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

ThemomentmapassociatedtotheGv-actiononthesymplecticvectorspaceEVisthemapψ:EV→glVwithi-componentψi:EV→EndVigivenby ε(h)xhxhψi(x)=¯.

h∈H,in(h)=iDe nethefunctionε:H→{ 1,1}byε(h)=1forall h∈ andε(h)= 1for¯allh∈ .LetV∈V.TheLiealgebraofGVisglV=iEnd(Vi)anditactsonEVby(a,x)=((ai),(xh))→[a,x]=(x′h)=(ain(h)xh xhaout(h)).Let ·,· bethenondegenerate,GV-invariant,symplecticformonEVwithvaluesinCde nedby x,y =ε(h)tr(xhyh¯).h∈H

De nition1.0.1([L1]).Anelementx∈EVissaidtobenilpotentifthereex-′′′istsanN≥1suchthatforanysequenceh′1,h2,...,hNinHsatisfyingout(h1)=′′′′in(h′x′...xh′:2),out(h2)=in(h3),...,out(hN 1)=in(hN),thecompositionxh′

1h2NVout(h′→Vin(h′iszero.1)N)

De nition1.0.2([L1]).LetΛVbethesetofallnilpotentelementsx∈EVsuchthatψi(x)=0foralli∈I.

1.1.TypeA∞.LetgbethesimpleLiealgebraoftypeA∞.LetI=Zbethesetofverticesofagraphwiththesetoforientededgesgivenby

H={i→j|i,j∈I,i j=1}∪{i←j|i,j∈I,i j=1}.

Wede netheinvolution¯:H→Hasthefunctionthatinterchangesi→jandi←j.Forh=(i→j),wesetout(h)=iandin(h)=jandforh=(i←j),wesetout(h)=jandin(h)=i.Let bethesubsetofHconsistingofthearrowsi→j.Proposition1.1.1([L1]).TheirreduciblecomponentsofΛVaretheclosuresoftheconormalbundlesofthevariousGV-orbitsinEV, .

Proof.ThecasewheregisoftypeAnisprovenin[L1].TheA∞casefollowsbypassingtothedirectlimit.

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

4IGORB.FRENKELANDALISTAIRSAVAGE

Fortwointegersk′≤k,de neV∞(k′,k)∈Vtobethevectorspacewithbasis{er|k′≤r≤k}.Werequirethaterhasdegreer∈I.Letx∞(k′,k)∈EV∞(k′,k), bede nedbyx∞(k′,k):er→er 1fork′≤r≤k,whereek′ 1=0.Itisclearthat(V∞(k′,k),x∞(k′,k))isanindecomposablerepresentationofourquiver.Conversely,anyindecomposable nite-dimensionalrepresentation(V,x)ofourquiverisisomorphictosome(V∞(k′,k),x∞(k′,k)). ∞bethesetofallLetZ∞bethesetofallpairs(k′≤k)ofintegersandletZ

functionsZ∞→Nwith nitesupport.

ItiseasytoseethatforV∈V,thesetofGV-orbitsinEV, isnaturallyindexed ∞ofZ ∞consistingofthosef∈Z ∞suchthatbythesubsetZV

f(k′,k)=dimVi

k′≤i≤k

foralli∈I.Herethesumisoverallk′≤ksuchthatk′≤i≤k.Correspondingtoagivenfistheorbitconsistingofallrepresentationsisomorphictoasumoftheindecomposablerepresentationsx∞(k′,k),eachoccuringwithmultiplicityf(k′,k). ∞.LetCfbetheconormalDenotebyOftheGV-orbitcorrespondingtof∈ZV¯fbeitsclosure.Wethenhavethefollowingproposition.bundletoOfandletC

¯fisaone-to-onecorrespondencebetweentheProposition1.1.2.Themapf→C ∞andthesetofirreduciblecomponentsofΛV.setZV

Proof.ThisfollowsimmediatelyfromProposition1.1.1.

