Representations of Quantum Affine Superalgebras

We study the quantum affine superalgebra $U_q(Lsl(M,N))$ and its finite-dimensional representations. We prove a triangular decomposition and establish a system of Poincar\'{e}-Birkhoff-Witt generators for this superalgebra, both in terms of Drinfel'd currents. We define the Weyl modules in the spirit of Chari-Pressley and prove that these Weyl modules are always finite-dimensional and non-zero. In consequence, we obtain a highest weight classification of finite-dimensional simple representations when $M \neq N$. Some concrete simple representations are constructed via evaluation morphisms.


Introduction
In this paper q ∈ C \ {0} is not a root of unity and our ground field is always C. We study a quantized version of the enveloping algebra of the affine Lie superalgebra Lsl(M, N ), which we denote by U q (Lsl(M, N )). Some properties of U q (Lsl(M, N)). For M, N ∈ Z ≥1 , the quantum affine superalgebra U q (Lsl(M, N )) is defined in terms of Drinfel'd currents. It is the superalgebra with (1) Drinfel'd generators X ± i,n , h i,s , K ±1 i for 1 ≤ i ≤ M + N − 1, n ∈ Z, s ∈ Z =0 ; (2) Z 2 -grading |X ± M,n | = 1, and |X ± i,n | = |K ±1 i | = |h i,s | = |K ±1 M | = |h M,s | = 0 for i = M ; (3) defining relations (.)-(.) (see section 3.1 for details). Informally, when q = 1, U q (Lsl(M, N )) becomes the universal enveloping algebra of the Lie superalgebra Lsl(M, N ) sl(M, N ) ⊗ C[t, t −1 ] with the convention that Let U + q (Lsl(M, N )) (resp. U − q (Lsl(M, N )), resp. U 0 q (Lsl(M, N ))) be the subalgebra of U q (Lsl(M, N )) generated by the X + i,n (resp. the X − i,n , resp. K ±1 i , h i,s ). Then the Chevalley relations imply that and that U 0 q (Lsl(M, N )) is a commutative algebra. When M = N , U q (Lsl(M, N )) has a Chevalley presentation [Ya99, Theorem 6.8.2], which endows U q (Lsl(M, N )) with a Hopf superalgebra structure. Using the coproduct, we can form the tensor product of two representations of U q (Lsl(M, N )). Note however that the coproduct formulae for X ± i,n , h i,s are highly non-trivial.
Backgrounds. In analogy with the applications of quantum affine algebras in solvable lattice models [JM99], quantum affine superalgebras also appear as the algebraic supersymmetries of some solvable models. In [BT08], the quantum affine superalgebra U q (Lsl(2, 1)), together with its universal R-matrix, which exists in the framework of Khoroshkin-Tolstoy [KT91,KT92], was used to define the Q-operators and to deduce their functional relations. These Q-operators were then applied in integrable models of statistic mechanics (3-state gl(2, 1)-Perk-Schultz model) and the associated quantum field theory. Here the functional relations come essentially from the tensor product decompositions of representations of U q (Lsl(2, 1)) and its Borel subalgebras. When M = N , U q ( sl(M, N )) (extended U q (Lsl(M, N )) with derivation) is the quantum supersymmetry analogue of the supersymmetric t − J model (with or without a boundary). A key problem is to diagonalize the commuting transfer matrices. In [Ko13] for example, Kojima proposed a construction of the boundary state using the machinery of algebraic analysis method. There to obtain the bosonization of the vertex operators [KSU97], one needs to work in some highest weight Fock representations of U q ( sl(M, N )).
In the case M = N = 2, the Lie superalgebra sl(2, 2) admits a two-fold non-trivial central extension. In [BK08], using the quantum deformation of this centrally extended algebra and its fundamental representations (which exist infinitely), Beisert-Koroteev deduced Shastry's spectral R-matrix R(u, v). Also it is found [Bei06] that the S-matrix of AdS/CFT enjoys a symmetry algebra : the conventional Yangian associated to the centrally extended algebra. Later, [BGM12] derived a quantum affine superalgebra Q depending essentially on two parameters, together with a Hopf superalgebra structure and its fundamental representations of dimension 4. This algebra is interesting itself as it is explained there to have two conventional "limits" : one is U q ( sl(2, 2)) ; the other is the Yangian limit. The two limiting processes carry over to the fundamental representations. Higher representations of this algebra are however still missing.
It is therefore worthwhile to study quantum (affine) superalgebras, Yangians, and their representations. For a symmetrizable quantum affine superalgebra U q (g), early in 1997, Ruibin Zhang has classified integrable irreducible highest weight representations [Zh97](here being symmetrizable excludes the existence of simple isotopic odd roots). Recently, in [HW12,KKO13], the authors obtained a (super)categorification of some quantum symmetrizable Kac-Moody superalgebras and their integrable highest weight modules from quiver Hecke superalgebras. However, the affine Lie superalgebras Lsl(M, N ) are not symmetrizable, as they contain simple isotopic odd roots. It is desirable to study U q (Lsl(M, N )) and their representations.
In the paper [Zh96], Zhang considered the gl(M, N ) super Yangian and its finite-dimensional representations. The super Yangian Y (gl(M, N )) can be viewed as a deformation of the universal enveloping superalgebra U (gl(M, N ) ⊗ C[t]). Zhang equipped the super Yangian with a Hopf superalgebra structure and wrote explicitly a Poincaré-Birkhoff-Witt (PBW for short) basis. From this PBW basis one reads a triangular decomposition. Zhang proved that all finite-dimensional representations of Y (gl(M, N )) are of highest weight with respect to this triangular decomposition, and parametrised these highest weights by polynomials (see section 6 below). The aim of this paper is to develop a similar highest weight representation theory for some quantum affine superalgebras.
