Cluster algebras and quantum affine algebras

55 Pages·2009·0.38 MB·English

Cluster algebras and quantum affine algebras
DavidHernandez and Bernard Leclerc
Abstract
Let C be the categoryof finite-dimensionalrepresentationsof a quantumaffine algebra
U (g) of simply-laced type. We introducecertain monoidalsubcategories C (ℓ∈N) of C
q ℓ
andwestudytheirGrothendieckringsusingclusteralgebras.
b
Contents
1 Introduction 1
2 Clusteralgebrasandtheirmonoidalcategorifications 5
3 Finite-dimensionalrepresentationsofUq(g) 7
4 Thecaseℓ=1:statementofresultsandconjectures 10
b
5 q-characters 14
6 Truncatedq-characters 21
7 F-polynomials 26
8 Atensorproducttheorem 31
9 Clusterexpansions 34
10 TypeA 35
11 TypeD 39
12 Applications 43
13 Generalℓ 48
1 Introduction
1.1 Letg be asimple Lie algebra of type A ,D or E , and letU (g)denote the corresponding
n n n q
quantum affine algebra, with parameter q∈C∗ not a root of unity. The monoidal category C of
finite-dimensional Uq(g)-modules has been studied by many authorsbfrom different perspectives
(see e.g.[AK, CP1, FR, GV, KS, N1]). In particular its simple objects have been classified by
Chari and Pressley, anbd Nakajima has calculated their character in terms of the cohomology of
certainquivervarieties.
Inspite of these remarkable results manybasic questions remain open, and inparticular little
isknownaboutthetensorstructure of C.
Wheng=sl ,ChariandPressley[CP2]haveshownthateverysimpleobjectisisomorphictoa
2
tensorproductofsimpleobjectsofaspecialtypecalledKirillov-Reshetikhinmodules. Conversely,
theyhaveshownthatatensorproductS ⊗···⊗S ofKirillov-Reshetikhinmodulesissimpleifand
1 k
1
onlyifS ⊗S issimpleforevery16i< j6k. Moreover,S ⊗S issimpleifandonlyifS andS
i j i j i j
are “in general position” (a combinatorial condition on the roots of the Drinfeld polynomials of
S and S ). Hence, the Kirillov-Reshetikhin modules can be regarded as the prime simple objects
i j
of C [CP6], and one knows which products of primes are simple. As an easy corollary, one can
seethatthetensorpowersofanysimpleobjectof C aresimple.
Forg6=sl ,thesituationisfarmorecomplicated. Thus,alreadyforg=sl ,wedonotknowa
2 3
generalfactorization theoremforsimpleobjects(see[CP6],whereatentativelistofprimesimple
objects is conjectured). In fact, it was shown in [L] that the tensor square of a simple object of
C isnotnecessarily simple ingeneral, sooneshould not expect results similartothe sl case for
2
otherLiealgebras g.
Because ofthese difficulties, wedecide inthis paper tofocus on somesmallersubcategories.
Weintroduce asequence
C ⊂C ⊂···⊂C ⊂···, (ℓ∈N),
0 1 ℓ
of full monoidal subcategories of C, whose objects are characterized by certain strong restric-
tions on the roots of the Drinfeld polynomials of their composition factors. By construction, the
Grothendieck ring R of C is a polynomial ring in n(ℓ+1) variables, where n is the rank of g.
ℓ ℓ
Ourstarting pointisthat R isnaturally equipped withthestructure ofaclusteralgebra.
