Extremal fundamental loop weight modules 10 4

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MATHIEU MANSUY

Abstract. We construct by fusion product new irreducible representations of the quantum affinizationUq( ˆsl∞). The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of typeA∞. We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebrasUq(sl^torn+1): we prove the conjectural link between Uq( ˆsl∞) andUq(sl^torn+1) stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtained in [23].

Contents

1. Introduction 1

2. Background 4

3. Extremal fundamental loop weight modules 10

4. Fusion product of extremal fundamental loop weight modules 19

5. Application to quantum toroidal algebras 25

References 28

1. Introduction

Let U_q(sl∞) be the quantum group associated to the infinite Dynkin diagram

It can be considered as limit of the quantum groups of typeAnand has been studied by various authors (see for example [1, 6, 8, 20] and references therein). In this paper we study the quantum affinizationU_q( ˆsl∞) ofU_q(sl∞) [12, 25]. This algebra is introduced in [14] and can be viewed as the limit of the quantum affine algebrasU_q( ˆsln+1) when n→ ∞.

The quantum affinizations, in particular the quantum affine algebras and the quantum toroidal algebras, have been intensively studied (see for example [3, 7, 9, 10, 12, 13, 14, 25] and references therein). In recent works [22, 23] we constructed new fam- ilies of integrable representations of the quantum toroidal algebra U_q(sl_n+1^tor ), called extremal loop weight modules, which generalize the `-highest weight modules: there are representations generated by an extremal vector for the horizontal quantum affine subalgebra in the sense of Kashiwara [16, 18]. The main motivation is the construction of finite-dimensional representations of the quantum toroidal algebra at roots of unity.

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The representation theory of the algebra U_q( ˆsl∞) is related to the one of quantum toroidal algebrasU_q(sl_n+1^tor ) in the following way (see [14]) : let us consider the morphism between the corresponding Dynkin diagrams of the algebrasU_q( ˆsl∞) andU_q(sl^tor_n+1)

φ_n:i∈Z7→i∈Z/(n+ 1)Z. It gives rise to a ring morphism

φ_n:Z[Y_i,a^±1]i∈Z,a∈C^∗ −→Z[Y_i,a^±1]i∈I_n,a∈C^∗.

Then the following combinatorial link between the algebras U_q( ˆsl∞) and U_q(sl^tor_n+1) is expected in [14].

Conjecture. [14, Conjecture 5.3] Let V be a simple U_q( ˆsl∞)-module in the category O_int. Then the image of the q–character of V by φn is also the q–character of a representation ofU_q(sl_n+1^tor ).

The main motivation in [14] is to predict q–character formulae for representations of U_q(sl^tor_n+1). This conjecture is proved for the class of Kirillov-Reshetikhin modules of U_q(sl^tor_n+1).

The aim of this article is twofold. First construct extremal loop weight modules forU_q( ˆsl∞): we obtain here a large class of such modules with basis labelled by semi- standard tableaux. Second by using the combinatorial link with the quantum toroidal algebras, construct extremal loop weight modules forU_q(sl_n+1^tor ): we prove the conjecture above for the particular family of extremal fundamental loop weight modules. We recover in this way the extremal loop weight modules defined in [23] for the quantum toroidal algebras.

Let us explain this in more details. The construction of extremal loop weight modules is inspired by the original Kashiwara’s study of the extremal weight modules (see [16]): we consider the fusion product of fundamental`-highest weight modules and fundamental`-lowest weight modules associated respectively to the fundamental weights Λ_` and −Λ₀ (`≥ 1) and to a non-zero complex parameter. The action of U_q( ˆsl∞) is defined by the Drinfeld coproduct, in the spirit of [23]. We show that we get in this way extremal loop weight modules associated to the weight Λ`−Λ0, called extremal fundamental loop weight modules. By fusion product ofkextremal fundamental loop weight modules, we obtain extremal loop weight modules associated to the weightkΛ`−kΛ0

with basis labelled by the set of semi-standard tableauxT[1,`]×[1,k]of shape (`×k). Fur- thermore, the action ofU_q( ˆsl∞) is explicitly describe on all these modules and involves theU_q(sl∞)-crystal structure onT[1,`]×[1,k].

Our main motivation are applications to quantum toroidal algebras. We prove the conjecture above for the family of extremal fundamental loop weight modules ofU_q( ˆsl∞) of weight Λ_`−Λ₀introduced in this paper: we determineq–character formulae for these representations. Furthermore, we show that their image by the morphismφn is in fact the q–character of a representation of U_q(sl^tor_n+1) constructed in [23]. Recall that our original goal is to construct extremal loop weight modules. We show that we get in

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this way extremal loop weight modules if and only if

(`= 1) or (n= 2r+ 1 and `=r+ 1).

Let us explain another motivation to consider the algebra U_q( ˆsl∞). Recall that it is defined as limit of the quantum affine algebras U_q( ˆsln+1) (n ≥ 2). For this reason the representation theory of this algebra is better understood than for general quantum affinizations: the fundamental modules we have to consider are thin, with basis labelled by semi-standard tableaux on which the action is explicitly known. It does not hold in the quantum toroidal case: in fact for U_q(sl^tor_n+1) the fundamental modules are not thin (see [14, Section 4.1]). This is one of the main reason of the met difficulties in the study of extremal loop weight modules for the quantum toroidal algebras (besides see [23, Remark 3.9]). We will see that more precise results can be obtained in our construction of extremal loop weight modules for the quantum affinizationU_q( ˆsl∞).

The paper is organized as follows.

