(m)
ThereisacloserelationshipbetweentherepresentationtheoryofAandAop.
foralla∈Aandm∈M.Conversely,givenanA-moduleM,themapM−→Mgivenbym7−→rm
islinear,andgivesthedesiredrepresentationf :A−→Endk(M). (cid:3) DEFINITION 1.2.10. LetM beafinitedimensional(left)A-module. Thenthedualmodule
NsuDchEFthINaItTfIO(aNm1).=2.4a.fA(mh)omfoormalolraph∈isAmabnedtwmee∈nMA-.mTohdiusliessaMniasnodmNoripshaislmineparercmisaeplyfw:hMen−t→he Mfor∗aisllthae∈duAa,lmve∈ctMorsapnadcfeH∈oHmokm(Mk(,Mk),kw)i.thBayrLigemhtmAa-m1o.2d.u3lethaisctgioivnegsiMve∗ntbhye(sftrau)c(tmur)e=off a(almef)t
Aop-module.
linearmapisabijection.
DEFINITION1.2.5. GivenanA-moduleM,asubmoduleofMisasubspaceNofMsuchthat TakingthedualofanAop-modulegivesanA-module,anditiseasytoverify(asforvector
foralln∈Nanda∈Awehavean∈N.(NotethatNisanA-moduleinitsownright.)Thequotient spaces)that
space
M/N={m+N:m∈M} LEMMA1.2.11. ForanyfinitedimensionalA-moduleMwehaveM∗∗∼=M.
1.3.QUIVERS 11 12 1.ALGEBRASANDMODULES
1.3. Quivers PROOF. TheassociativityofmultiplicationinkQisstraightforward. Nextnotethattheele-
mentseisatisfy
DEFINITION1.3.1. AquiverQisadirectedgraph. WewilldenotethesetofverticesbyQ0, eie j=dijei
qaunidvethr.esTehteofuenddgeersly(iwnghigchrawphecQ¯alolfararoqwusiv)ebryQQ1i.sItfheQ0graanpdhQo1btaarienebdotfhrofimnitQetbhyenfoQrgiesttainfignaitlel andhenceformasetoforthogonalidempotents.Further,foranypathp∈kQwehaveeip=pif
orientationsofedges. pendsatvertexiand0otherwise.HenceifQ0isfinitethen
ApathoflengthninQisasequencep=a 1a 2...a nwhereeacha iisanarrowanda istarts (cid:229) eip=p.
atthevertexwherea i+1ends. Foreachvertexi,thereisapathoflength0,whichwedenoteby i∈Q0
ei.Aquiverisacycliciftheonlypathswhichstartandendatthesamevertexhavelength0,and Similarly
connectedifQ¯isaconnectedgraph. (cid:229) pei=p
EXAMPLE1.3.2. (a)ForthequiverQgivenby i∈Q0
andhence
•1 1= (cid:229) ei
a i∈Q0
g
b 55•(cid:15)(cid:15)2 d ////•3oo r • isthCeounnviteirnseklyQ,.supposethatQ0isinfiniteand1∈kQ. Then1=(cid:229) l ipiforsome(finite)setof
thesetofpathsoflengthgreaterthan1isgivenby pathspiandscalarsl i.Pickavertexjsuchthatforallithepathpidoesnotendatj.Thene j1=0,
whichgivesacontradiction.
{b n+2,b n+1a ,gb n+1,db n+1,gb na ,db na :n≥0}.
Finally,ifQ0orQ1isnotfinitethenkQisclearlynotfinitedimensional. Givenafiniteset
ofverticeswithfinitelymanyedges,thereareonlyfinitelymanypathsbetweenthemunlessthe
(b)ForthequiverQgivenby
quivercontainsacycle. (cid:3)
a •1 b
55 ii
thesetofpathscorrespondstowordsina andb (alongwiththetrivialword). EXAMPLE1.3.5. EachofthequiversinExample1.3.2isfinite,andsothecorrespondingkQ
containsaunit. However,thepathalgebrascorrespondingto1.3.2(a)and1.3.2(b)arenotfinite
(c)ForthequiverQgivenby dimensional. Indeed,itiseasytoseethatthepathalgebrafor(b)isisomorphictokhx,yi,under
themaptakinga toxandb toy.Thepathalgebrafor1.3.2(c)isan8-dimensionalalgebra.
a b g
•1 //•2 //•3oo •4
BecauseofLemma1.3.4wewillonlyconsiderfinitequiversQ,sothatthecorrespondingpath
thesetofpathsis
algebrasareunital.
