Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators

We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra U_q(gl(M|N)^). This is characterized by Drinfeld rational fractions. In particular, we consider contractions of U_q(gl(M|N)) in the FRT formulation and obtain explicit solutions of the graded Yang-Baxter equation in terms of q-oscillator superalgebras. These solutions correspond to L-operators for Baxter Q-operators. We also discuss an extension of these representations to the ones for contracted algebras of U_q(gl(M|N)^) by considering the action of renormalized generators of the other side of the Borel subalgebra. We define model independent universal Q-operators as the supertrace of the universal R-matrix and write universal T-operators in terms of these Q-operators based on shift operators on the supercharacters. These include our previous work on U_q(sl(2|1)^) case [arXiv:0805.4274] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T-and Q-functions in [arXiv:0906.2039, arXiv:1109.5524].


Introduction
The Baxter Q-operators were introduced [4] by Baxter when he solved the 8-vertex model. Nowadays his method of Q-operators is recognized as one of the most powerful tools in quantum integrable systems. In particular, Bazhanov, Lukyanov and Zamolodchikov [5] defined Q-operators as the trace of the universal R-matrix over q-oscillator representations of the Borel subalgebra of the quantum affine algebra U q (ŝl (2)). Their work based on the q-oscillator algebra was generalized and developed for various directions [6,7,8,9,1,10].
In our previous paper [1], we gave Q-operators for the quantum affine superalgebra U q (ŝl(2|1)). Our Q-operators in [1] are universal in the sense that they do not depend on the quantum space and can be applied for both lattice models and quantum field theoretical models as well. We also proposed [2] an idea that there are 2 M +N kind of Baxter Q-functions for U q (ĝl(M|N)) case and gave Wronskian-like formulas on T-and Q-functions for finite [2] and infinite [3] dimensional representations for any (M, N) 1 . The Q-function in [2] is labeled by the index set I, which is a subset of the set {1, 2, . . . , M + N}. In this paper, we continue these our previous works and define model independent universal Q-operators for U q (ĝl(M|N)) (or U q (ŝl(M|N))) as the supertrace of the universal R-matrix for any (M, N). This gives a cue for the operator realization of the Wronskian-like formulas in [2,3].
In section 2, we define the quantum affine superalgebra (or rather quantum loop superalgebra) U q (ŝl(M|N)) in terms of the Chevalley generators and the universal R-matrix associated with it. We also mention their extension to U q (ĝl(M|N)). Our task is basically evaluate this universal R-matrix for q-oscillator representations of the Borel subalgebra. As is well known, the Yang-Baxter equation follows from the defining relations of the universal R-matrix. The images of the universal R-matrix for particular representations give the so-called L-operators and R-matrices. The Yang-Baxter equations for the L-operators and the R-matrix (RLL = LLR relations), which are also image of the Yang-Baxter equation for the universal R-matrix, give another realization of the quantum affine superalgebra (FRT realization, [12]). In accordance with the quantum affine superalgebra, the quantum (finite) superalgebra U q (gl(M|N)) also have these two realizations. In section 3, we consider 2 M +N kind of contractions of the L-operator for U q (gl(M|N)), which define contracted algebras U q (gl(M|N; I)). A preliminary form of these contractions for (M, N) = (3, 0) case was previously considered in [13]. We also reported such contractions for (M, N) = (2, 1) case in conferences [14].