In Section 2 we recall the main definitions of the quantum group U_q(sl∞) and its quantum affinization U_q( ˆsl∞) and we briefly review their representation theory. In particular one defines the extremal weight modules and we introduce the notion of extremal loop weight modules forU_q( ˆsl∞). In Section 3 we construct the fusion product of fundamental`-highest weight modules and fundamental`-lowest weight modules associated respectively to the weights Λ` and −Λ₀ (`≥1) and to a non-zero complex parameter. We define aU_q( ˆsl∞)-module structure on it by using the Drinfeld coproduct (Proposition 3.2). We obtain (as submodule of the fusion product) an extremal loop weight module associated to the weight Λ`−Λ0 (Theorem 3.14), called the extremal fundamental loop weight module of weight Λ_` −Λ₀. Furthermore we explicit the action ofU_q( ˆsl∞) on it (Theorem 3.10). Section 4 is devoted to the study of fusion products of extremal fundamental loop weight modules. When the non-zero complex parameters are chosen generic, we get extremal loop weight modules (Theorem 4.3).

In the non-generic case we recover all the extremal fundamental loop weight modules from the fusion product of the extremal fundamental loop weight modules associated to the weight Λ1 −Λ0 and to a q-segment (Theorem 4.6). By fusion product of k extremal fundamental loop weight modules of weight Λ_`−Λ₀, we obtain extremal loop weight modules of the weights kΛ_`−kΛ₀ (`, k ≥ 1) (Theorem 4.9). In Section 5, we discuss the applications to quantum toroidal algebras. We show that the q–character formulae obtained from the extremal fundamental loop weight U_q( ˆsl∞)-modules are theq–character of representations of the quantum toroidal algebraU_q(sl_n+1^tor ) (Theorem 5.3). Furthermore we determine when these representations give extremal loop weight modules (Theorem 5.4).

Acknowledgements : I would like to thank my advisor David Hernandez for his encouragements and for his precious comments. I am grateful to Nicolas Jacon for his interest in my work. I want also to thank Alexandre Bouayad and Dragos Fratila for their accurate remarks.

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2. Background

2.1. Cartan matrix. Let C = (Ci,j)i,j∈Z be the Cartan matrix of type A∞, that is (i, j∈Z)

Ci,i = 2 , Ci,i+1=Ci+1,i=−1 andCi,j = 0 otherwise.

Consider the vector space

h=M

i∈Z

Qhi

and the linear functions Λ_i (the fundamental weights) on hgiven by Λi(hj) =δi,j for all i, j∈Z.

Fori∈Z, setα_i= 2Λ_i−Λ_i−1−Λ_i+1. Denote by Π ={α_i, i∈Z}the set of simple roots and Π^∨={h_i, i∈Z}the set of simple coroots. LetP =L

i∈ZZΛ_ibe the weight lattice andP⁺=L

i∈ZNΛi the semigroup of dominant weights. LetQ=L

i∈IZαi⊆P (the root lattice) andQ⁺=L

i∈INαi ⊆Q. Forλ, µ∈h^∗, writeλ≥µifλ−µ∈Q⁺. Denote by W the Weyl group of type sl∞: it is the subgroup of transformations stabilizing P generated by the simple reflections si defined by si(λ) = λ−λ(hi)αi

(i∈Z, λ∈P).

2.2. Quantum group U_q(sl∞). In this article q = e^t ∈ C^∗ (t ∈ C) is not a root of unity and is fixed. Forl∈Z, r≥0, m≥m⁰ ≥0 we set

[l]q = q^l−q^−l

q−q⁻¹ ∈Z[q^±1], [r]q! = [r]q[r−1]q. . .[1]q, m

m⁰

q

= [m]q! [m−m⁰]_q![m⁰]_q!. Definition 2.1. The quantum group U_q(sl∞) is the C-algebra with generators k_h (h∈h), x^±_i (i∈Z) and relations

k_hk_h⁰ =k_h+h⁰, k₀ = 1, k_hx^±_j k−h =q^±α^j^(h)x^±_j , [x⁺_i , x⁻_j ] =δ_i,jk_i−k⁻¹_i

q−q⁻¹ ,

(x^±_i )⁽²⁾x^±_i+1−x^±_i x^±_i+1x^±_i +x^±_i+1(x^±_i )⁽²⁾ = 0, where we use the notationsk^±_i =k±h_i and for all r≥0, (x^±_i )^(r)= ^(x

± i)^r [r]q! .

For J = [a, b] ⊂Z denote by U_q(sl∞)_J the subalgebra of U_q(sl∞) generated by the x^±_i , kh fori∈J, h∈ ⊕_j∈[a,b]Qhj. Then U_q(sl∞)J is isomorphic to the quantum group U_q(slb−a+2).

LetU_q(sl∞)⁺(resp. U_q(sl∞)⁻,U_q(h)) be the subalgebra ofU_q(sl∞) generated by the x⁺_i (resp. the x⁻_i , resp the kh). We have a triangular decomposition ofU_q(sl∞) (see [21]):

Theorem 2.2. We have an isomorphism of vector spaces U_q(sl∞)' U_q(sl∞)⁻⊗ U_q(h)⊗ U_q(sl∞)⁺.

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2.3. Representations of U_q(sl∞). ForV a representation ofU_q(sl∞) andν ∈P, the weight spaceV_ν ofV is

V_ν ={v∈V|k_i·v=q^ν(hⁱ⁾v,∀i∈Z}.

Definition 2.3. A representation V is said to be in the categoryO if (i) it admits a weight space decomposition V =L

ν∈PV_ν, (ii) Vν is finite-dimensional for all ν,

(iii) {ν|V_ν 6={0}} ⊂ ∪_j=1,···_,N{ν|ν ≤λ_j} for some λ₁,· · · , λ_N ∈P.

For λ ∈ P, a representation V is said to be of highest weight λ if there is v ∈ V_λ such that for all i∈I, x⁺_i ·v= 0 and U_q(sl∞)·v=V. Such a representation is in the category O. Furthermore there is a unique simple highest weight module of highest weightλ.

Definition 2.4. A representation V is said to be integrable if (i) it admits a weight space decomposition V =L

ν∈PVν, (ii) all the x^±_i (i∈Z) are locally nilpotent.