{e1,e2,e3,e4,a ,b ,g,ba }.
DEFINITION1.3.6. GivenafinitequiverQ,theidealRQofkQgeneratedbythearrowsinQ
iscalledthearrowidealofkQ. ThenRmistheidealgeneratedbyallpathsoflengthminQ. An
Wewouldliketoassociateanalgebratoaquiver;however,weneedtotakealittlecare. Q
idealIinkQiscalledadmissibleifthereexistsm≥2suchthat
ofpaDthEsFIiNnIQTI.OMNu1lt.i3p.l3ic.aTtihoenpiastvhiaalcgoenbcraatkeQnaotifoanqoufipvearthQs:iisftphe=k-av1eact2o.r..sapancaenwdiqth=babs1ibs2th..e.bsemt RmQ⊆I⊆R2Q.
then IfIisadmissiblethen(Q,I)iscalledaboundquiver,andkQ/Iisaboundquiveralgebra.
pq=a 1a 2...a nb 1b 2...b m
ifa nstartsatthevertexwhereb 1ends,andis0otherwise. isgrNeaotteertthhaatnifthQemisafixniimteaalnpdatahclyecnlgicththiennQa.nyidealcontainedinR2Qisadmissible,asRmQ=0ifm
Wehavenotyetcheckedthattheabovedefinitiondoesinfactdefineanalgebrastructureon EXAMPLE 1.3.7. LetQbeasinExample1.3.2(b),andletI=hba ,b 2i. Thisisnotanad-
kQ. missibleidealinkQasitdoesnotcontaina mforanym≥1,andsodoesnotcontainRmforany
Q
m≥2.
LEMMA 1.3.4. LetQbeaquiver. ThenkQisanassociativealgebra. FurtherkQhasan
identityelementifandonlyifQ0isfinite,andisfinitedimensionalifandonlyifQisfiniteand PROPOSITION1.3.8. LetQbeafinitequiverwithadmissibleidealIinkQ.ThenkQ/Iisfinite
acyclic. dimensional.
1.4.REPRESENTATIONSOFQUIVERS 13 14 1.ALGEBRASANDMODULES
PROOF. AsIisadmissiblethereexistsm≥2suchthatRmQ⊆I.Hencethereisasurjectiveal- Thishasarepresentation
gtawsebothrDiaenrhEeQoFaImsNrueoIcTmohInOotlhryNpafiht1ina.s3ilmtl.e9plf.yarotAmhmsarenkhlQyaavt/pieoRanttmQhhiseononskaftoQmlekeniQsgs/ttaahIr.filteBnvsiuestertttthelhiaxneneafamnordr.mcthoeemrasblaignmeabetrieoannidsovcfeleprataertlxhy.sfiIofnf{itrleejnd:gimtjh∈eanJts}iloeinas(cid:3)asatl k(=01=)=(=10=)===//(cid:30)(cid:30)kk22(cid:127)((cid:127)(cid:127)11(cid:127)10(cid:127)()(cid:127)01(cid:127)//??11k)2(oo102110)k3
setofrelationsinkQsuchthattheidealgeneratedbythesetisadmissiblethenwesaythatkQis
boundbytherelations. Noticehoweasyitwastogivearepresentation: therearenocompatibilityrelationstobe
checked(apartfromthatthelinearmapsgobetweentheappropriatedimension)soexamplescan
EXAMPLE1.3.10. ConsiderthequiverinExample1.3.2(a)andtherelations
beeasilygeneratedforanypathalgebra.Thisisverydifferentfromwritingdownexplicitmodules
{gb 2a −da ,gb +db ,b 5}. foranalgebra(ingeneral).