ℓ
Recall that cluster algebras were introduced by Fomin and Zelevinsky [FZ1] as a combi-
natorial device for studying canonical bases and total positivity. They found immediately lots
of applications, including a proof of a conjecture of Zamolodchikov concerning certain discrete
dynamical systems arising from the thermodynamic Bethe ansatz, called Y-systems [FZ2]. As
observed byKuniba,NakanishiandSuzuki[KNS],Y-systemsarestronglyrelatedwiththerepre-
sentation theory ofU (g)viasomeothersystems offunctional relations called T-systems. Itwas
q
conjectured in [KNS] that the characters of the Kirillov-Reshetikhin modules are solutions of a
T-system,andthiswasblaterprovedbyNakajima[N2]inthesimply-lacedcase,andbyHernandez
in the general case [H3]. Nowit iseasy to notice that in the simply-laced case the equations ofa
T-systemareexactlyofthesameformastheexchangerelationsinaclusteralgebra. Thisledusto
introduceaclusteralgebrastructureonR byusinganinitialseedconsistingofachoiceofn(ℓ+1)
ℓ
Kirillov-Reshetikhin modulesinC . Theexchangematrixofthisseedencodesnℓequationsofthe
ℓ
T-system satisfied by these Kirillov-Reshetikhin modules. (Note that the seed contains n frozen
variables – or coefficients – in the sense of [FZ1].) By definition of a cluster algebra, one can
obtain new seeds by applying sequences of mutations to the initial seed. Then one of our main
conjectures isthatallthenewclustervariables produced inthiswayareclasses ofsimpleobjects
ofC . (Notethatingeneral, thesesimpleobjectsarenolongerKirillov-Reshetikhin modules.)
ℓ
1.2 Forℓ=0,theclusterstructureofR istrivial: thereisauniqueclusterconsistingentirelyof
0
frozenvariables.
Thecaseℓ=1isalready veryinteresting, andmostofthispaperwillbedevoted toit. Recall
thatFominandZelevinskyhaveclassifiedtheclusteralgebraswithfinitelymanyclustervariables
in terms of finite root systems [FZ3]. It turns out that for every g the ring R has finitely many
1
clustervariables, andthatitsclustertypecoincides with therootsystemofg. Therefore, onemay
expectthatthetensorstructureofthesimpleobjectsofthecategoryC canbedescribedin“afinite
1
way”. In fact we conjecture that forevery g the category C behaves as nicely as the category C
1
forsl ,andweproveitforgoftypeA andD .
2 n 4
Moreprecisely, wesingle outafinitesetofsimpleobjects of C whoseDrinfeldpolynomials
1
are naturally labeled by the set of almost positive roots of g (i.e., positive roots and negative
2
simple roots). Recall that the almost positive roots are in one-to-one correspondence with the
cluster variables [FZ3], so weshall call these objects the cluster simple objects. Tothese objects
weadd ndistinguished simple objects whichwecallfrozen simpleobjects. Ourfirstclaim isthat
theclassesoftheseobjects inR coincidewiththeclustervariablesandfrozenvariables.
1
Recall also that the cluster variables are grouped into overlapping subsets of cardinality n
called clusters [FZ1]. The number of clusters is a generalized Catalan number, and they can be
identified with the faces of the dual of a generalized associahedron [FZ2]. Our second claim is
that a tensor product of cluster simple objects is simple if and only if all the objects belong to a
commoncluster. Moreover, thetensorproduct ofafrozensimpleobjectwithanysimpleobjectis
againsimple. ItfollowsthateverysimpleobjectofC isatensorproductofclustersimpleobjects
1
and frozen simple objects. As a consequence, the tensor powers of any simple object of C are
1
simple.
Toprovethis, weshow that atensorproduct S ⊗···⊗S of simpleobjects of C issimple if
1 k 1
andonlyifS ⊗S issimpleforevery i6= j. Thisresultisproveduniformlyforalltypes.
i j
Notethat only the frozen objects and the cluster objects attached to positive simple roots and
negative simple roots are Kirillov-Reshetikhin modules. The remaining cluster objects (labelled
bythepositivenonsimpleroots)probably deservetobestudiedmoreclosely.
1.3 When ℓ> 1 the ring R has in general infinitely many cluster variables, grouped into in-
ℓ
finitelymanyclusters. Anotableexception isthecaseg=sl ,forwhichR isaclusteralgebraof
2 ℓ
finitetype A intheclassification of[FZ3]. Inthisspecial caseitfollowsfrom [CP2]that, again,
ℓ
theclassesinR ofthesimpleobjects ofC areprecisely theclustermonomialsof R .