Theorem 2.5. [21, 14] The simple highest weight module of highest weight λ is integrable if and only if λis dominant. It is denotedV(λ).

Now we recall the definition and some properties of extremal weight modules for the quantum groupU_q(sl∞) [16].

Definition 2.6. For an integrable U_q(sl∞)-module V and λ ∈ P, a vector v ∈ V_λ is called i-extremal if x⁺_i ·v= 0 or x⁻_i ·v= 0. In this case we set

S_i(v) = (x⁻_i )^(λ(hⁱ⁾⁾·v or S_i(v) = (x⁺_i )^(−λ(hⁱ⁾⁾·v.

A vectorv∈V_λis called extremal of weightλif, for anyl≥1,S_i₁· · ·S_i_l(v)isi-extremal for anyi, i1,· · ·, il ∈Z.

Forv an extremal vector of weightλ, we set

W ·v={S_i₁· · ·S_i_l(v)|l∈N, i₁,· · ·i_l∈Z}.

Definition 2.7. For λ ∈ P, the extremal weight module of extremal weight λ is the U_q(sl∞)-module generated by a vector v_λ with the defining relations thatv_λ is extremal of weightλ. It is denotedV(λ).

Example 2.8. If λ is dominant, V(λ) is the simple highest weight module of highest weightλ.

Theorem 2.9. [16] For λ∈P, the module V(λ) is integrable and has a crystal basis B(λ).

2.4. Quantum affinizationU_q( ˆsl∞). We recall the definition of the quantum affiniza- tionU_q( ˆsl∞) (without central charge) in terms of currents.

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Definition 2.10. The quantum affinization U_q( ˆsl∞) is the C-algebra with generators x^±_i,r (i∈ Z, r ∈ Z), k_h (h ∈ h), hi,m (i∈ Z, m∈ Z− {0}) and the following defining relations (i, j∈Z, h, h⁰ ∈h):

k_hk_h⁰ =k_h+h⁰, k₀ = 1,

φ^±_i (z)φ^±_j (w) =φ^±_j (w)φ^±_i (z) , φ⁻_i (z)φ⁺_j (w) =φ⁺_j (w)φ⁻_i (z), (1) (w−q^±Cîjz)φ⁺_i (z)x^±_j (w) = (q^±Cîjw−z)x^±_j (w)φ⁺_i (z), (2) (w−q^±Cîjz)φ⁻_i (z)x^±_j (w) = (q^±Cîjw−z)x^±_j (w)φ⁻_i (z),

[x⁺_i (z), x⁻_j (w)] = δi,j

q−q⁻¹(δ w z

φ⁺_i (w)−δ z w

φ⁻_i (z)), (z−q^±C^ijw)x^±_i (z)x^±_j (w) = (q^±a^ijz−w)x^±_j (w)x^±_i (z), x^±_i (z₁)x^±_i (z₂)x^±_i±1(w)−(q+q⁻¹)x^±_i (z₁)x^±_i±1(w)x^±_i (z₂)

+x^±_i±1(w)x^±_i (z₁)x^±_i (z₂) + (z₁ ↔z₂) = 0, and[x^±_i (z), x^±_j(w)] = 0for i6=j, j±1.

Here we use the formal power series δ(z) =P

s∈Zz^s and x^±_i (z) =X

r∈Z

x^±_i,rz^r,

φ^±_i (z) = X

m≥0

φ^±_i,±mz^±m=k_i^±1exp(±(q−q⁻¹) X

m⁰≥1

hi,±m⁰z^±m⁰).

There is an algebra morphism U_q(sl∞) → U_q( ˆsl∞) defined by k_h 7→ k_h, x^±_i 7→ x^±_i,0 (h∈h, i∈Z). Its image is called the horizontal quantum affine subalgebra of U_q( ˆsl∞) and is denoted byU_q^h( ˆsl∞). In particular, a U_q( ˆsl∞)-module V has also a structure of aU_q(sl∞)-module. We denote by Res(V) the restrictedU_q(sl∞)-module obtained from V.

For J = [a, b]⊆Z, let us denote by U_q( ˆsl∞)_J the subalgebra of U_q( ˆsl∞) generated by thex^±_i,r,k_h,hi,m (i∈J,r ∈Z,m∈Z− {0},h∈ ⊕_j∈JQhj). It is isomorphic to the quantum affine algebraU_q( ˆslb−a+2)⁰ [2, 5].

For i∈ Z, the subalgebra ˆU_i generated by the x^±_i,r, h_i,m, k_ph_i (r ∈ Z, m ∈ Z− {0}, p∈Q) is isomorphic to U_q( ˆsl₂)⁰.

We have a triangular decomposition of U_q( ˆsl∞).

Theorem 2.11. [24, 25]We have an isomorphism of vector spaces U_q( ˆsl∞)' U_q( ˆsl∞)⁻⊗ U_q(ˆh)⊗ U_q( ˆsl∞)⁺,

where U_q( ˆsl∞)^± (resp. U_q(ˆh)) is generated by the x^±_i,r (resp. the k_h, the h_i,m).

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2.5. Representations of U_q( ˆsl∞).

Definition 2.12. A representation V of U_q( ˆsl∞) is said to be integrable (resp. in the categoryO) ifRes(V)is integrable (resp. in the category O) as aU_q(sl∞)-module. One denotes byO_int the category of integrable representations ofU_q( ˆsl∞) in the category O.

Definition 2.13. A representation V of U_q( ˆsl∞) is said to be of `-highest weight if there is v∈V such that

(i) V =U_q( ˆsl∞)⁻·v, (ii) U_q(ˆh)·v=Cv,

(iii) for any i∈Z, r∈Z, x⁺_i,r·v= 0.

Such a representation is in the categoryO. For γ ∈Hom(U_q(ˆh),C), by Theorem 2.11 we have a corresponding Verma moduleM(γ) and a simple representationV(γ) which are`-highest weight.