We remark that Zhang's proof of the classification result relied on the coproduct structure ∆ and on some superalgebra automorphisms φ s of the super Yangian. For the quantum affine superalgebra U q (Lsl(M, N )) defined in terms of Drinfel'd generators, the coproduct structure is highly non-trivial (its existence is not clear a priori), and we do not have the analogue of the automorphisms φ s . To overcome such difficulties we propose the PBW argument in this paper, which is independent of coproduct structures.
Main results. In this paper, we study finite-dimensional representations of the quantum affine superalgebras Furthermore, the three subalgebras above admit presentations as superalgebras.
With respect to this triangular decomposition, we can define the Verma modules M(Λ), which are parametrised by the linear characters Λ on U 0 q (Lsl(M, N )), and are isomorphic to U − q (Lsl(M, N )) as vector superspaces. These Verma modules are important as it is shown that when M = N , all finite-dimensional simple U q (Lsl(M, N ))modules are their quotients up to modification by one-dimensional modules. We are led to consider the existence of finite-dimensional non-zero quotients of M(Λ), the so-called modules of highest weight Λ.
Let V be a finite-dimensional quotient of M(Λ), with a non-zero even highest weight vector v Λ . When 1 ≤ i ≤ M + N − 1 and i = M , the subalgebra U i generated by X ± i,n , K i , h i,s for n ∈ Z, s ∈ Z =0 is isomorphic to U q i (Lsl 2 ). As U i v Λ is finite-dimensional, from the highest weight representation theory of U q (Lsl 2 ) we conclude that there exists a Drinfel'd polynomial P i ∈ 1 + zC[z] such that (see section 2.1 below for the φ ± i,n ) On the other hand, the subalgebra U M is no longer U q (Lsl 2 ), but a superalgebra with simpler structure. As U M v Λ is finite-dimensional, we can eventually find another Drinfel'd polynomial then we see that The linear character Λ is completely determined by P = (P i ) and (f, c) in view of equations (.)-(.). We come out with the set R M,N of highest weights consisting of Λ = (P , f, c) such that there exists Q(z) satisfying relations (.)-(.). For such (Λ, Q), motivated by the theory of Weyl modules for quantum affine algebras [CP01], we define the Weyl module W(Λ; Q) as the quotient of M(Λ) by relations (.)-(.). Hence all finite-dimensional non-zero quotients of M(Λ), if exist, should be quotients of W(Λ; Q) for some Q. The sufficiency of restrictions (.)-(.) on the linear characters is guaranteed by Theorem 4.5. For all Λ = (P , f, c) ∈ R M,N and Q ∈ 1 + zC[z] such that Qf = 0, deg Q < dim W(Λ; Q) < ∞.
In consequence, when M = N , as remarked above, up to modification by some one-dimensional modules, finite-dimensional simple U q (Lsl(M, N ))-modules are parametrised by their highest weights Λ ∈ R M,N .
The first inequality deg Q < dim W(Λ; Q) comes from a detailed analysis of some weight subspaces of W(Λ; Q), using firmly the triangular decomposition theorem 3.3. Indeed, we shall see that the U q (Lsl(M, N ))-module structure on W(Λ; Q) determines the parameter (Λ; Q) uniquely, which justifies the definition of a highest weight. For the proof of W(Λ; Q) being finite-dimensional, we argue by induction on (M, N ) (this explains the reason for considering also M = N ). We use a system of linear generators for the vector superspace U q (Lsl(M, N )), the so-called PBW generators, to control the size of the Weyl modules. To be more precise, denote by the set of positive roots of the Lie superalgebra sl(M, N ) with the ordering : N )) as quantum brackets in such a way (definition 3.11) that finally The proof of the PBW theorem above is a combinatorial argument by inductions on (M, N ) and on the length of weights. We have not considered the problem of linear independence, which is beyond the scope of this paper.
We remark that equation (.) is by no means superficial. Indeed, for Λ ∈ R M,N , the quotient of M(Λ) by relation (.), noted W(Λ), is infinite-dimensional. We call W(Λ) the universal Weyl module in the sense that all integrable quotients of M(Λ) remain quotients of W(Λ). In particular, contrary to the case of quantum affine algebras, integrable highest weight finite-dimensional highest weight.
The paper is organised as follows. In section 2, we remind the notion of a Weyl module for the quantum affine algebra U q (Lsl N ), and that of a Kac module for the quantum superalgebra U q (gl(M, N )). In section 3, we define the quantum affine superalgebra U q (Lsl(M, N )) and its enlargement U q (L ′ sl(M, N )) in terms of Drinfel'd currents, following Yamane [Ya99]. Here, the enlargement is needed to avoid the problem of linear dependence among the simple roots of sl(M, N ). We prove a triangular decomposition (theorem 3.3) in terms of Drinfel'd currents, following the argument of [Ja96,He05]. Then we define the root vectors (definition 3.11) and prove the PBW theorem 3.12. In section 4, the notion of a highest weight, the Verma modules M(Λ), the Weyl modules W(Λ; Q), and the relative simple modules S ′ (Λ) are defined. We prove that the Weyl modules are always finite-dimensional and non-zero (theorem 4.5) by using the triangular decomposition and the PBW theorem. When M = N , we conclude the highest weight classification of finite-dimensional simple U q (Lsl(M, N ))-modules (proposition 4.15). The universal Weyl modules are introduced to study integrability property.
In section 5, we recall Yamane's isomorphism (theorem 5.2) between Drinfel'd and Chevalley presentations for U q (Lsl(M, N )) in the case M = N . From this isomorphism, we deduce a Hopf superalgebra structure on U q (Lsl(M, N )), a formula for the highest weight of the tensor product of two highest weight vectors (corollary 5.5), and henceforth a commutative monoid structure on the set R M,N of highest weight. From Zhang's evaluation morphisms (proposition 5.6) we construct explicitly some simple U q (Lsl(M, N ))-modules (proposition 5.9). Section 6 is left to further discussions. We include in the two appendixes the related calculations that are needed in the triangular decomposition and the coproduct formulae for some Drinfel'd currents. Acknowledgement : the author is grateful to his supervisor David Hernandez for the discussions, to Marc Rosso for kindly giving his preprint [Ros], and to Xin Fang, Jyun-Ao Lin and Mathieu Mansuy for the discussions and for pointing out useful references.