Anypathoflengthatleast7mustcontainb 5,andsoQisboundbythissetofrelations. Definition1.4.1looksratherdifferent fromthat foran algebra. However, thenextlemma
showsthatrepresentationsofQcorrespondtokQ-modulesinanaturalway.
Infacttheaboveexamplegeneralises:itiseasytoseethatanyidealIinR2 isadmissibleifit
Q LEMMA 1.4.4. LetMbearepresentationofafiniteacyclicquiverQ. Considerthevector
containseachcycleinQtosomepower.Further,wehave
space
PROPOSITION1.3.11. LetQbeafinitequiver.EveryadmissibleidealinkQisgeneratedbya M′= Ma.
finitesequenceofrelationsinkQ. aM∈Q0
ThiscanbegiventhestructureofakQ-modulebydefiningforeacha :i−→jamapfa′ :M−→M
PROOF. (Sketch)ItiseasytocheckthateveryadmissibleidealIisfinitelygeneratedbysome by
set{a1,...,an}(asRmQandI/RmQarefinitelygenerated). However,ingeneralasetofgenerators fa′(m1,...,mn)=(0,...,0,fa (mi),0,...0)
forIwillnotbeasetofrelations,asthepathsineachaimaynotallhavethesamestartvertexand wherethenon-zeroentryisinposition j,andforeachi∈Q0amapei:M−→Mby
endvertex.However,thenon-zeroelementsintheset ei(m1,...,mn)=(0,...,0,mi,0,...,0)
{exaiey:1≤i≤n,x,y∈Q0} wherethenon-zeroentryisinpositioni. Conversely,supposethatN isakQ-module. Thenwe
areallrelations,andthissetgeneratesI. (cid:3) obtainarepresentationofQbysettingNa=eaNanddefiningfa fora :a−→btobetherestriction
oftheactionofa ∈kQtoNa.
1.4. Representationsofquivers PROOF. CheckingthattheabovedefinitionsgiveakQ-moduleandarepresentationofQre-
spectivelyisroutine. (cid:3)
DEFINITION1.4.1. LetQbeafinitequiver.ArepresentationMofQoverkisacollectionofk-
vectorspaces{Ma:a∈Q0}togetherwithalinearmapfa :Ma−→Mbforeacharrowa :a−→b Wealsoneedthenotionofarepresentationofaboundquiver. Notethatwedonotneedto
inQ1.TherepresentationMisfinitedimensionalifalltheMaarefinitedimensional. assumethatQisacyclichere,asadmissibleidealsguaranteethattheassociatedquotientalgebrais
DEFINITION1.4.2.GiventworepresentationsMandM′ofafinitequiverQ,ahomomorphism finitedimensional.
fromMtoNisacollectionoflinearmaps fi:Mi−→Mi′suchthatforeacharrowa :i−→ jwe DEFINITION1.4.5. Givenapathp=a 1a 2...a ninafinitequiverQfromatobandarepre-
havefa′ fi=fjfa . sentationMofQwedefinethelinearmapf pfromMatoMbby
Whengivingexamplesofrepresentationsofquiverswewillusuallyfixbasesofeachofthe f p=fanfan−1...fa1.
vectorspaces,andrepresentthemapsbetweenthembymatriceswithrespecttocolumnvectorsin Ifr isalinearcombinationofpathspiwiththesamestartvertexandthesameendvertexthenfr
thesebases. wisedseafiynethdattoMbeisthbeocuonrdrebsypoInidfifnrg=lin0eaforrcaolmlrbeinlaattiioonnsorft∈heI.f pi.GivenanadmissibleidealIinkQ
EXAMPLE1.4.3. Considerthequiver
EXAMPLE 1.4.6. ConsidertherepresentationinExample1.4.3. Let p=ba andq=rd .