Theorem 2.14. [14] The simple representationsV(γ)ofU_q( ˆsl∞)are integrable if there is(Pi)i∈Z∈(1 +uC[u])⁽^Z⁾ satisfying for i∈Z the relation in C[[z]] (resp. in C[[z⁻¹]])

γ(φ^±_i (z)) =q^deg(Pⁱ⁾Pi(zq⁻¹) P_i(zq) .

The polynomialsP_i are called Drinfeld polynomials and the representationV(γ) is then denoted byV((P_i)i∈Z).

The Kirillov-Reshetikhin module associated tok≥0,a∈C^∗and`∈Z, is the simple integrable representation defined by then–tuple

P_i(u) =

(1−ua)(1−uaq²)· · ·(1−uaq^2(k−1)) for i=`, 1 for i6=`.

Ifk= 1, it is also called fundamental module.

Theorem 2.15. [14] Let v be a `-highest weight vector of the simple U_q( ˆsl∞)-module V((Pi)i∈Z). Then

V((P_i)i∈Z) = [

n≥0

V_n((P_i)i∈Z) where V_n((P_i)i∈Z) =U_q( ˆsl∞)_[−n,n]·v.

Furthermore each Vn((Pi)i∈Z) is a simple finite-dimensional U_q( ˆsl∞)[−n,n]-module.

Consider an integrable representation V of U_q( ˆsl∞). As the subalgebra U_q(ˆh) is commutative, the weight spacesVν split in simultaneous generalized eigenspaces

V_ν = M

γ∈Hom(Uq(ˆh),C)

V_γ, where

Vγ={x∈V /∃p∈N,∀i∈Z,∀m≥0,(φ^±_i,±m−γ(φ^±_i,±m))^p·x= 0}.

IfV_γ 6={0},γ is called an`-weight of V andV_γ is an `-weight space.

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Definition 2.16. A U_q( ˆsl∞)-moduleV is weighted if the Cartan subalgebra U_q(ˆh) acts onV by diagonalizable operators. The module V is thin if it is weighted and the joint spectrum is simple.

Definition 2.17. [10, 12, 25] The q–character of an integrable representation V of U_q( ˆsl∞) with finite-dimensional`-weight spaces is defined by the formal sum

χq(V) = X

γ∈Hom(Uq(ˆh),C)

dim(Vγ)e(γ).

As in [7], we set

ψ(z) = q−q⁻¹z 1−z .

Proposition 2.18. [10, 14] Let V be an integrable representation of U_q( ˆsl∞). An

`-weight γ of V satisfies the property

(1) for all i∈Z, there exist k, l ∈N and a_1,i,· · ·, a_k,i, b_1,i,· · · , b_l,i ∈C^∗ such that in C[[z]] (resp. in C[[z⁻¹]]):

X

m≥0

γ(φ^±_i,±m)z^±m = Y

1≤u≤k

ψ(a_u,iqz) Y

1≤v≤l

ψ(b_v,iqz)⁻¹= Y

1≤u≤k

ψ(a_u,iqz) Y

1≤v≤l

ψ(b⁻¹_v,iqz⁻¹) Furthermore ifV is in the category O, then

(2) there exist ω∈P⁺, α∈Q⁺ satisfying ν=ω−α.

Consider formal variables Y_i,a^±1 (i∈Z,a∈C^∗) and letA be the group of monomials of the form

m= Y

i∈Z,a∈C^∗

Y_i,a^u^i,a^(m), u_i,a(m)∈Z. Set

Ai,a=Y_i,aq^l−1Y_i,aql+1Y_i−1,aq⁻¹ lY_i+1,aq⁻¹ l∈A.

A monomial m is said to be J-dominant (J ⊂ Z) if for all j ∈J and a∈C^∗ we have uj,a(m)≥0. A Z-dominant monomial is said to be dominant.

Let V be an integrable U_q( ˆsl∞)-module. For γ an `-weight of V, one defines the monomialm_γ =Q

i∈Z

Q

ai∈C^∗Y_i,a_iQ

bi∈C^∗Y_i,b⁻¹

i where X

m≥0

γ(φ^±_i,±m)z^±m =Y

ai

ψ(aiqz)Y

bi

ψ(biqz)⁻¹.

We denote Vγ = Vmγ. We rewrite the q–character of an integrable representation V with finite-dimensional`-weight spaces by the formal sum

χ_q(V) =X

m

dim(V_m)m∈Z^A.

Let us denote byM(V) the set of monomials occurring in χ_q(V).

By this correspondence between`-weights and monomials due to Frenkel-Reshetikhin [10], theZ-tuple of Drinfeld polynomials are identified with the dominant monomials.

In particular for a dominant monomialm, one denotes byV(m) the simple module of

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`-highest weight m. For example V(Y`,aY_`,aq². . . Y_`,aq^2(k−1)) is the Kirillov-Reshetikhin module associated to`∈Z,k≥0 and a∈C^∗, and V(Y_`,a) is the fundamental module associated to`∈Z anda∈C^∗.

In general no formula is known for the q–character of a `-highest weight module of U_q( ˆsl∞). But in the special case of Kirillov-Reshetikhin modules, the q–character can be explicitly given as a tableaux sum expression. For that we set for all a∈ C^∗ and j∈Z

j a=Y_j−1,aq⁻¹ jY_j,aq^j−1.

ForI, J ⊂Z, a tableaux (T_i,j)i∈I,j∈J is said to be semi-standard if it has coefficients in Zwhich increase relatively toj and strictly increase relatively toi.

Consider`∈Zandk≥0. LetT_`,kbe the set of semi-standard tableaux (T_i,j)i≤`,1≤j≤k

such that for any 1≤j≤k,

T_i,j =ifori <<0.

Theorem 2.19. [14] We have the following formula χ_q(V(Y_`,a· · ·Y_`,aq2(k−1))) = X

T∈T_`,k

m_T (3)

where m_T =Q

i≤`,1≤j≤k T^i,jaq^{`−1+2(j−i)}.