Preliminaries
We recall the highest weight representation theories for the quantum affine algebra U q (Lsl N ) and the quantum superalgebra U q (gl(M, N )). Here we use gl instead of sl to avoid the problem of linear dependence among simple roots when M = N (see notation 3.10).
2.1 Weyl modules for the quantum affine algebra U q (Lsl N ) Following Drinfel'd, the quantum affine algebra U q (Lsl N ) is an algebra with [Be94a, Theorem 4.7] : (a) generators Note that U q (Lsl N ) has a structure of Hopf algebra from its Chevalley presentation. [CP01,section 4], or the review [CH10, section 3.4] where we borrow the notations) : Let V be an U q (Lsl N )-module. We say that V is integrable if the actions of X ± i,0 for 1 ≤ i ≤ N − 1 are locally nilpotent. We say that V is of highest weight Λ if V is generated by a vector v satisfying relations (.)-(.).
We reformulate [CP95,Theorem 3.3] and [CP01, Proposition 4.6] in the case of sl N as follows.
(c) All integrable modules of highest weight Λ are quotients of W(Λ), in particular, they are finite-dimensional.
The Weyl modules W(Λ) are generally non-simple, due to the non-semi-simplicity of the category of finitedimensional U q (Lsl N )-modules, a phenomenon that appears also in the classical case U (Lsl N ).
2.2 Kac modules for the quantum superalgebra U q (gl(M, N )) From this section on, we consider superalgebras. By definition, a superalgebra is an (associative and unitary) algebra A with a compatible Z 2 -grading We remark that a superalgebra can be defined by a presentation : generators, their Z 2 -degrees, and their defining relations.
Given a vector superspace V = V 0 ⊕ V 1 , we endow the algebra of endomorphisms End(V ) with the following canonical superalgebra structure : By a representation of a superalgebra A, we mean a couple (ρ, V ) where V is a vector superspace and ρ : A −→ End(V ) is a homomorphism of superalgebras. Call V an A-module in this case. When A is a Hopf superalgebra, given two representations (ρ i , V i ) i=1,2 , we can form another representation ((ρ 1 ⊗ρ 2 )∆, V 1 ⊗V 2 ). Here ⊗ means the super tensor product and ∆ : A −→ A⊗A is the coproduct.
On the other hand, a Lie superalgebra is by definition a vector superspace When A is a superalgebra, [a, b] = ab − (−1) ij ba for a ∈ A i , b ∈ A j makes A into a Lie superalgebra. In particular, when V is the vector superspace with V 0 = C M and V 1 = C N , we write End(V ) as gl(M, N ) to emphasis its Lie superalgebra structure. There is a super-trace on gl(M, N ) given by And sl(M, N ) ker(str) is a sub-Lie-superalgebra of gl(M, N ). We refer to [Ka77,Sc79] for the classification of finite-dimensional simple Lie superalgebras in terms of Dynkin diagrams and Cartan matrices.
Zǫ i with the following bilinear from We remark that the subalgebra U q (sl(M, N )), generated by The following lemma is needed later.
Let Λ ∈ S M,N . The Kac module, K(Λ), is the U q (gl(M, N ))-module generated by v Λ , with Z 2 -grading 0, and relations [Zh93, section 3] We call it Kac module as it is a generalisation of Kac's induction module construction for Lie superalgebras [Ka78, Proposition 2.1]. Note that we also have the notion of integrable modules (actions of the e ± j being locally nilpotent) and highest weight modules. The Kac modules in the category of finite-dimensional U q (gl(M, N ))-modules play the same role as the Weyl modules in that of finite-dimensional U q (Lsl N )-modules.
(b) All finite-dimensional simple U q (gl(M, N ))-modules are of the form L(Λ)⊗C θ where Λ ∈ S M,N and C θ is a one-dimensional U q (gl(M, N ))-module.
(c) All integrable modules of highest weight Λ are quotients of K(Λ), in particular, they are finite-dimensional.
3 The quantum affine superalgebra U q (Lsl(M, N )) In this section, we recall the Drinfel'd realization of the quantum affine superalgebra U q (Lsl(M, N )) following Yamane [Ya99, Theorem 8.5.1]. We prove a triangular decomposition for this superalgebra. Then we give a system of linear generators of PBW type in terms of Drinfel'd currents. These turn out to be crucial for the development of finite-dimensional representations in the next section.
Hence (l i c i,j ) can be viewed as a Cartan matrix for the Lie superalgebra sl(M, N ).
Definition 3.1. [Ya99,Theorem 8.5.1] U q (Lsl(M, N )) is the superalgebra generated by X ± i,n , K ±1 i , h i,s for 1 ≤ i ≤ M + N − 1, n ∈ Z, s ∈ Z =0 , with the Z 2 -grading |X ± M,n | = 1 for n ∈ Z and 0 for other generators, and with the following relations : 1 ≤ i, j ≤ M + N − 1, m, n, k, u ∈ Z, s, t ∈ Z =0 where the φ ± i,n are given by the generating series We understand that U q (Lsl(M, 0)) = U q (Lsl M ) and U q (Lsl(0, N )) = U q −1 (Lsl N ). We also need an extension of the superalgebra U q (Lsl(M, N )). For this, note that there is an action of the group algebra C[K 0 , K −1 0 ] on it : One can see informally U q (L ′ sl(M, N )) as a deformation of the universal enveloping algebra of the Lie superalgebra L ′ sl(M, N ) where When M = N , the Lie superalgebra L ′ sl(M, N ) is nothing but (sl(M, N ) (1) ) H in Yamane's notation [Ya99, 1.5].