•1BBdBBaBBBB// ••25||||b|r|||//>>•3oo g •4. Tanhdensothisrefppre=se(cid:18)nta11tion10i(cid:19)sb(cid:18)ou10nd(cid:19)b=yth(cid:18)e11ide(cid:19)alhba f−q=rd (cid:18)i.01 11 (cid:19)(cid:18) 01 (cid:19)=(cid:18) 11 (cid:19)
1.5.EXERCISES 15 16 1.ALGEBRASANDMODULES
ItiseasytoverifythatthecorrespondencebetweenrepresentationoffiniteacyclicQandkQ- (a) GivenvectorspacesNa≤Ma,whatconditionsmustbesatisfiedfor(Na,fa)tobea
modulesgiveninLemma1.4.4extendstoacorrespondencebetweenrepresentationsoffiniteQ subrepresentationNofQ?
boundbyIandkQ/I-modules. (b) SupposethatMisarepresentationofQboundbyanadmissibleidealI. Showthat
therepresentationNisalsoboundbyI.
Inthiscourseweareavoidingthelanguageofcategorytheory. Thisismainlyduetolack
(c) IfQhasnvertices,givennon-isomorphicsimplerepresentationsofkQ,andalsoof
of time: the language of categories and functors is a very powerful one, and many results in
representationtheoryarebeststatedinthisway.Roughly,acategoryisacollectionofobjects(e.g. kQ/I.(Hint:whatconditiononthedimensionsoftheNaguaranteestheabsenceofa
propersubrepresentation?)
kQ-modules)andmorphisms(e.g. kQ-homomorphisms),andtheideaistostudythecategoryas
(d) IfQisacyclicthenwewillseeinChapter2thattheseexamplesformacompleteset
awholeratherthanjusttheobjectsormorphismsseparately. Afunctoristhenamapfromone
ofsimplerepresentations.However,itisalsopossibletoshowthisdirectly.Suppose
categorytoanotherwhichtransportsbothobjectsandmorphismsinasuitablycompatibleway.
InthislanguagetheaboveresultrelatingboundrepresentationsofQandkQ/I-modulesgivesan thatMisarepresentationofanacyclicQsuchthatmorethanoneMaisnon-zero.
ShowthatMhasapropersubrepresentation.
equivalencebetweenthecorrespondingcategories.
(e) SupposethatQisfinitebutcontainssomecycle. ShowthatQnowhasinfinitely
manynon-isomorphicsimplerepresentationsoverC.
1.5. Exercises
(7) InthisexercisewewillclassifytheindecomposablerepresentationsofthequiverQgiven
(1) SupposethatIisanidealinanalgebraA. by
(a) ShowthatA/Ihasanalgebrastructuresuchthatthereisasurjectivehomomorphism •1 a1 //•2 a2 //•3 a3 //... an−2//•n−1an−1 //•n.
fromAtoA/I. LetM=(Mi,fi)beanindecomposablerepresentationofQ.
(b) SupposethatAisanalgebrawithidealI,andthatMisanA/I-module.ShowthatM (a) ShowthatiffiisnotinjectivethenMj=0for j>i.
canbegiventhestructureofanA-module. (b) SimilarlyshowthatiffiisnotsurjectivethenMj=0for j≤i.
(c) IfMisanA-module,whatconditionmustitsatisfytobeanA/I-module? (c) DeducethatMisisomorphictoarepresentationoftheform
0 //... //0 //k id //... id //k //0 //... //0.
(2) Supposethat(P,≤)isapartiallyorderedsetofcardinalityn, anddefinekPtobethe
subsetofMn(k)givenby (d) Showthatthen(n+1)suchmodulesarepairwisenon-isomorphic.
2
kP={M=(mij):mij=0if i6≤j}. WewillseeinChapter4thatthisexampleispartofamoregeneralpicture.
(a) ShowthatkPisasubalgebraofMn(k)(thisiscalledtheincidencealgebraof(P,≤)).
(b) ShowthatPcanbeidentifiedwiththeset{1,...n}insuchawaythatkPcanbe (8) LetS3denotethesymmetricgrouponthreesymbols.DecomposethegroupalgebraCS3
identified withasubalgebraofthealgebraLTn(k)oflowertriangularmatricesin intoadirectsumofsimplerepresentationsforS3.(Youmayfinditconvenienttoidentify
Mn(k). CS3withaspaceofpermutationmatrices.)