2.6. Extremal loop weight modules. The definition of extremal loop weight modules is given in [14] for the quantum toroidal algebras and studied in [22, 23]. We introduce its definition in the context of the quantum affinizationU_q( ˆsl∞).

Definition 2.20. An extremal loop weight module of U_q( ˆsl∞) of `-weight m is an integrable representation V such that there is v∈Vm satisfying

(i) U_q( ˆsl∞)·v=V,

(ii) v is extremal for U_q^h( ˆsl∞),

(iii) U_q( ˆsl∞)_J ·w is finite-dimensional for allw∈V andJ = [a, b]⊂Zfinite.

Proposition 2.21. Let m be a dominant monomial. Then the simple`-highest weight moduleV(m) is an extremal loop weight module of U_q( ˆsl∞).

Proof. Let v be a `-highest weight vector of V(m). Denote its weight by λ∈P⁺. As it is recalled above,V(m) = U_q( ˆsl∞)·v and is integrable. Furthermore v is a highest weight vector forU_q^h( ˆsl∞) which satisfies for all i∈Z,

(x⁻_i )^1+λ(hⁱ⁾·v= 0.

HenceU_q^h( ˆsl∞)·vis isomorphic to the simple highest weightU_q(sl∞)-moduleV(λ), and vis extremal of weight λfor the horizontal quantum affine subalgebra.

It remains to show that for all w∈V(m) and J = [a, b]⊂Z finite,U_q( ˆsl∞)_J·w is finite-dimensional. But there existsn∈Nsuch that J ⊂[−n, n] and

U_q( ˆsl∞)_J·w⊂ U_q( ˆsl∞)_[−n,n]·w⊂Vn(m) =U_q( ˆsl∞)_[−n,n]·v.

AsV_n((P_i)i∈Z) is finite-dimensional by Theorem 2.15, it is also the case ofU_q( ˆsl∞)_J·w.

So theU_q( ˆsl∞)-module V(m) is an extremal loop weight module of `-weight m.

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In this paper we construct extremal loop weight modules ofU_q( ˆsl∞) by fusion product of `-highest weight modules and `-lowest weight modules. This process is introduced in [23] in the toroidal case, and is motivated by the original study of extremal weight modules of quantum Kac-Moody algebras [16]: in fact to study these representations, tensor products of highest weight modules and lowest weight modules are considered (see [16, Lemma 8.2.1]).

2.7. Drinfeld coproduct. Let ∆ be the coproduct defined by (i∈Z) (4) ∆(x⁺_i (z)) =x⁺_i (z)⊗1 +φ⁻_i (z)⊗x⁺_i (z),

(5) ∆(x⁻_i (z)) =x⁻_i (z)⊗φ⁺_i (z) + 1⊗x⁻_i (z), (6) ∆(φ^±_i (z)) =φ^±_i (z)⊗φ^±_i (z).

Note that this map does not define a coproduct in the usual sense because (4) and (5) involve infinite sums and are not elements of U_q( ˆsl∞)⊗ U_q( ˆsl∞) (they take values in a completion of it). However it can be used to define a structure of U_q( ˆsl∞)-module on (a subspace of) tensor products ofU_q( ˆsl∞)-modules when the summations are well- defined.

Lemma 2.22. [7]Let V, W be U_q( ˆsl∞)-modules. LetU ⊆V ⊗W be a linear subspace such that for anyu∈U,∆(x⁺_i (z))·u is well-defined inU (i∈Z). Then the coproduct

∆endowsU with a structure ofU_q( ˆsl∞)-module, called the fusion product ofV andW. Lemma 2.23. Assume further that V and W satisfies U_q( ˆsl∞)J ·v and U_q( ˆsl∞)J·w are finite-dimensional for all v ∈ V, w ∈ W and all J = [a, b] ⊂ Z finite in the last lemma. Then the U_q( ˆsl∞)-moduleU satisfies also

U_q( ˆsl∞)_J ·u is finite-dimensional for all u∈U and J = [a, b]⊂Z finite.

In particular if V andW are integrable, U is an integrableU_q( ˆsl∞)-module.

Proof. Set J = [a, b]⊂Z finite and let u =v⊗W ∈ V ⊗W. By the formulas of the coproduct, we have

U_q( ˆsl∞)J ·u⊂

U_q( ˆsl∞)J ·v

⊗

U_q( ˆsl∞)J·v

.

The result follows.

3. Extremal fundamental loop weight modules

In this section, we define a U_q( ˆsl∞)-module structure on the tensor product of fundamental modules

V(Y_`,a)⊗V(Y_0,aq⁻¹_`) with`≥1 anda∈C^∗

by using the Drinfeld coproduct ∆. We obtain in that way new integrable modules V(Y_`,aY_0,aq⁻¹_`) with basis labelled by semi-standard tableaux of shape (`). We show in particular that these modules are`-extremal. We call them the extremal fundamental loop weight modules ofU_q( ˆsl∞).

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The fusion product process used here is inspired to the one in [23] for the toroidal case.

As it is said in the introduction, the representation theory ofU_q( ˆsl∞) and of quantum toroidal algebras are very different: the fundamentalU_q( ˆsl∞)-modules are thin, which is not the case for the quantum toroidal algebrasU_q(sl^tor_n+1) (see [14, Section 4.1]). Let us point out that these differences have consequences in our work: we obtain here more precise results for the fusion product ofU_q( ˆsl∞)-modules. Another difficulty met in [23]

is that the cyclicity of the Dynkin diagram associated toU_q(sl_n+1^tor ) bring some restricted conditions in the construction of extremal loop weight modules (see [23, Proposition 3.11]). We will see that it is possible here to associate extremal fundamental loop weightU_q( ˆsl∞)-modules for all the nodes of the Dynkin diagram of type A∞.