Triangular decomposition
There is an injection of superalgebras U q (Lsl(M, N )) ֒→ U q (L ′ sl(M, N )) given by x → x ⋊ 1. Identify U q (Lsl(M, N )) with a subalgebra of U q (L ′ sl(M, N )) from now on. N )) be the subalgebra of U q (L ′ sl(M, N )) generated by U 0 q (Lsl(M, N )) and K ±1 0 . These subalgebras are clearly Z 2 -homogeneous.
Theorem 3.3. We have the following triangular decomposition for U q (Lsl(M, N )) : (a) the multiplication m : ) is isomorphic to the algebra with generators X + i,n (resp. X − i,n ) and As an immediate consequence, we obtain also a triangular decomposition for U q (L ′ sl(M, N )).
Corollary 3.4. The multiplication below Another consequence is the existence of (anti-)isomorphisms of superalgebras.
(1) There is an isomorphism of superalgebras τ 1 : In view of theorem 3.3 about presentations of algebras, it is enough to prove that τ 1 , τ 2 respect the defining relations (.)-(.).
Remark 3.6. (1) The triangular decomposition will be used to construct the Verma modules and to argue that the Weyl modules are non-zero. See section 3.2.
(2) There are two types of triangular decomposition : one is in terms of Chevalley generators, the other Drinfel'd currents. For the Chevalley type, the triangular decomposition for quantum Kac-Moody algebras was proved in [Ja96, Theorem 4.21]. For the Drinfel'd type, Hernandez proved the triangular decomposition for general quantum affinizations [He05, Theorem 3.2]. Their ideas of proof are essentially the same, which we shall follow below.
(3) For g a simple finite-dimensional Lie algebra, as demonstrated by Grossé [Gr07, Proposition 8], one can realize the quantum affine algebra U q ( g) as a quantum double by introducing topological coproducts on the Borel subalgebras with respect to Drinfel'd currents. In this way, the Drinfel'd type triangular decomposition follows automatically and a topological Hopf algebra structure is deduced on U q ( g). We believe that analogous results hold for U q (Lsl(M, N )). In particular, U q (Lsl(M, N )) could be endowed with a topological Hopf superalgebra structure (with coproduct being Drinfel'd new coproduct).
Proof. We argue that this follows essentially from [He05, Theorem 3.2]. If i / ∈ {k, u}, then it is clear that . Writing φ ± k,m as a product of K ±1 k , h k,s and using the relations We want to write this vector as a product of the formṼ −Ṽ 0Ṽ + by using only the following relations We are in the same situation as U q k ( sl 3 ), when showing that the Drinfel'd relations of degree 2 respect the triangular decomposition. It follows from Theorem 3.2 and the technical lemmas in section 3.3.1 of [He05] that k,u , X ∓ k,0 ] = 0 when c k,u = 0 and k = u. For the second part, note that the I ± i,j are stable by the [h u,s , ]. Relation (.) applies. This means that the Drinfel'd relations of degree 2 respect the triangular decomposition. Let V be the quotient ofṼ by the two-sided ideal generated by the I + + I − . Then where the isomorphism is induced by the triangular decomposition forṼ . Let π 1 :Ṽ −→ V be the canonical projection. By abuse of notation, write . The above identifications say that (V − , V 0 , V + ) forms a triangular decomposition for V . Moreover, the projection π 1 induces isomorphisms When c i,j = ±1 and i = M , denote by J ± i,j the subspace of V ± generated by the LHS of relation (.) with ± for all m, n, k ∈ Z. Denote by J + (resp. J − ) be the sum of the J + i,j (resp. the J − i,j ). Using Theorem 3.2 and the technical lemmas in section 3.3.1 of [He05] we deduce that (the same argument as lemma 3.7 above) In other words, the Serre relations of degree 3 respect the triangular decomposition. Suppose now M, N > 1. Denote by O + (resp. O − ) the subspace of V + (resp. V − ) generated by the LHS of relation (.) with + (resp. with −) for all m, n, k, u ∈ Z.
Sketch of proof. When i / ∈ {M − 1, M, M + 1}, this is clear from relation (.). We are reduced to the case M = N = 2. The related calculations are carried out in appendix A.
By definition the superalgebra U q (Lsl(M, N )) is the quotient of V by the two-sided ideal N generated by J + + J − + O + + O − . Now from the two lemmas above we get from which theorem 3.3 follows.

Linear generators of PBW type
We shall find a system of linear generators for the vector superspace U + q (Lsl(M, N )). In view of corollary 3.5, this will produce one for U − q (Lsl(M, N )). Notation 3.10.
(1) For simplicity, in this section, denote with the following total ordering : Following [HRZ08, Definition 3.9], we can now define the root vectors with the convention that X α i (n) = X + i,n = X i (n). Similar to the quantum affine algebra U r,s ( sl n ) [HRZ08, Theorem 3.11], we have Theorem 3.12. The vector space U M,N is spanned by vectors of the form Remark 3.13. (1) The above generators are called of Poincaré-Birkhoff-Witt type because on specialisation q = 1 they degenerate to PBW generators for universal enveloping algebra of Lie superalgebras [Mu12, Theorem 6.1.1]. This PBW theorem will be used to argue that the set of weights of a Weyl module is always finite.
(2) We believe that the vectors in theorem 3.12 with the following conditions form a basis of U M,N : for Indeed, in the paper [HRZ08], the PBW basis theorem 3.11 was obtained for the two-parameter quantum affine algebra U r,s ( sl n ), with the linear independence among the PBW generators following from a general argument of Lyndon words [Ros]. Hu-Rosso-Zhang called this PBW basis the quantum affine Lyndon basis.
(3) For g a simple finite-dimensional Lie algebra, Beck has found a convex PBW-type basis for the quantum affine algebra U q ( g) in terms of Chevalley generators (see [Be94a,Proposition 6.1] and [Be94b, Proposition 3]). When g = sl 2 , the Drinfel'd type Borel subalgebra of U q (Lsl 2 ) can be realized as the Hall algebra of the category of coherent sheaves on the projective line P 1 (F q ). In this way, the Drinfel'd type PBW basis follows easily ([BK01, Proposition 25]).