ℓ ℓ ℓ
We conjecture that for arbitrary g and ℓ, every cluster monomial of R is the class of a sim-
ℓ
ple object. We also conjecture that, conversely, the class of a simple object S in C is a cluster
ℓ
monomial if and only if S⊗S is simple. In this case, following [L], we call S a real simple ob-
ject. We believe that real simple objects form an interesting class of irreducible U (g)-modules,
q
and the meaning ofourpartial results and conjectures isthat their characters are governed by the
combinatorics ofclusteralgebras. b
1.4 Letusnowdescribe thecontentsofthepaperinmoredetail.
In Section 2 we recall the definition of a cluster algebra, and we introduce the new notion of
monoidal categorification of a cluster algebra (Definition 2.1). We show (Proposition 2.2) that
the existence of a monoidal categorification gives an immediate answer to some important open
problems in the theory of cluster algebras, like the linear independence of cluster monomials, or
thepositivity ofthecoefficients oftheirexpansion withrespecttoanarbitrary cluster.
In Section 3 we briefly review the theory of finite-dimensional representations of U (g) and
q
weintroduce thecategories C . Wealso recall the definition ofthe Kirillov-Reshetikhin modules
ℓ
andwereviewtheT-systemofequations thattheysatisfy. b
In Section 4 we introduce some simple objects S(a ) of C attached to the almost positive
1
roots a ,andweformulate ourconjecture (Conjecture 4.6)forthecategory C . Itstates thatC is
1 1
a monoidal categorification of a cluster algebra A with the same Dynkin type as g, and that the
S(a )aretheclustersimpleobjects. Weillustrate theconjecture intypeA .
3
In Section 5 we review the definition and main properties of the q-characters of Frenkel-
Reshetikhin. One of the main tools to calculate them is the Frenkel-Mukhin algorithm which
werecallandillustrate withexamples.
In Section 6, we introduce some truncated versions of the q-characters for C . These new
1
truncatedcharactersaremucheasiertocalculateandtheycontainalltheinformationtodetermine
3
the composition factors ofan object of C . Themainresult ofthis section (Proposition 6.7) isan
1
explicitformulaforthetruncated q-characterof S(a )whena isamultiplicity-free positiveroot.
In Section 7, we review following [FZ2, FZ5] the F-polynomials of the cluster algebra A.
These are variants of the Fibonacci polynomials of [FZ2], which are the building blocks of the
general solution ofaY-system. Theysatisfy afunctional equation similartoaT-system andeach
clustervariablecanbeexpressedintermsofitsF-polynomialinasimpleway(Equation(32)). We
showthatConjecture4.6(i)isequivalenttothefactthatthe(normalized)truncatedq-charactersof
theclustersimpleobjectsareequaltotheF-polynomials, andweproveitforthemultiplicity-free
roots(Theorem7.8).
InSection8,weproveanimportanttensorproducttheoremforthecategoryC (Theorem8.1):
1
ifS ,...,S aresimpleobjectsofC ,thenS ⊗···⊗S issimpleifandonlyifS ⊗S issimplefor
1 k 1 1 k i j
everyi6= j.
InSection9,weintroducefollowing[FZ2]thenotionsofcompatiblerootsandclusterexpan-
sion. Because of Theorem 8.1 and of the existence and uniqueness of a cluster expansion [FZ2],
wereduce Conjecture 4.6(ii)foragiven gtoafinitecheck: onehastoverify that S(a )⊗S(b )is
simpleforeverypair (a ,b )ofcompatible roots.
In Section 10 and Section 11 we prove Conjecture 4.6 in type A and D . Conjecture 4.6 (i)
n 4
isproved moregenerally intype D , byshowing that the truncated q-characters ofcluster simple
n
objectsaregivenbytheexplicitcombinatorial formulaof[FZ2]fortheFibonaccipolynomials of
2-restricted roots.
In Section 12 we present some applications of our results for C . First we show that the q-
1
charactersofthesimpleobjectsofC aresolutionsofasystemoffunctionalequationssimilartoa
1
periodicT-system. Secondly,weexplainthatthel-weightmultiplicitiesappearinginthetruncated
q-characters of the cluster simple objects are equal to some tensor product multiplicities. This is
reminiscent of the Kostka duality for representations of sl , but in our case it is not limited to
n
type A. Thirdly, we exploit some known geometric formulas for F-polynomials due to Fu and
Keller [FK] to express the coefficients of the truncated q-characters of the simple objects of C
1
as Euler characteristics of some quiver grassmannians. This is similar to the Nakajima character
formulaforstandardmodules, butourformulaworksforsimplemodules.