In the first part, we give explicit formulae of the action on the fundamental U_q( ˆsl∞)-modules, using the U_q(sl∞)-crystal structure on the set of semi-standard tableaux. In the second part, we construct the fusion product of theU_q( ˆsl∞)-modules V(Y`,a) andV(Y_0,aq⁻¹`), the action of U_q( ˆsl∞) being defined via the Drinfeld coproduct (Proposition 3.2). We consider a submodule V(Y`,aY_0,aq⁻¹`) of it, which we call the extremal fundamental loop weight module. We study theU_q( ˆsl∞)-modulesV(Y`,aY_0,aq⁻¹`) in the third part: these representations are integrable with basis labelled by semi- standard tableaux of shape (`). The action of U_q( ˆsl∞) on it is described by the U_q(sl∞)-crystal structure on the set of semi-standard tableaux of shape (`) (Theo- rem 3.10). Furthermore V(Y_`,aY_0,aq⁻¹_`) is an extremal loop weight module of `-weight Y_`,aY_0,aq⁻¹_` (Theorem 3.14) which is irreducible as a U_q( ˆsl∞)-module and as a U_q^h( ˆsl∞)- module (Proposition 3.12).

3.1. Action on the fundamental U_q( ˆsl∞)-modules. In this section we give explicit formulae of the action on the fundamentalU_q( ˆsl∞)-modulesV(Y`,a) (`∈Z, a∈C^∗).

Let T_`⁺=T_`,1 be the set of semi-standard tableaux

T = (· · ·< i`−2 < i`−1< i_`) such thati_j =j forj <<0.

We endow T_`⁺ with a structure of U_q(sl∞)-crystal: the weight function is defined by setting

wt(T) =X

j≤`

Λij−Λij−1

for allT = (· · ·< i`−2 < i`−1< i`)∈ T_`⁺. The Kashiwara operators ˜ei, ˜fi are described in terms of tableaux: fori∈Zwe have ˜ei·T =T⁰ or 0. HereT⁰ is obtained from T by replacingi+ 1 by i. If it is not possible (i.e. when we have both i+ 1 and i in T or wheni+ 1 does not occur inT), then it is zero. Similarly ˜f_i·m_T_;j =m_T⁰⁰_;j or 0, where T⁰⁰ is given by replacing iby i+ 1.

Proposition 3.1. Let `∈Zanda∈C^∗ and consider the fundamentalU_q( ˆsl∞)-module V(Y_`,a). Then there exists a basis of V(Y_`,a) indexed by T_`⁺ such that the action of U_q( ˆsl∞) is given by (for convenience in the calculations, we remind the complex number

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a∈C^∗ in subscript) x⁺_i (z)·T_a = X

p

δ_{i_p_=i+1}δ(aq^i+`+1−2pz)(˜e_i·T)_a, x⁻_i (z)·T_a = X

p

δ_{i_p_=i}δ(aq^i+`+1−2pz)( ˜f_i·T)_a,

φ^±_i (z)·Ta =











ψ(aq^i+`+3−2pz)⁻¹·Ta

if there exists p≤` such that i_p =i+ 1 andip−1 6=i, ψ(aq^i+`+1−2pz)·T_a if there exists p≤` such that

if ip =iand ip+1 6=i+ 1,

Ta otherwise.

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where T = (· · ·< i`−2 < i`−1 < i_`) is a semi-standard tableaux in T_`⁺. In particular for allT ∈ T_`⁺,T_a is an `-weight vector of weight m_T.

Proof. This result is known for the fundamental U_q( ˆsl_n+1)-modules (see [22, Theorem 4.11]). We obtain the result forU_q( ˆsl∞) by inductive limit, using Theorem 2.15.

We have an analogue result for the`-lowest weight modules. For that letT_s⁻ (s∈Z) be the set of semi-standard tableaux

T = (i_s+1 < i_s+2< i_s+3<· · ·) such that i_j =j forj >>0.

Then (T_s⁻,wt,e˜i,f˜i) is a U_q(sl∞)-crystal. And for all a ∈ C^∗, there exists a basis of V(Y_s,a⁻¹) indexed by T_s⁻ such that the action of U_q( ˆsl∞) on it is given for all T = (is+1 < is+2 < is+3<· · ·)∈ T_s⁻ by

x⁺_i (z)·Ta = X

p

δ_{i_p_=i+1}δ(aq^i+s+1−2pz)(˜ei·T)a, x⁻_i (z)·Ta = X

p

δ_{i_p_=i}δ(aq^i+s+1−2pz)( ˜fi·T)a,

φ^±_i (z)·Ta =











ψ(aq^i+s+3−2pz)⁻¹·Ta

if there existsp≥s+ 1 such that ip =i+ 1 andip−1 6=i, ψ(aq^i+s+1−2pz)·T_a if there existsp≥s+ 1 such that

ifip=iandip+1 6=i+ 1,

Ta otherwise.

3.2. Fusion product of fundamental modules.

Proposition 3.2. Let`≥1anda∈C^∗. The coproduct∆is well-defined on the tensor product V(Y_`,a) ⊗ V(Y_0,aq⁻¹_`) and it endows it with a structure of integrable U_q( ˆsl∞)-module.

Remark 3.3. We have introduced a fusion product process in[23, Section 3] to define new integrable modules for the quantum toroidal algebras. This process can also be used in our situation to show Proposition 3.2. We give here a direct proof of it for the

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convenience of the reader. As it is pointed out above, the computations are simpler in that case, stemming from the thin property of the fundamentalU_q( ˆsl∞)-modules.