Denote by U ′ M,N the vector subspace of U M,N spanned by the vectors in theorem 3.12. As these vectors are all P M,N -homogeneous, Proof. This comes essentially from Proposition 3.10 of [HRZ08], whose proof relied only on the Drinfel'd relations of degree 2.
Proof of theorem 3.12. This is divided into three steps.
Step 1 : induction hypotheses. We shall prove U M,N = U ′ M,N by induction on (M, N ). This is true when Step 2 : consequences of these hypotheses. To simplify notations, denote X γ n∈Z CX γ (n) for γ ∈ ∆ M,N and Proof. According to hypothesis B, it suffices to verify that We are reduced to consider the subalgebra of U M,N generated by the X s (n) with 2 ≤ s ≤ M + N − 1 and n ∈ Z, which is canonically isomorphic to U M −1,N (theorem 3.3). Hypothesis A applies.
Step 3 : demonstration of theorem 3.12. Now we are ready to show that ( where the second equality comes from hypothesis B applied to β − α i . We are led to verify that for 2 ≤ i ≤ M + N − 1 and 1 ≤ s ≤ M + N − 1. Assume furthermore β = α i + (α 1 + · · · + α s ) (using the same argument as one in the proof of the first case of Claim 3), so that k = s + 1. When i ≥ s + 1, thanks to proposition 3.15 and the Drinfel'd relation (.), it is clear that X i X α 1 +···+αs ⊆ (U ′ M,N ) β . Suppose i ≤ s. If s < M + N − 1, then we are working in the subalgebra of U M,N generated by the X i (n) with 1 ≤ i ≤ M + N − 2 and n ∈ Z, hypotheses A applied. Thus assume s = M + N − 1 and we are to show (a) Suppose that i = 1. In view of relation (.), The second term of the RHS is contained in (U ′ M,N ) β thanks to Claim 2. For the first term, from proposition 3.15 (or hypotheses B applied to β − α 1 ), we get Following the proof of case (a), it is enough to verify that M,N ) β thanks to Claim 2. (c) Suppose at last 1 < i < M + N − 1. As in the cases (a) and (b), it suffices to verify that for a, b ∈ Z. An argument of Drinfel'd relation (.) shows that Next, using Drinfel'd relation (.) between X i and X i+1 , together with Serre relations around X i of degree 3 and oscillation relation (.) of degree 4 when i = M (which guarantees M, N > 1), we get Now an argument of Drinfel'd relation (.) and Claim 2 ensure that This completes the proof of theorem 3.12. In this section, we consider the analogy of theorem 2.1 for quantum affine superalgebras. As we shall see, in the super case, corresponding to the odd isotopic root α M , Drinfel'd polynomials have to be replaced by formal series with torsion.

Highest weight representations
Note that M(P , f, c) has a natural U q (L ′ sl(M, N ))-module structure by demanding  Simple modules. Let (P , f, c) ∈ R M,N . From the isomorphism (.) we see that the U q (L ′ sl(M, N ))-module M(P , f, c) has a weight space decomposition (see notation 3.10) In particular, (M(P , f, c)) λ (P ,f,c) = Cv (P ,f,c) is one-dimensional. In consequence, there is a unique quotient of M(P , f, c), which is a simple U q (L ′ sl(M, N ))-module. This leads to the following

Main result
In this section, we shall see that the Weyl modules we defined before are always finite-dimensional and non-zero, a generalisation of theorem 2.1 (a). More precisely, we have  In particular, S ′ (P , f, c) is finite-dimensional.
To prove theorem 4.5, one can assume M > 1, N ≥ 1 due to the following : Lemma 4.8. Suppose M N > 0. The following defines a superalgebra isomorphism : Here p ∈ hom Z (P M,N , Z 2 ) is the parity map in remark 3.13.
Proof. This comes directly from definition 3.1 of U q (Lsl(M, N )).
We remark that the isomorphism π M,N respects the corresponding triangular decompositions of U q (Lsl(M, N )) and U q (Lsl(N, M )). Hence, π * M,N of a Verma/Weyl module over U q (Lsl(N, M )) is again a Verma/Weyl module over U q (Lsl(M, N )). Furthermore, π M,N • π N,M is the identity map. Proof of theorem 4.5. This is divided into two parts. We fix notations first. Let Cv i ) ⊆ J. We want to find (J 1 ) λ and (J 1 ) λ−α M . Indeed where X + i n∈Z CX + i,n (we have used these X + i in the proof of theorem 3.12). Claim. (J 1 ) λ = 0. Next, given 1 ≤ i ≤ M + N − 1 with i = M , let U i be the subalgebra of U q (L ′ sl(M, N )) generated by the X ± i,m , K ±1 i , h i,s with m ∈ Z and s ∈ Z =0 . Then U i v becomes a quotient of the Weyl module over U q i (Lsl 2 ) of highest weight P i . Theorem 2.1 (a) forces that (X + i ) 1+deg P i v i = 0. From the triangular decomposition of U q (L ′ sl(M, N )), we see that J = U − q (Lsl(M, N ))U 0 q (L ′ sl(M, N ))J 1 and   (2) The proposition above also says that the polynomial Q(z) can be reconstructed from the U q (L ′ sl(M, N ))module structure on W. We proceed to verifying (2). First, by using the isomorphism τ 1 : U + q (Lsl(M, N )) −→ U − q (Lsl(M, N )) in corollary 3.5 and the root vectors in definition 3.11, we define : for β ∈ ∆ M,N and n ∈ Z Theorem 3.12 says that (Lsl(M, N )) generated by the X − i,n with 2 ≤ i ≤ M + N − 1 and n ∈ Z. According to theorem 3.3, U − M −1,N is isomorphic to U − q (Lsl (M − 1, N )) as superalgebras.