(c) DeducethatifQisafiniteacyclicquiverwithatmostonearrowbetweeneachpair
ofvertices,thenkQisasubalgebraofLTn(k)forsomen.
(d) IllustrateyourlastconstructioninthecaseofthequiverinExample1.3.2(c).
(e) WhichquivercorrespondtothewholeofLTn(k)?
(3) SupposethatQisaquiver,andletQopbethequiverobtainedbyreversingallthearrows.
Showthatthereisanisomorphismofalgebrask(Qop)∼=(kQ)op.
(4) SupposethatGisagroup.ShowthatkG∼=(kG)op.
(5) ClassifythesimplemodulesforthecyclicgroupCnoveranalgebraicallyclosedfieldof
characteristicp≥0.
(6) SupposethatM=(Ma,fa)isarepresentationofsomefinitequiverQ.
18 2.SEMISIMPLICITYANDSOMEBASICSTRUCTURETHEOREMS
PROOF. (Sketch)Notethat(a)implies(b)and(b)implies(c)areclear. For(c)implies(a)
considerthesetofsubmodulesofAwhoseintersectionwithN is0. Pickonesuch, Lsay, of
maximaldimension;ifN⊕L6=MthenthereissomesimpleSinMnotinN⊕L.Butthiswould
CHAPTER 2 implythatS+Lhasintersection0withA,contradictingthemaximalityofL. (cid:3)
LEMMA2.1.5. IfMisasemisimpleA-modulethensoiseverysubmoduleandquotientmodule
Semisimplicityandsomebasicstructuretheorems ofM.
PROOF. (Sketch)IfNisasubmodulethenM=N⊕LforsomeLbytheprecedingLemma.
Inthischapterwewillreviewsomeoftheclassicalstructuretheoremsforfinitedimensional ButthenM/L∼=N,andsoitisenoughtoprovetheresultforquotientmodules.
algebras. Inmostcasesresultswillbestatedwithonlyasketchoftheproof. Henceforthwewill IfM/Lisaquotientmoduleconsidertheprojectionhomomorphismp fromMtoM/L.Write
restrictourattentiontofinitedimensionalmodules. MasasumofsimplemodulesSiandverifythatp (S)iseithersimpleor0.ThisprovesthatM/L
isasumofsimplemodules,andsotheresultfollowsfromtheprecedinglemma. (cid:3)
2.1. Simplemodulesandsemisimplicity
Toshowthatanalgebraissemisimple,wedonotwanttohavetochecktheconditionforevery
possiblemodule.Fortunatelywehave
RecallthatasimplemoduleisamoduleSsuchthattheonlysubmodulesareSand0. These
formthebuildingblocksoutofwhichallothermodulesaremade: PROPOSITION2.1.6. EveryfinitedimensionalA-moduleisisomorphictoaquotientofAnfor
somen.HenceanalgebraAissemisimpleifandonlyifAissemisimpleasanA-module.
LEMMA2.1.1. IfMisafinitedimensionalA-modulethenthereexistsasequenceofsubmod-
ules
PROOF. (Sketch)SupposethatMisafinitedimensionalA-module,spannedbysomeelements
0=M0⊂M1⊂···⊂Mn=M m1,...,mn.Wedefineamap
suchthatMi/Mi−1issimpleforeach1≤i≤n.SuchaseriesiscalledacompositionseriesforM. f :⊕ni=1A−→M
by
ofmPiRnOimOaFl.dPimroecneseidonb,ywinhdicuhctiisonneocnesthsaerdiliymseinmspiolen.oNfMow.IdfimM(iMsn/oMt1s)im<pdlei,mpMick,aansdubsomtohdeurleesMul1t f ((a1,...,an))=(cid:229)n aimi.
followsbyinduction. (cid:3) i=1
ItiseasytocheckthatthisisahomomorphismofA-modules,andsobytheisomorphismtheorem
wehavethat
Moreover,wehave
M∼=⊕n A/kerf .
i=1
THEOREM2.1.2(Jordan-Hölder). SupposethatMhastwocompositionseries Theresultnowfollowsfromtheprecedinglemma. (cid:3)
0=M0⊂M1⊂···⊂Mm=M, 0=N0⊂N1⊂···⊂Nn=M. ForfinitegroupswecansayexactlywhenkGissemisimple:
Thenn=mandthereexistsapermutations of{1,...n}suchthat
THEOREM2.1.7(Maschke). LetGbeafinitegroup.ThenthegroupalgebrakGissemisimple
Mi/Mi+1∼=Ns (i)/Ns (i)+1. ifandonlyifthecharacteristicofkdoesnotdivide|G|,theorderofthegroup.