(1)(1) 1.2.TypeAn.Letgbethea neLiealgebraoftypeAn,thatis,theLiealgebra

generatedbythesetofelementsEk,Fk,Hk(k=0,1,...,n)anddsatisfyingthefollowingrelations:

[Ek,Fl]=δklHk,[Hk,El]=aklEl,[Hk,Fl]= aklFl,

fork=l.[d,Ek]=δk0Ek,(adEk)1 aklEl=0,

Here

akl=2δ(k,l) δ(k,l+1) δ(k,l 1),

whereδ(k,l)=1ifk≡lmod(n+1)andisequaltozerootherwise.

LetI=Z/(n+1)Zbethesetofverticesofagraphwiththesetoforientededgesgivenby

H={i→j|i,j∈I,i j=1}∪{i←j|i,j∈I,i j=1}.

Fortwointegersk′≤k,de neV(k′,k)∈Vtobethevectorspacewithbasis{er|k′≤r≤k}.Werequirethaterhasdegreei∈Iwherer≡i(modn+1).Letx(k′,k)∈EV(k′,k), bede nedbyx(k′,k):er→er 1fork′≤r≤k,whereek′ 1=0.Itisclearthat(V(k′,k),x(k′,k))isanindecomposablerepresentationofourquiverandthatx(k′,k)isnilpotent.Also,theisomorphismclassofthisrepresentationdoesnotchangewhenk′andkaresimultaneouslytranslatedbyamultipleofn+1.Conversely,anyindecomposable nite-dimensionalrepresentation(V,x)ofourquiver,withxnilpotent,isisomorphictosome(V(k′,k),x(k′,k))wherek′andkareuniquelyde neduptoasimultaneoustranslationbyamultipleofn+1.

LetZbethesetofallpairs(k′≤k)ofintegersde neduptosimultaneous bethesetofallfunctionsZ→Nwithtranslationbyamultipleofn+1andletZ

nitesupport.[d,Fk]= δk0Fk,(adFk)1 aklFl=0

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

AFFINELIEALGEBRAS,QUIVERVARIETIESANDSTATISTICALMECHANICS5ItiseasytoseethatforV∈V,thesetofGV-orbitsonthesetofnilpotent VofZ consistingofthoseelementsinEV, isnaturallyindexedbythesubsetZ suchthatf∈Z f(k′,k)#{r|k′≤r≤k,r≡i(modn+1)}=dimVi

k′≤k

foralli∈I.Herethesumistakenoverallk′≤kuptosimultaneoustranslationbyamultipleofn+1.Correspondingtoagivenfistheorbitconsistingofallrepresentationsisomorphictoasumoftheindecomposablerepresentationsx(k′,k),eachoccuringwithmultiplicityf(k′,k).DenotebyOftheGV-orbitcorresponding V.tof∈Z Visaperiodicifforanyk′≤k,notallf(k′,k),f(k′+1,k+1),Wesaythatf∈Z V,letCfbetheconormal...,f(k′+n,k+n)aregreaterthanzero.Foranyf∈Z¯fbeitsclosure.bundleofOfandletC

V.ThefollowingtwoconditionsareProposition1.2.1([L1,15.5]).Letf∈Z

equivalent.

(1)Cfconsistsentirelyofnilpotentelements.

(2)fisaperiodic.

¯fisa1-1correspondencebetweenProposition1.2.2([L1,15.6]).Themapf→C VandthesetofirreduciblecomponentsofΛV.thesetofaperiodicelementsinZ

′′′Proposition1.2.3([L1,12.8]).Letx′∈EV, andx′′∈EV, ¯.Thenψi(x+x)=

0foralli∈Iifandonlyifx′′isorthogonalwithrespectto , tothetangentspacetotheGV-orbitofx′,regardedasavectorsubspaceofEV, .

1.3.gln+1Case.Sincegln+1isnotaKac-Moodyalgebrainastrictsense,thiscaseisnotcoveredbyLusztig’stheoryandrequirescertainmodi cations.Wepreservethenotationoftheprevioussubsection.

Vbethesetofallelementsx=x′+x′′,wherex′∈EV, …… 此处隐藏：5910字，全部文档内容请下载后查看。喜欢就下载吧 ……