Proof. Let U M −1,N be the subalgebra of U q (Lsl(M, N )) generated by the −1, N )) as superalgebras. Moreover, U M −1,N v can be realised as a quotient of the the Weyl module W((P i ) i≥2 , f, c; Q) over U q (Lsl(M − 1, N )). From the induction hypothesis, On the other hand, using the anti-automorphism τ 2 of corollary 3.5, we can also write (Lsl(M, N )) generated by the X − i,n with 1 ≤ i ≤ M + N − 2 and n ∈ Z. Similar argument as in the proof of the Claim above shows that U − M,N −1 v is a finite-dimensional subspace of W.
for some e ′ i ≥ 0 and 0 ≤ f ′ j < C 2 . It follows that Now inequalities (.) and (.) imply that In particular, µ + sα 1 / ∈ P when |s| ≫ 0. Hence, X + 1,0 and X − 1,0 are locally nilpotent operators on W. Denote by U 0 the subalgebra of U q (L ′ sl(M, N )) generated by the X ± 1,0 , K ±1 i with 0 ≤ i ≤ M + N − 1. Then U 0 is an enlargement of U q (sl 2 ). From the theory of integrable modules over U q (sl 2 ) we see that µ ∈ P =⇒ s 1 (µ) ∈ P . Here, for In view of (.), Now the three inequalities (.)-(.) say that all the u i are bounded by a constant. In other words, P is finite. This completes the proof of theorem 4.5.
(1) Our proof relied heavily on the theory of Weyl modules over U q (Lsl 2 ). Using PBW generators, we deduced the integrability property of Weyl modules : the actions of X ± i,0 for 1 ≤ i ≤ M + N − 1 are locally nilpotent. Even in the non-graded case of quantum affine algebras considered in [CP01], the integrability property (theorem 2.1) is non-trivial (see the references therein).
(2) From integrability, we get an action of Weyl group on the set P of weights [Lu93, Section 41.2]. In the non-graded case, the action of Weyl group already forces that P be finite (argument of Weyl chambers). In our case, the Weyl group, being S M × S N , is not enough to ensure the finiteness of P . And once again, we used PBW generators to obtain further information on P .

Classification of finite-dimensional simple representations
In this section, we show that all finite-dimensional simple modules of U q (L ′ sl(M, N )) (or U q (Lsl(M, N )   Then P is a finite set. There exists λ 0 ∈ P such that λ 0 + α i / ∈ P for all 1 ≤ i ≤ M + N − 1. (Here we really need the fact that these α i are linearly independent.) Note that (V ) λ 0 is also Z 2 -graded. Moreover, (V ) λ 0 is stable by the commutative subalgebra U 0 q (L ′ sl(M, N )). One can therefore choose a non-zero Z 2 -homogeneous vector v ∈ (V ) λ 0 which is a common eigenvector of U 0 q (L ′ sl(M, N )). In particular, X + i,n v = 0 for all 1 ≤ i ≤ M + N − 1 and n ∈ Z, and K 0 v = λ 0 (K 0 )v. When i = M , let U i be the subalgebra generated by the X ± i,n , K ±1 i , h i,s with n ∈ Z and s ∈ Z =0 . Then U i ∼ = U q i (Lsl 2 ) as algebras, and U i v is a finite-dimensional highest weight U q i (Lsl 2 )-module. One can thus find (ε i , P i ) satisfying the above relation thanks to theorem 2.1 (b).
On the other hand, as X − M v is finite-dimensional, there exists m ∈ Z, d ∈ Z ≥0 and a 0 , · · · , a d ∈ C such that a 0 a d = 0, a 0 = 1,  N )) of the following forms : Note that such an automorphism always preserves U q (Lsl(M, N )). Denote by D the set of superalgebra automorphisms of U q (Lsl(M, N )) of the form π| Uq (Lsl(M,N )) with π ∈D. Definition 4.14. Let (P , f, c) ∈ R M,N and let V be a U q (Lsl(M, N ))-module. We say that V is of highest weight (P , f, c) if there is an epimorphism of U q (Lsl(M, N ))-modules : M(P , f, c) ։ V .
One can now have a super-version of theorem 2.1 (b). Denote by ι : U q (Lsl(M, N )) ֒→ U q (L ′ sl(M, N )) the canonical injection defined in section 2.2.

Integrable representations
This section deals with generalisations of theorem 2.1 (c). We shall see that, for all Λ ∈ R M,N , there exists a largest integrable module of highest weight Λ. However, such modules turn out to be infinite-dimensional, contrary to the quantum affine algebra case.
Definition 4.16. Call a U q (Lsl(M, N ))-module integrable if the actions of X ± i,0 are locally nilpotent for 1 ≤ i ≤ M + N − 1.
Note that the actions of X ± M,0 are always nilpotent. From the representation theory of U q (sl 2 ) [Ja96, Chapter 2], we see that finite-dimensional U q (Lsl(M, N ))-modules are always integrable. In particular, the Weyl modules and all their quotients are integrable.
In particular, W(P , f, c) is integrable.
Proof. The idea is similar to that of the proof of theorem 4.5 : to use PBW generators to deduce restrictions on the set of weights. One also needs the isomorphism π M,N : U q (Lsl(M, N )) −→ U q (Lsl(N, M )) to change the forms of the PBW generators.
Thus, for (P , f, c) ∈ R M,N , the universal Weyl module W(P , f, c) is the largest integrable highest weight module of highest weight (P , f, c). Note however that W(P , f, c) is by no means finite-dimensional when M N > 0. Indeed, for all (P , f, c; Q), we have an epimorphism of U q (Lsl(M, N ))-modules W(P , f, c) ։ W(P , f, c; Q). It follows from proposition 4.9 that dim W(P , f, c) > deg Q. As deg Q can be chosen arbitrarily large, W(P , f, c) is infinite-dimensional.