Proof. Let us consider

T = (· · ·< i`−2 < i`−1< i`)∈ T_`⁺ and T⁰ = (j1 < j2 < j3<· · ·)∈ T₀⁻. We study the action ofx⁺_i (z) on T_a⊗T_aq⁰ ` ∈V(Y_`,a)⊗V(Y_0,aq⁻¹_`):

x⁺_i (z)·(Ta⊗T_aq⁰ `) =P

rδ_{i_r_=i+1}δ(aq^i+`+1−2rz)(˜ei·T)a⊗T_aq⁰ `+ X

r,s

h

1 +δ_{i_r_=i+1}δ_{i_r−1_6=i}

ψ(q^2(s−r)+2)⁻¹−1

+δ_{i_r_=i}δ_{i_r+1_6=i+1}

ψ(q^2(s−r))−1i

×δ_{j_s_=i+1}δ(aq^i+`+1−2rz)T_a⊗(˜e_i·T⁰)_aq`.

By definition of T_`⁺ and T₀⁻, we have r ≤ i, s ≥ i+ 1, and s−r ≥ 1. Hence the coefficients of the series x⁺_i (z)·(Ta⊗T_aq⁰ `) are well-defined. The result follows by

Lemma 2.22.

Let us fix `∈Z. ForT = (· · ·< i`−2 < i`−1 < i_`) a semi-standard tableaux in T_`⁺, we define

αT = sup{p≤`|i_p =p}.

Note that αT is well-defined by definition of T_`⁺. In the same way, we define for all T = (i1 < i2< i3 <· · ·)∈ T₀⁻

βT = inf{p≥1|i_p=p}.

Assume that ` ≥ 1. We have endowed the sets T_`⁺ and T₀⁻ with a structure of U_q(sl∞)-crystal. By the tensor product rules (see [16, 17]), T_`⁺ ⊗ T₀⁻ is a U_q(sl∞)-crystal. Let us denote byT_` the subset ofT_`⁺⊗ T₀⁻ consisting of

T⊗T⁰ withT ∈ T_`⁺, T⁰ ∈ T₀⁻ and αT ≥βT⁰−1.

Proposition 3.4. The setT_`is the connected sub-U_q(sl∞)-crystal ofT_`⁺⊗T₀⁻generated by (· · ·< `−2< `−1< `)⊗(1<2<3<· · ·).

Proof. SetT ⊗T⁰ ∈ T_` and i∈Z. Let us show that ˜e_i·(T ⊗T⁰)∈ T_`∪ {0} (the proof for the Kashiwara operators ˜f_i is analogous). We have the following equalities

α_e_˜_i·T =

(α_T if ˜e_i·T⁰ 6= 0 and i > α_T + 1, αT + 1 ifi=αT + 1,

β_e_˜_i·T⁰ = (

β_T⁰ if ˜ei·T⁰ 6= 0 andi < β_T⁰ −1, β_T⁰+ 1 ifi=β_T⁰ −1.

In particular α_e_˜_i·T ≤ α_T. So ˜e_i·(T ⊗T⁰) belongs to T_`∪ {0}, except perhaps when αT =βT⁰ −1 =i. But in that case, we have by the tensor product rules

˜

ei·(T⊗T⁰) = (˜ei·T)⊗T⁰= 0∈ T_`∪ {0}.

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Hence T_` is a sub-U_q(sl∞)-crystal ofT_`⁺⊗ T₀⁻. And one can show by straightforward computations that it is the connected sub-U_q(sl∞)-crystal of T_`⁺⊗ T₀⁻ generated by

(· · ·< `−2< `−1< `)⊗(1<2<3<· · ·).

Denote by V(Y_`,aY_0,aq⁻¹_`) (` ≥ 1, a ∈ C^∗) the subvector space of V(Y_`,a)⊗V(Y_0,aq⁻¹_`) generated byT_a⊗T_aq⁰ ` withT ⊗T⁰ ∈ T_`.

Proposition 3.5. The coproduct ∆ endows V(Y_`,aY_0,aq⁻¹_`) with a structure of U_q( ˆsl∞)-module. We call it the extremal fundamental loop weight module.

Proof. The action of x^±_i (z) (i∈Z) is well-defined on V(Y`,aY_0,aq⁻¹`) by Proposition 3.2.

By Lemma 2.22, it remains to show that x⁺_i (z)·(Ta ⊗T_aq⁰ `) ∈ V(Y`,aY_0,aq⁻¹`) for all T⊗T⁰ ∈ T_` (the proof being similar for x⁻_i (z)). As in the last proof, it holds except perhaps whenαT =βT⁰ −1 =i. In that case we have

x⁺_i (z)·Ta⊗T_aq⁰ ` =ψ(q²)δ(aq^`−i−1z)·Ta⊗(˜ei·T⁰)_aq`= 0∈V(Y_`,aY_0,aq⁻¹`) Hence the coproduct ∆ endowsV(Y`,aY_0,aq⁻¹`) with a structure ofU_q( ˆsl∞)-module.

3.3. Study of the extremal fundamental loop weight module V(Y`,aY_0,aq⁻¹`). Let T_[1,`] be the set of semi-standard tableaux T = (i₁ < i₂ < · · · < i_`) of shape (`). We consider the application Π :T_`→ T_[1,`]defined for all T⊗T⁰ ∈ T_` by

Π(T⊗T⁰) = (j1< j2 <· · ·< jαT < iαT+1 <· · ·< i`)

whereT = (· · ·< i`−2 < i`−1 < i`) and T⁰ = (j1 < j2 < j3 <· · ·). This application is bijective, its inverse sendingT = (i1 < i2 <· · ·< i_`)∈ T_[1,`] on

(· · ·<0<max(i1,1)<· · ·<max(i`, `))⊗(min(i1,1)<· · ·<min(i`, `)< `+ 1<· · ·).

Proposition 3.6. The application Π : T_` → T_[1,`] is an isomorphism of U_q(sl∞)- crystals.