Finally in Section 13 we state our conjectures for C (arbitrary ℓ) and illustrate them for g=
ℓ
sl (arbitrary ℓ) where they follow from [CP2], and g = sl (ℓ = 2). We also explain how our
2 3
conjecture for g=sl (arbitrary ℓ)would essentially follow from ageneral conjecture of [GLS2]
r
abouttherelationbetweenLusztig’sdualcanonical anddualsemicanonical bases.
1.5 Kedem [Ke] and Di Francesco [DFK] have studied another connection between quantum
affine algebras and cluster algebras, based on othertypes of functional equations (Q-systems and
generalized T-systems). Keller [Kel2] has obtained a proof of the periodicity conjecture for Y-
systemsattachedtopairsofsimply-lacedDynkindiagramsusing2-Calabi-Yaucategorificationsof
cluster algebras. More recently, Inoue, Iyama, Kuniba, Nakanishi and Suzuki [IIKNS] have also
studied the connection between Y-systems, T-systems, Grothendieck rings of U (g) and cluster
q
algebras, motivated by periodicity problems. These papers do not study the relations between
clustermonomialsandirreducibleUq(g)-modules. b
After this paper was submitted for publication, Nakajima [N5] gave a geometric proof of
Conjecture 4.6 for all types A,D,E, ubsing a tensor category of perverse sheaves on quiver va-
rieties. This category also makes sense for non Dynkin quivers and Nakajima showed that its
Grothendieck ring has a cluster algebra structure and that all cluster monomials are classes of
simple objects. Thanks to Proposition 2.2 below, this yields strong positivity results for every
acyclic cluster algebra with a bipartite seed. To the best of our knowledge, our conjecture for
4
ℓ>1remainsopen.
We have presented our main results in several seminars and conferences in 2008 and 2009
(IHP Paris (BL), MSRI Berkeley (BL), NTUA Athens (BL), ETH Zurich (DH), UNAM Mexico
(DH),Math. InstituteOxford(DH),MIOOberwolfach(BL)).Wethanktheseinstitutionsfortheir
kind invitations. Special thanks are due to Arun Ram and MSRI for organizing in spring 2008
a program on combinatorial representation theory where a large part of this work was done. We
also thank Keller for his preliminary Oberwolfach report on this work [Kel1]. Finally we thank
Nakajimaforhelpful commentsandstimulating discussions.
2 Cluster algebras and their monoidal categorifications
2.1 Wereferto[FZ4]foranexcellent surveyonclusteralgebras. Hereweonlyrecallthemain
definitions andresults.
2.1.1 Let 06n< r be some fixed integers. If B=(b ) is an r×(r−n)-matrix with integer
ij
entries, then the principal part B of Bis the square matrix obtained from B by deleting the last n
rows. Givensomek∈[1,r−n]defineanewr×(re−n)-matrix m (B)=(b′ )by
k ij
e e
−bij ifi=korej=k,
b′ = |b |b +b |b | (1)
ij ik kj ik kj
(bij+ otherwise,
2
where i∈[1,r] and j∈[1,r−n]. One calls m (B) the mutation of the matrix B in direction k. If
k
Bisanintegermatrix whoseprincipal part isskew-symmetric, thenitiseasy tocheck that m (B)
k
is also an integer matrix with skew-symmetric perincipal part. We will assumeefrom now on that
Behas skew-symmetric principal part. In this case, one can equivalently encode B by a quivereG
with vertex set {1,...,r} and with b arrows from j to i ifb >0 and −b arrows from i to j if
ij ij ij
be <0. NotethatG hasnoloopnor2-cycle. e
ij
NowFominand Zelevinsky define acluster algebra A(B)asfollows. LetF =Q(x ,...,x )
1 r
be the fieldof rational functions in r commuting indeterminates x=(x ,...,x ). Onecalls (x,B)
1 r
theinitialseedofA(B). For16k6r−ndefine e
e
e x∗= (cid:213) bik>0xbiik+(cid:213) bik<0x−i bik. (2)
k x
k
Thepair (m (x),m (B)), where m (x) isobtained from xbyreplacing x by x∗, isthe mutation of
k k k k k
the seed (x,B) in direction k. One can iterate this procedure and obtain new seeds by mutating
(m (x),m (B))inanyedirectionl∈[1,r−n]. LetS denotethesetofallseedsobtainedfrom(x,B)
k k
by any finiteesequence of mutations. Each seed of S consists of an r-tuple of elements of F
calledacluseter,andofamatrix. Theelementsofaclusterareitsclustervariables. Everyseedheas
r−n neighbours obtained by a single mutation in direction 1 6k6r−n. One does not mutate
the last n elements of a cluster; they are called frozen variables and belong to every cluster. We
thendefinethecluster algebraA(B)asthesubringofF generated byalltheclustervariables of
theseedsofS. Theintegerr−niscalledtherankofA(B).