PROOF. Theproofissimilartothatforgroups. (cid:3) PROOF. (Sketch)Firstsupposethatthecharacteristicofkdoesnotdivide|G|.Wemustshow
thateverykG-submoduleMofkGhasacomplementasamodule. Clearlyasvectorspaceswe
Lifewouldbe(relatively)straightforwardifeverymodulewasadirectsumofsimplemodules. canfindNsuchthatM⊕N=kG. Letp :kG−→Mbetheprojectionmapp (m+n)=mforall
m∈Mandn∈N.Wewanttomodifyp sothatitisamodulehomomorphism,andthenshowthat
DEFINITION2.1.3. AmoduleMissemisimple(orcompletelyreducible)ifitcanbewrittenas thekernelisthedesiredcomplement.
adirectsumofsimplemodules.AnalgebraAissemisimpleifeveryfinitedimensionalA-module
issemisimple. DefineamapTp :kG−→Mby
LEMMA2.1.4. IfMisafinitedimensionalA-modulethenthefollowingareequivalent: Tp (m)= 1 (cid:229) g(p (g−1m)).
|G|
(a)IfNisasubmoduleofMthenthereexistsLasubmoduleofMsuchthatM=L⊕N. g∈G
(b)Missemisimple. Notethatthisispossibleas|G|−1existsink. ItisthenroutinetocheckthatTp isakG-module
(c)Misa(notnecessarilydirect)sumofsimplesubmodules. map.
17
2.2.SCHUR’SLEMMAANDTHEARTIN-WEDDERBURNTHEOREM 19 20 2.SEMISIMPLICITYANDSOMEBASICSTRUCTURETHEOREMS
NowletK=ker(Tp),whichisasubmoduleofkG.WewanttoshowthatkG=M⊕K.First GivenanA-moduleMweset
rsahnokw-ntuhlalittyTpthaecotrsemasftohreliidneenartitmyaopns,MkG, w=hiMch+imKp.liCesomthbaitnMing∩tKhes=e0tw.oNfeaxcttsnwoteedthedatucbeytthhaet EndA(M)={f :M−→M|f isanA-homomorphism}.
kG=M⊕Kasrequired. ThisisasubalgebraofEndk(M).Moregenerally,ifMandNareA-modulesweset
Forthereverseimplication,considerw=(cid:229) g∈Gg∈kG.Itiseasytocheckthateveryelement HomA(M,N)={f :M−→N|f isanA-homomorphism}.
ofgfixesw,andhencewspansaone-dimensionalsubmoduleMofkG.Nowsupposethatthereis
acomplementarysubmoduleNofkG,anddecompose1=e+fwhereeandfaretheidempotents ArguingasintheproofofLemma2.2.1aboveweobtain
correspondingtoMandNrespectively. Wehavee=l wforsomel ∈k,ande2=e=l 2w2. It
iseasytocheckthatw2=|G|wandhencel w=l 2|G|wwhichimpliesthat1=l |G|. Butthis LEMMA2.2.3(Schur). IfkisalgebraicallyclosedandSandTaresimpleA-modulesthen
contradictsthefactthat|G|=0ink. (cid:3) HomA(S,T)∼=(cid:26) k0 oifthSe∼=rwTise.
Thenextresultwillbeimportantinthefollowingsection.
LEMMA2.1.8. ThealgebraMn(k)issemisimple. Wecannowgiveacompleteclassificationofthefinitedimensionalsemisimplealgebras.