Evaluation morphisms
Throughout this section, we assume M > 1, N ≥ 1 and M = N . After [Ya99, Theorem 8.5.1], there is another presentation of the quantum superalgebra U q (Lsl(M, N )). From this new presentation, we get a structure of Hopf superalgebra on U q (Lsl(M, N )) (in the usual sense). Using evaluation morphisms between U q (Lsl(M, N )) and U q (gl(M, N )), we construct certain finite-dimensional simple U q (Lsl(M, N ))-modules.  N )) is the superalgebra generated by E ± i , K ±1 i for 0 ≤ i ≤ M + N − 1 with the Z 2 -grading |E ± 0 | = |E ± M | = 1 and 0 for other generators, and with the following relations : (M, N )). We reformulate part of [Ya99, Theorem 8.5.1] : Theorem 5.2. There exists a unique superalgebra homomorphism Φ : U ′ q ( sl(M, N )) −→ U q (Lsl(M, N )) such that : from definition 3.11 and the proof of theorem 4.5.
From the construction of U ′ q ( sl(M, N )) in section 6.1 of [Ya99], we see that U ′ q ( sl(M, N )) is endowed with a Hopf superalgebra structure : for 0 Here the coproduct formula is consistent with that of (.). Under this coproduct, ∆(c) = c ⊗ c. Hence, ker Φ = U ′ q ( sl(M, N ))(c − 1) becomes a Hopf ideal of U ′ q ( sl(M, N )), and Φ induces a Hopf superalgebra structure on U q (Lsl(M, N )).
f ±s z ±s and where the neutral element is (1, 0, 1). From the commutativity of (R M,N , * ) we see also that : if the tensor product S 1 ⊗S 2 of two finite-dimensional simple U q (Lsl(M, N ))-modules remains simple, then so does S 2 ⊗S 1 and S 1 ⊗S 2 ∼ = S 2 ⊗S 1 as U q (Lsl(M, N ))-modules.

Evaluation morphisms
In this section, we construct some simple modules of U q (Lsl(M, N )) via evaluation morphisms. As in the case of quantum affine algebras U q ( sl n ), we also have evaluation morphisms : Proposition 5.6. There exists a morphism of superalgebras ev : Remark 5.7. ev(E ± 0 ) appeared implicitly in [Zh92,Lemma 4]. Note however that Zhang did not verify the degree 5 relations in the case (M, N ) = (2, 1). A lengthy calculation shows that this is true, and that ev is always welldefined. The modified element K = t 1 t −1 M +N is needed to ensure that Ke ± i K −1 = e ± i for 2 ≤ i ≤ M + N − 2 and Ke ± i K −1 = q ±1 e ± i when i = 1, M + N − 1. If 0 < |M − N | ≤ 2, then K can be chosen so that K ∈ U q (sl (M, N )). It is clear that ev(K 0 · · · K M +N −1 ) = 1. This implies that ev : U q ( sl(M, N )) −→ U q (gl(M, N )) factorizes through Φ : U q ( sl(M, N )) ։ U q (Lsl(M, N )). Denote by ev ′ : U q (Lsl(M, N )) −→ U q (gl(M, N )) the superalgebra morphism thus obtained.
Remark that from the definition 3.1 of quantum affine superalgebras U q (Lsl(M, N )) admit naturally a Zgrading provided by the second index (the first index gives P M,N -grading). From this Z-grading, we construct for each a ∈ C \ {0}, a superalgebra automorphism Φ a : U q (Lsl(M, N )) −→ U q (Lsl(M, N )) defined by :  N )). In particular, one obtains some finite-dimensional simple modules in this way. Take a ∈ C \ {0}. Lemma 5.8 together with theorem 2.3 leads to the following proposition : Example 5.10. When Λ i = δ i,1 , we get the fundamental representation of U q (gl(M, N )) on the vector superspace V = V 0 ⊕ V 1 with dim V 0 = M, dim V 1 = N (see for example [BKK00, Section 3.2] for the actions of Chevalley generators). We obtain therefore finite-dimensional simple U q (Lsl(M, N ))-modules corresponding to (P , 0, 1) ∈ R M,N where P i (z) = 1 − qδ i,1 az with a ∈ C \ {0}.

Further discussions
Representations of quantum superalgebras. As we have seen in section 5.2, for a ∈ C \ {0} there exists a superalgebra homomorphism (assuming M = N ) ev a : U q (Lsl(M, N )) −→ U q (gl (M, N )).
One can pull back representations of U q (gl(M, N )) to get those of U q (Lsl(M, N )).
In 2000, Benkart-Kang-Kashiwara [BKK00] proposed a subcategory O int of finite-dimensional representations of U q (gl(M, N )) over the field Q(q) to study the crystal bases. Using the notations in section 2.2, one has also the finite-dimensional simple modules The simple module L(Λ) in O int always admits a polarizable crystal base (Theorem 5.1), with the associate crystal B(Λ) being realised as the set of all semi-standard tableaux of shape Y Λ (Definition 4.1). Here Y Λ is a Young diagram constructed from Λ. In this way one gets a combinatorial description of the character and the dimension for the simple module L(Λ).
Explicit constructions of representations of the quantum superalgebra U q (gl(M, N )) are also of importance to us. In [KV04], Ky-Van constructed finite-dimensional representations of U q (gl(2, 1)) and studied their basis with respect to its even subalgebra U q (gl(2) ⊕ gl(1)). Early in [Ky94,KS95], certain finite-dimensional representations of U q (gl(2, 2)) were constructed together with their decomposition into simple modules with respect to the subalgebra U q (gl(2) ⊕ gl(2)). In 1991, Palev-Tolstoy [PT91] deformed the finite-dimensional Kac/simple modules of U (gl (N, 1)) to the corresponding modules of U q (gl(N, 1)) and wrote down the actions of the algebra generators in terms of Gel'fand-Zetlin basis. Later, Palev-Stoilova-Van der Jeugt [PSV94] generalised the above constructions to the quantum superalgebra U q (gl (M, N )). However, their methods applied only to a certain class of irreducible representations, the so-called essentially typical representations. Recently, the the coherent state method was applied to construct representations of superalgebras and quantum superalgebras. In [KKNV12], Kien-Ky-Nam-Van used the vector coherent state method to construct representations of U q (gl(2, 1)). However, for quantum superalgebras U q (gl(M, N )) of higher ranks, the analogous constructions are still not explicit.