A cluster monomial is a monomeial in the cluster variables of a single cluster. Note that the
exchange relation(2)isoftheform e
x x∗=m +m (3)
k k + −
wherem andm aretwoclustermonomials.
+ −
5
2.1.2 Thefirstimportantresultofthetheoryisthateverycluster variablezofA(B)isaLaurent
polynomialinxwithcoefficientsinZ. Itisconjecturedthatthecoefficientsarepositive. Notethat
because of(2), everycluster variable can bewritten asasubtraction freerational exepression in x,
butthisisnotenoughtoensurethepositivity ofitsLaurent expansion.
The second main result is the classification of cluster algebras of finite type, i.e.with finitely
many different cluster variables. Fomin and Zelevinsky proved that this happens if and only if
thereexistsaseed(z,C)suchthatthequiverattachedtotheprincipalpartofC isaDynkinquiver
(that is, an arbitrary orientation of a Dynkin diagram of type A,D,E). In this case, the cluster
monomialsformadisteinguished subsetofA(B),whichisconjectured tobeaeZ-basis[FZ5,§11].
IfA(B)isnotoffiniteclustertype,theclustermonomialsdonotspanit,butitisconjectured
that they are linearly independent. It is an inteeresting open problem to specify a “canonical” Z-
basisofAe(B)containing theclustermonomials.
2.2 We noew propose a natural framework which would yield positive answers to the above
questions. We say that a simple object S of a monoidal category is prime if there exists no non
trivialfactorization S∼=S ⊗S . WesaythatSisrealifS⊗Sissimple.
1 2
Definition2.1 Let A be a cluster algebra and let M be an abelian monoidal category. We say
that M is a monoidal categorification of A if the Grothendieck ring of M is isomorphic to A,
andif
(i) thecluster monomialsofA aretheclassesofalltherealsimpleobjectsofM;
(ii) the cluster variables of A (including the frozen ones) are the classes of all the real prime
simpleobjectsofM.
Theexistence of a monoidal categorification of a cluster algebra is a very strong property, as
shownbythefollowingresult.
Proposition 2.2 Suppose thattheclusteralgebraA hasamonoidalcategorification M. Then
(i) every cluster variable of A has aLaurent expansion withpositive coefficients withrespect
toanycluster;
(ii) thecluster monomialsofA arelinearlyindependent.
Proof— Ifmisaclustermonomial,wedenotebyS(m)thesimpleobjectwithclass[S(m)]=m.
Letzbeaclustervariable, andlet
N(x ,...,x )
1 n
z=
xd1···xdr
1 r
denote its cluster expansion with respect to the cluster x = (x ,...,x ). Here the numerator
1 r
N(x ,...,x ) is a polynomial with coefficients in Z. Multiplying both sides by the denominator,
1 r
weseethatN(x ,...,x )istheclassofthetensorproduct
1 r
P:=S(z)⊗S(x )⊗d1⊗···⊗S(x )⊗dr.
1 r
Moreover, since xisacluster, everymonomial m=xk1···xkr istheclassofasimpleobject
1 r
S =S(x )⊗k1⊗···⊗S(x )⊗kr
1 r
6
ofM. HencethecoefficientofminN(x ,...,x )isequaltothemultiplicityofS asacomposition
1 r
factorofP,thusitisnonnegative. Thisproves(i).