THEOREM 2.2.4 (Artin-Wedderburn). Let Abea finitedimensionalalgebraover an alge-
PROOF. LetEijdenotethematrixinA=Mn(k)consistingofzeroseverywhereexceptforthe braicallyclosedfieldk.ThenAissemisimpleifandonlyif
(i,j)thentry,whichis1.Wefirstnotethat
1=E11+E22+···+Enn A∼=Mn1(k)⊕Mn2(k)⊕···⊕Mnt(k)
is an orthogonal idempotentdecompositionof 1, and hence A decomposes as a direct sum of forsomet∈Nandn1,...,nt∈N.
modulesoftheformAEii.Wewillshowthatthesesummandsaresimple.
PROOF. (Sketch)WesawinLemma2.1.8thatMn(k)isasemisimplealgebra,andifAandB
FirstobservethatAEiiisjustthesetofmatriceswhicharezeroexceptpossiblyincolumni. aresemisimplealgebras,thenitiseasytoverifythatA⊕Bissemisimple.
Pthieckmxat∈rixAExiwinhoicnh-ziesron;onw-ezemrou.sBtsuhtotwhenthatAx=AEii.Asxisnon-zerothereissomeentryxmiin ForthereverseimplicationsupposethatMandNareA-modules,withM=⊕ni=1MiandN=
⊕mi=1Ni.ThefirstclaimisthatHomA(M,N)canbeidentifiedwiththespaceofmatrices
andhenceEji∈Axforall1≤j≤n.BEutjmthxis=imxmpilEiejsi∈thaAtxAx=AEiiasrequired. (cid:3) {(fij)1≤i≤n,1≤j≤m|fi,j:Mj−→NianA-homomorphism}
andthatifM=NwithMi=Niforallithenthisspaceofmatricesisanalgebrabymatrixmulti-
plication,isomorphictoEndA(M).Thisfollowsbyanelementarycalculation.
2.2. Schur’slemmaandtheArtin-Wedderburntheorem
NowapplythistothespecialcasewhereM=N=A,and
WebeginwithSchur’slemma,whichtellsusaboutautomorphismsofsimplemodules. A=(S1⊕S2⊕···⊕Sn1)⊕(Sn1+1⊕···⊕Sn1+n2)⊕···⊕(Sn1+n2+···+nt−1+1⊕···⊕Sn1+n2+···+nt)
LEMMA2.2.1(Schur). LetSbeasimpleA-moduleandf :S−→Sanon-zerohomomorphism. isadecompositionofAintosimplessuchthattwosimplesareisomorphicifandonlyiftheyoccur
Thenf isinvertible. inthesamebracketedterm. BySchur’sLemmaaboveweseethatfijinthisspecialcaseis0if
f 6=P0R,OsoOFM. L=et0Mand=fkeisrfinjaencdtivNe.=Siimmifla;rltyhewseesaereetbhoatthNsu=bmS,osdouflesisosfuSrj.ecBtiuvte,Sainsdshimenpcleefanids iSwsioeamhnadovrSpehjiasrmeionfdHifofemreAn(tAb,rAa)ckweittehdMtenr1m(ks),a⊕nd··i·s⊕soMmnte(kl)i.j∈Fiknaoltlhye,rwweisne.otTehtehraetifsotrhaennyanalogbebvrioauAs
invertible. (cid:3) EndA(A,A)∼=Aop
LEMMA2.2.2. IfkisalgebraicallyclosedandSisfinitedimensionalwithnon-zeroendomor- andhence
phismf ,thenf =l .idS,forsomenon-zerol ∈k. A=(Aop)op∼=Mn1(k)op⊕···⊕Mnt(k)op.
PROOF. AskisalgebraicallyclosedanddimS<¥ themapf hasaneigenvaluel ∈k.Then ButitiseasytoseethatMn(k)∼=Mn(k)opviathemaptakingamatrixXtoitstranspose,andso
f −l idSisanendomorphismofSwithnon-zerokernel(containingalleigenvectorswitheigen- wearedone. (cid:3)
value l ). Arguing as in the preceding lemma we deduce that ker(f −l idS)=S, and hence
f =l idS. (cid:3) Wecanalsodescribeallthesimplemodulesforsuchanalgebra.