Bases of representations of type A affine Lie algebras via q(3)

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,QUIVERVARIETIESANDSTATISTICALMECHANICS3

vectorsx=ForanysubsetHofH,letEV,H′bethesubspaceofEVconsistingofall ′(xh)suchthatxh=0wheneverh∈H.ThealgebraicgroupGV=iAut(Vi)

actsonEVandEV,H′by

1(g,x)=((gi),(xh))→gx=(x′h)=(gin(h)xhgout(h)).h.Wede netheinvolution¯:H→Htobethefunctionwhichtakesh∈HtotheelementofHconsistingofthethesameedgewithoppositeorientation.An¯=Handorientationofourgraphisachoiceofasubset Hsuchthat ∪ ¯= . ∩ LetVbethecategoryof nite-dimensionalI-gradedvectorspacesV=⊕i∈IVioverCwithmorphismsbeinglinearmapsrespectingthegrading.ThenV∈VshalldenotethatVisanobjectofV.ThedimensionofV∈Visgivenbyv=dimV=(dimV0,...,dimVn).Weidentifythisdimensionwiththeelement(dimV0)α0+···+(dimVn)αnoftherootlatticeofg.Heretheαiarethesimplerootscorrespondingtotheverticesofourquiver(graphwithorientation),whoseunderlyinggraphistheDynkingraphofg.GivenV∈V,let EV=Hom(Vout(h),Vin(h)).h∈H′

NotethatEVcanbeconsideredasthecotangentspaceofEV, underthisform.

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.

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