Bydefinitionofamonoidalcategorification, theclustermonomialsformasubsetofthesetof
classesofallsimpleobjectsofM,whichisaZ-basisoftheGrothendieckgroup. Thisproves(ii).
2
Remark2.3 (i)Inrecentyears,manyexamplesofcategorifications ofclusteralgebrashavebeen
constructed (see e.g.[MRZ, BMRRT, CC, GLS2, BIRS, CK, GLS4]). They are quite different
from the monoidal categorifications introduced in this paper. Indeed, these categories are only
additive andhave notensoroperation. Themultiplication ofthe clusteralgebra reflectsthe direct
sum operation of the category. We shall call these categorifications additive. Note that there is
no analogue of Proposition 2.2 foradditive categorifications. Although additive categorifications
have been helpful for proving positivity of cluster expansions or linear independence of cluster
monomials in some cases, this always requires some additional work, for example to show the
positivity of some Euler characteristics. Finally, to recover the cluster algebra from its additive
categorification one does not consider the Grothendieck group (which would be too small) but a
kind of “dual Hall algebra” constructed via a cluster character. This is in general a complicated
procedure.
(ii)Inviewofthestrength andsimplicity ofProposition 2.2,onemightwonderwhetherthere
existanyexamplesofmonoidalcategorifications ofclusteralgebras. Oneoftheaimsofthispaper
istoproduce someexamplesusingrepresentations ofquantum affinealgebras.
(iii)LetM beanabelian monoidal category. IfM isamonoidal categorification ofacluster
algebra A,thenwegetnewcombinatorial insights about thetensorstructure ofM. Inparticular
ifA hasfiniteclustertype,wecanexpressanysimpleobjectofM asatensorproduct offinitely
manyprimeobjects,andthisyieldsacombinatorialalgorithmtocalculatethecompositionfactors
of a tensor product of simple objects of M. So this can be a fruitful approach to study certain
interesting monoidalcategories M. Thisisthepointofviewweadoptinthispaper.
3 Finite-dimensional representations ofU (g)
q
Inthissectionwebrieflyreviewsomeknownresultsintherepresentationtheoryofquantumaffine
algebras. Formoredetailedsurveyswereferthereadertothembonograph[CP1,chap. 12]andthe
recentpaper[CH].
3.1 LetgbeasimpleLiealgebraoverCoftypeA ,D orE . WedenotebyI=[1,n]thesetof
n n n
vertices oftheDynkindiagram, by A=[a ] theCartanmatrixof g,byhtheCoxeternumber,
ij i,j∈I
byP ={a |i∈I}thesetofsimpleroots, byW theWeylgroup, withlongestelement w .
i 0
LetU (g)denote thecorresponding quantum affinealgebra, withparameter q∈C∗ notaroot
q
ofunity. U (g)hasaDrinfeld-Jimbo presentation, whichisaq-analogue oftheusualpresentation
q
of the Kac-bMoody algebra g. It also has a second presentation, due to Drinfeld, which is better
suitedtostudbyfinite-dimensional representations. There areinfinitelymanygenerators
b
x+, x−, h , k, k−1, c, c−1, (i∈I, r∈Z, s∈Z\{0}),
i,r i,r i,s i i
and a list of relations which we will not repeat (see e.g.[FR]). Remember that g can be realized
as a central extension of the loop algebra g⊗C[t,t−1]. If x+,x−,h (i∈I) denote the Chevalley
i i i
generators of g then x±i,r isaq-analogue of x±i ⊗tr, hi,s is aq-analogue ofhi⊗ts,bki stands forthe
q-exponential ofh ≡h ⊗1,andcfortheq-exponential ofthecentralelement.
i i
7
Forevery a∈C∗ thereexistsanautomorphism t ofU (g)givenby
a q
t (x±)=arx±, t (h )=ash , t (k±1)=k±1, t (c±1)=c±1.
a i,r i,r a i,s i,s a i b i a
Therealsoexistsaninvolutive automorphism s givenby
s (x±)=x∓ , s (h )=−h , s (k±1)=k∓1, s (c±1)=c∓1.
i,r i,−r i,s i,−s i i
3.2 We consider the category C of finite-dimensional U (g)-modules (of type 1). It is easy to
q
see that if V is an object of C, then c acts on V as the identity, and the generators h act by
i,s
pairwisecommutingendomorphisms ofV. b
SinceU (g)isaHopf algebra, C is anabelian monoidal category. Itiswell-known that C is
q
notsemisimple.