Relations with Yangians. In the paper [Zh96], Zhang developed a highest weight theory for finite-dimensional representations of the super Yangian Y (gl(M, N )), and obtained a classification of finite-dimension simple modules (Theorems 3, 4). Here the set T M,N of highest weights consists of Λ = (Λ i : where Q,Q ∈ 1 + zC[z] are co-prime polynomials of the same degree, P i (z) ∈ C[z] are polynomials of leading Other simple modules S(Λ) can always be realised as subquotients of n s=1 φ * s S(Λ (s) ) where Λ (s) ∈ T 0 M,N and the φ s are some superalgebra automorphisms of Y (gl (M, N )).
In the case of quantum affine superalgebra U q (Lsl(M, N )), the set of highest weights is R M,N , which is a commutative monoid. It is not easy to see, in view of definition 4.1 and corollary 5.5, that as monoids In the following, we investigate the monoid structure of R 1,1 . As we shall see, R 1,1 is almost T 1,1 . Thus, informally speaking, the two monoids R M,N and T M,N are almost equivalent, and the finite-dimensional representation theories for U q (Lsl(M, N )) and for Y (gl(M, N )) should have some hidden similarities.
Recall that R 1,1 is the set of couples (f, c) where f is a formal series with torsion, with c−c −1 q−q −1 being the constant term. Denote respectively by ι ± : defines an isomorphism of monoids.
Proof. First, ι +,− is well-defined, as the formal series in the RHS of (.) is killed by P (z). Moreover, ι +,− respects the monoid structures in view of corollary 5.5.
Consider the formal power series It is clear that a ′ 0 = 1 and that This says that Q(z) ∈ 1 + zC[z] is of degree d and that lim z→∞ We remark that f (z) is completely determined by f 0 , f 1 , · · · , f d−1 , which in turn are determined by Q(z). This forces ).
In the paper [GT13], Gautam-Toledano Laredo constructed an explicit algebra homomorphism from the quantum loop algebra U (Lg) to the completion of the Yangian Y (g) with respect to some grading, where g is a finite-dimensional simple Lie algebra. Also, they are able to construct a functor from a subcategory of finitedimensional representations of Y (g) to a subcategory of finite-dimensional representations of U (Lg). It is hopeful to have some generalisations for quantum affine superalgebras and super Yangians.
Other quantum affine superalgebras. Our approach to the theory of Weyl/simple modules of U q (Lsl(M, N )) is quite algebraic, without using evaluation morphisms and coproduct, and is less dependent on the actions of Weyl groups. In general, for a quantum affine superalgebra, if we know its Drinfel'd realization and its PBW generators in terms of Drinfel'd currents, it will be quite hopeful that we arrive at a good theory of finite-dimensional highest weight modules (Weyl/simple modules).
We point out that in the paper [AYY11], Azam-Yamane-Yousofzadeh classified finite-dimensional simple representations of generalized quantum groups using the notion of a Weyl groupoid. We remark that Weyl groupoids appear naturally in the study of quantum/classical Kac-Moody superalgebras due to the existence of non-isomorphic Dynkin diagrams for the same Kac-Moody superalgebra. Roughly speaking, a Weyl groupoid is generated by even reflections similar to the case of Kac-Moody algebras, together with odd reflections in order to keep track of different Dynkin diagrams. Note that early in [Ya99], Yamane generalised Beck's argument [Be94a] by using the Weyl groupoids instead of the Weyl groups to get the two presentations of U q (Lsl(M, N )). Later in [HSTY08], similar arguments of Weyl groupoids led to Drinfel'd realizations of the quantum affine superalgebras U q (D (1) (2, 1; x)). Also, in the paper [Ser], Serganova studied highest weight representations of certain classes of Kac-Moody superalgebras with the help of Weyl groupoids. We believe that Weyl groupoids should shed light on the structures of both quantum affine superalgebras themselves and their representations.
Affine Lie superalgebras. Consider the affine Lie superalgebra Lsl(M, N ) with M = N . As we have clearly the triangular decomposition and the PBW basis, we obtain a highest (l)-weight representation theory for U (Lsl(M, N )) similar to that of Chari [Ch86]. Here, the set W M,N of highest weights are couples (P , f ) where (a) P ∈ (1 + zC[z]) M +N −1 (corresponding to even simple roots) ; (b) f ∈ C[[z, z −1 ]] such that Qf = 0 for some Q ∈ 1 + zC[z] (corresponding to the odd simple root). Finite-dimensional simple Lsl(M, N )-modules are parametrized by their highest weight. Furthermore, for (P , f ) ∈ W M,N and Q ∈ 1 + zC[z] such that Qf = 0, we have also the Weyl module W(P , f ; Q) defined by generators and relations. We remark the recent work [Ra13] of S. Eswara Rao on finite-dimensional modules over multi-loop Lie superalgebras associated to sl(M, N ). In that paper, a construction of finite-dimensional highest weight modules analogous to that of Kac induction was proposed. In this way, the character formula for these modules is easily deduced once we know the character formulae [CL06] for Weyl modules over Lsl n . It is an interesting problem to compare Rao's highest weight modules with our Weyl modules, as they both enjoy universal properties.

A Oscillation relations and triangular decomposition
In this appendix we finish the proof of lemma 3.9 : the oscillation relations of degree 4 respect the Drinfel'd type triangular decomposition for U q (Lsl(2, 2)). As indicated in the proof of the main result theorem 4.5, the triangular decomposition is needed to deduce the non-triviality of the Weyl modules. In the following, we carry out the related calculations.
When i ≥ 3, the corresponding term in the above summation becomes 0. Similar argument leads to the first part of lemma 5.3 for ∆(h i,1 ).