Proof. LetT = (· · ·< i`−2< i`−1 < i_`) andT⁰ = (j₁ < j₂< j₃ <· · ·) be semi-standard tableaux such thatT⊗T⁰ ∈ T_`, and i∈Z. We have to consider the following cases

- i > α_T: then we haveβ_T⁰ ≤iandi, i+ 1∈T⁰. By the tensor product rules, we have

f˜i·(T ⊗T⁰) = ( ˜fi·T)⊗T⁰. Hence,

Π( ˜f_i·(T ⊗T⁰)) =







(j1 <· · ·< ip+ 1<· · ·< i`) if there existsp > αT such that ip=iandip+1 6=i+ 1, 0 otherwise

= ˜fi·Π(T ⊗T⁰)

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- i < αT: then we have i, i+ 1∈T and by the tensor product rules, f˜i·(T ⊗T⁰) =T ⊗( ˜fi·T⁰).

Furthermore,j_α_T ≤j_α_T₊₁−1 =α_T. Hence,

Π( ˜f_i·(T ⊗T⁰)) =







(j1 <· · ·< jp+ 1<· · ·< i`) if there exists 1≤p < αT such that ip=iand ip+16=i+ 1, 0 otherwise

= ˜fi·Π(T ⊗T⁰)

- i=α_T: then we have i+ 1∈/ T and β_T⁰ ≤i+ 1.

Assume thatβ_T⁰ < i+ 1. In that case, i, i+ 1∈T⁰ and we have f˜i·(T ⊗T⁰) = ( ˜fi·T)⊗T⁰.

Furthermore α_f_˜

i·T =α_T −1 andj_α_T =i. So we have

Π( ˜f_i·(T ⊗T⁰)) = (j₁ <· · ·< j_α_T−1 < i+ 1< i_α_T₊₁ <· · ·< i_`)

= ˜f_i·(j₁ <· · ·< j_α_T−1 < i=j_α_T < i_α_T₊₁ <· · ·< i_`)

= ˜f_i·Π(T ⊗T⁰).

Ifβ_T⁰ =i+ 1, we havei /∈T⁰. By the tensor product rules we get f˜_i·(T ⊗T⁰) =T ⊗( ˜f_i·T⁰) = 0.

Furthermore we have αT = β_T⁰ − 1 and jαT < αT = i < iαT+1. Then i /∈Π(T ⊗T⁰) and ˜fi·Π(T ⊗T⁰) = 0.

As the application Π : T_` → T_[1,`] is also bijective, this is an isomorphism of

U_q(sl∞)-crystals.

Remark 3.7. Let us describe this isomorphism at the level of the monomial crystals (see [19, 22, 26] for its definition). For that denote by M_` the sub-U_q(sl∞)-crystal of M(Y_`,a)⊗ M(Y⁻¹

0,aq^`) generated by Y_`,a⊗Y_0,aq⁻¹`. Let us consider also M(Y_`,aY_0,aq⁻¹`) the monomial crystal generated byY_`,aY_0,aq⁻¹`. By that we have done above, we have

M(Y_`,aY_0,aq⁻¹`) ={m_T = Y

1≤j≤`

ij

aq^`+1−2j |T = (i1< i2 <· · ·< i`)∈ T_[1,`]} and

M_` ={m_T ⊗m_T⁰|T ⊗T⁰ ∈ T_`}.

In the monomial realization,Π is simply given by the product in A Π(m₁⊗m₂) =m₁×m₂ withm₁⊗m₂ ∈ M_`. As a direct consequence, we have

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Proposition 3.8. Let ` ≥ 1 and a∈ C^∗. The U_q( ˆsl∞)-module V(Y_`,aY_0,aq⁻¹_`) is thin, and we have

(8) χ_q(V(Y_`,aY_0,aq⁻¹_`)) = X

T∈T_[1,`]

m_T

where m_T = Y

1≤j≤`

ij

aq^`+1−2j.

Proof. Let us recall that we have a basis of`-weight vectors ofV(Y_`,aY_0,aq⁻¹`): it is given by theT_a⊗T_aq⁰ ` in T_`. Furthermore by (6) and the remark above, the`-weight of the vectorTa⊗T_aq⁰ ` is

m_T ×m_T⁰ =m_Π(T_⊗T⁰₎. Hence we have

χ_q(V(Y_`,aY_0,aq⁻¹_`)) = X

T∈T_[1,`]

m_T.

Remark 3.9.In particular we have shown here that the monomial crystalM(Y_`,aY_0,aq⁻¹`) generated byY_`,aY_0,aq⁻¹` is closed in the sense of [22, Definition 3.6].

By the bijection Π, we parametrize the basis of V(Y_`,aY_0,aq⁻¹`) by the U_q(sl∞)-crystal T_[1,`]. Let us explicit formulae of the action on it.

Theorem 3.10. The action of U_q( ˆsl∞) on V(Y_`,aY_0,aq⁻¹`) is given for all T = (i1 < i2<· · ·< i`)∈ T_[1,`] by

x⁺_i (z)·T_a = X

1≤p≤`

δ_{i_p_=i+1}δ(aq^i+`+1−2pz)(˜e_i·T)_a, x⁻_i (z)·Ta = X

1≤p≤`

δ{i_p=i}δ(aq^i+`+1−2pz)( ˜fi·T)a,

φ^±_i (z)·Ta =











ψ(aq^i+`+3−2pz)⁻¹·Ta

if there exists 1≤p≤` such that ip =i+ 1 andip−1 6=i, ψ(aq^i+`+1−2pz)·T_a if there exists 1≤p≤` such that

if i_p =iand i_p+1 6=i+ 1,

Ta otherwise.

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Remark 3.11. Note that these formulae are identical to the one (7) obtained for the fundamental U_q( ˆsl∞)-modules. Actually this result generalizes [22, Theorem 4.11] for the extremal fundamental loop weight modules of U_q(sl^tor_n+1). In fact one can rewrite the action on V(Y`,aY_0,aq⁻¹`) in the context of monomial realizations: it is completely describe by the monomialU_q(sl∞)-crystal M(Y_`,aY_0,aq⁻¹`) and the formulae given in [22, Theorem 4.11].