Foran objbectV in C and a∈C∗, wedenote byV(a)the pull-back ofV under ta, and byVs
thepull-back ofV unders . ThemapsV 7→V(a)andV 7→Vs giveauto-equivalences ofC.
3.3 LetIdenotethesetofverticesoftheDynkindiagramofg. ItwasprovedbyChariandPress-
leythatthesimpleobjectsSofC areparametrizedbyI-tuplesofpolynomialsp =(p (u); i∈I)
S i,S
inoneindeterminate uwithcoefficientsinCandconstantterm1,calledtheDrinfeldpolynomials
of S. In particular, for a∈C∗ and i∈I we have a fundamental moduleV which is the simple
i,a
objectwithDrinfeldpolynomials:
1−au if j=i,
p (u)=
j,Vi,a 1 if j6=i.
(cid:26)
3.4 LetS∗ denote the dual ofS. The Drinfeld polynomials of S∗ are easily deduced from those
ofS,namely
p i,S∗(u)=p i∨,S(q−hu), (i∈I),
wherei7→i∨ denotes theinvolution onI givenbyw (a )=−a .
0 i i∨
The Drinfeld polynomials behave simply under the action of the automorphisms t , namely,
a
forasimpleobject SofC wehave
p (u)=p (au), (i∈I).
i,S(a) i,S
They also behave simply under the action of the involution s [CP3, Prop. 5.1] [H4, Cor. 4.11],
namely, if
p (u)=(cid:213) (1−a u),
i,S k
k
then
p i∨,Ss (u)=(cid:213) (1−a−k1q−hu).
k
Wealsohavethefollowing compatibility withtensorproducts. IfS andS aresimple,andif
1 2
SisthesimpleobjectwithDrinfeldpolynomials p :=p p (i∈I),thenSisasubquotient of
i,S i,S1 i,S2
thetensorproduct S ⊗S .
1 2
3.5 LetRdenotetheGrothendieck ringof C. Itisknown[FR,Cor. 2]that Risthepolynomial
ringoverZintheclasses[V ](i∈I, a∈C∗)ofthefundamental modules.
i,a
8
3.6 Since the Dynkin diagram of g is a bipartite graph, we have a partition I = I ⊔I such
0 1
that every edge connects a vertex of I with a vertex of I . The following notation will be very
0 1
convenient andusedinmanyplaces. Fori∈I weset
x = 0 ifi∈I0, e =(−1)xi. (4)
i 1 ifi∈I , i
1
(cid:26)
Clearly,themapi7→x iscompletelydeterminedbythechoiceofx ∈{0,1}forasinglevertexi ,
i i0 0
hencethereareonlytwopossible suchmaps.
3.7 LetC bethefullsubcategory ofC whoseobjectsV satisfy:
Z
for every composition factor S ofV and every i∈I, the roots of the Drinfeld polynomial
p i,S(u)belongtoq2Z+xi.
TheGrothendieckringRZofCZisthesubringofRgeneratedbytheclasses[Vi,q2k+xi](i∈I, k∈Z).
ItisknownthateverysimpleobjectSofC canbewrittenasatensorproduct S (a )⊗···⊗S (a )
1 1 k k
forsomesimpleobjects S ,...,S ofC andsomecomplexnumbersa ,...,a suchthat
1 k Z 1 k
a
i 6∈q2Z, (16i< j6k).
a
j
(This follows, for example, from the fact that such a tensor product satisfies the irreducibility
criterion in [Cha2].) Therefore, the description of the simple objects of C essentially reduces to
thedescription ofthesimpleobjects ofC .
Z
3.8 WewillnowintroduceanincreasingsequenceofsubcategoriesofC . Letℓ∈N. (Notethat
Z
inthispaperNdenotesthesetofnonnegative integers, thatis,0∈N.)
Definition3.1 The category C is the full subcategory of C consisting of those objectsV which
ℓ Z
satisfy:
for every composition factor S of V and every i∈I, the roots of the Drinfeld polynomial
p i,S(u)belongto{q−2k−xi |06k6ℓ}.
NotethatforeverysimpleobjectSofC ,thereexistsk∈ZsuchthatS(qk)belongstoC forsome
Z ℓ
ℓ∈N.
Proposition 3.2 C isanabelian monoidalcategory, withGrothendieck ringthepolynomial ring
ℓ
R =Z [V ]; 06k6ℓ .
ℓ i,q2k+xi
h i
The proof of Proposition 3.2 will be given in 5.2.4. It is an easy consequence of the theory of
q-characters.
Example3.3 Thesimpleobjects ofthecategory C haveasimpledescription. Indeed, itfollows
0
from [FM, Prop. 6.15] that all tensor products of fundamental modules of C are simple, hence
0
everysimpleobjectofC isisomorphictothetensorproductoffundamentalmodulescorrespond-
0
ing tothe irreducible factors of its Drinfeld polynomials. Moreover any tensor product ofsimple
objects ofC issimple.
0
9
3.9 Thedefinition ofC depends onthemapi7→x whichcanbechosenintwodifferent ways.
ℓ i
However, ifgisnotoftypeA ,theimageofC undertheauto-equivalence V 7→Vs (qh+2ℓ+1)is
2n ℓ
thecategorydefinedlikeC butwiththeopposite choiceofx . IfgisoftypeA ,theimageofC
ℓ i 2n ℓ
underV 7→V∗(qh) is the category defined like C but with the opposite choice of x . This shows
ℓ i
thatthechoiceofx isinfactirrelevant.
i
3.10 Fori∈I,k∈N∗ anda∈C∗,thesimpleobjectW(i) withDrinfeldpolynomials
k,a
(1−au)(1−aq2u)···(1−aq2k−2u) if j=i,
p =
j,W(i) 1 if j6=i,
k,a (cid:26)
(i)
iscalledaKirillov-Reshetikhin module. Inparticularfork=1,W coincideswiththefundamen-
1,a
(i)
talmoduleV . Byconvention,W isthetrivialrepresentation forevery ianda.
i,a 0,a
Theclasses [W(i)]inRsatisfythefollowingsystemofequations indexedbyi∈I,k∈N∗,and
k,a
a∈C∗,calledtheT-system:
(i) (i) (i) (i) (cid:213) (j)
[W ][W ]=[W ][W ]+ [W ], (5)
k,a k,aq2 k+1,a k−1,aq2 k,aq
j
where in the right-hand side, the product runs over all vertices j adjacent to i in the Dynkin
diagram. This was conjectured in [KNS] and proved in [N2, H3]. Using these equations, one
(i) (i)
can calculate inductively the expression of any [W ] as a polynomial in the classes [W ] of the
k,a 1,a
fundamental modules.
(i)
Example3.4 Forg=sl ,wehaveI={1},andwemaydroptheindexiin[W ]. TheT-system
2 k,a
reads
[W ][W ]=[W ][W ]+1, (a∈C∗, k∈N∗).
k,a k,aq2 k+1,a k−1,aq2
Thiseasilyimpliesthat[W ]isgivenbythek×kdeterminant:
k,a
[W ] 1 0 ··· 0
1,a
(cid:12) 1 [W1,aq2] 1 ... ... (cid:12)
(cid:12) (cid:12)
[Wk,a]=(cid:12)(cid:12)(cid:12) 0 1 [W1,aq4] ... ... (cid:12)(cid:12)(cid:12). (6)
(cid:12)(cid:12) ... ... ... ... ... (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) 0 ··· 0 1 [W ](cid:12)
(cid:12) 1,aq2k−2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 The case ℓ = 1: sta(cid:12)tement of results and conjectur(cid:12)es
Wenowfocusonthesubcategory C .
1
4.1 Let Q=ZP be the root lattice of g. Let F ⊂Q be the root system. Following [FZ2] we
denotebyF thesubsetofalmostpositiveroots, thatis,
>−1
F >−1=F >0∪(−P )
consists of the positive roots together with the negatives of the simple roots. In this section, we
attachtoeachb ∈F >−1 asimpleobjectS(b )ofC1.
10