Next, we consider q-oscillator realizations of these contracted algebras. These induce representations of the Borel subalgebra of the quantum affine superalgebra (or q-superYangian) via the evaluation map. We remark that these representations can not be straightforwardly extended to the full quantum affine superalgebra. These are examples of asymptotic representations characterized by the Drinfeld rational fractions 2 [15]. They are certain limits of the Kirillov-Reshetikhin modules (or their extension). The hart of an idea is to synchronize the highest weight of the representations and automorphisms of the algebra in the limit so that one can obtain finite quantities. In this way, we obtain spectral parameter dependent L-operators whose matrix elements are written in terms of the q-oscillator superalgebras. Similar L-operators for (M, N) = (3, 0) were previously considered in [16] and [10]. We also reported such L-operators for (M, N) = (2, 1) in [14,1]. All these L-operators satisfy the defining relations of the universal R-matrix (mentioned in section 2) evaluated by the tensor product of the q-oscillator representations and the fundamental representation of the Borel subalgebras. It should be remarked here that the above q-oscillator representations of the Borel subalgebra can be extended to those of contracted algebras U q (ĝl(M|N; I)) of U q (ĝl(M|N)). For example for U q (ĝl(2)) case, the contracted algebra U q (ĝl(2; {1})) in terms of the Chevalley generators is defined by the following commutation relations 3 : The generators of U q (ĝl(2; {1})) automatically satisfy the defining relations of the Borel subalgebras of U q (ĝl (2)). The restriction of the above relations to the generators {e 1 , f 1 , k 1 , k 2 } gives U q (gl(2; {1})). Then we can consider evaluation representations of U q (ĝl(2; {1})) in terms of the representations of U q (gl(2; {1})). The q-oscillator representations of the Borel subalgebra of U q (ŝl(2)) introduced by Bazhanov, Lukyanov and Zamolodchikov [5] are special cases of this type of representations.
In section 4, we define the universal Q-operators as the supertrace of the universal R-matrix over the q-oscillator representations defined in the previous section. The T-operators are written in terms of these Q-operators. In the same way as previous paper [1], our Q-operators here are universal in the sense that they do not depend on the quantum space. As an example, we write Q-operators whose quantum space is the fundamental representation on each lattice site based on the L-operators derived in section 3. Section 5 is devoted to concluding remarks. Technical details are tucked into the appendices and a number of footnotes.
There are many literatures on Q-operators related to sl(2), which we could not refer. However there are not so many references for the higher rank case or superalgebras case, which are our main subjects of this paper; and here we only mention some of them for rational models. In the rational limit (q → 1; after multiplying diagonal matrices for the renormalization), our L-operators naturally reduce to Loperators which are similar to the ones proposed recently in [17] for rational lattice models. However, our L-operators are not simple generalization of the rational ones since many of the non-zero matrix elements of our L-operators become zero in the rational limit. Thus the q-deformation of the rational L-operators is not trivial. There are also Q-operators for infinite dimensional representations on the quantum space [18,19]. It will be interesting to evaluate our universal Q-operators for infinite dimensional representations on the quantum space and to see how (or if) their formulas are lifted to the trigonometric case. We also proposed [20] Q-operators based on the co-derivative [21] on the supercharacters of gl(M|N). This construction of the Q-operators is useful to discuss [20,22] functional relations among T-and Qoperators and embed them into the soliton theory. It is desirable to generalize this for the trigonometric case.
2 The quantum affine superalgebra U q (ŝl(M |N )) and the universal R-matrix, and their extension to U q (ĝl(M |N )) Let us introduce a grading parameter p(i) = 0 for i ∈ {1, 2, . . . , M} and p(i) = 1 for i ∈ {M + 1, M + 2, . . . , M + N}. The quantum affine superalgebra U q (ŝl(M|N)) [23] (see also [24]) is a Z 2 -graded Hopf algebra generated by the generators 4 e i , f i , h i , where i ∈ {0, 1, . . . , M + N − 1}. We assign the parity for these generators so that p(e 0 ) = p(e M ) = p(f 0 ) = p(f M ) = 1 for MN = 0 and p(X) = 0 for all the other generators X. For any X, Y ∈ U q (ŝl(M|N)), we define , the defining relations of the algebra U q (ŝl(M|N)) are given by In this paper, we do not use the degree operator d.
[e i , e j ] = [f i , f j ] = 0 for a ij = 0, where (a ij ) 0≤i,j≤M +N −1 is the Cartan matrix In addition to the above relations, there are Serre relations and also for the superalgebra case (MN = 0), the extra Serre relations 5 : In this paper, we consider the case where the following central element is zero (level zero condition): The algebra has the co-product ∆ : We heard from Hiroyuki Yamane that we will need infinitely many Serre relations for M = N = 2 case, due to the Lusztig isomorphism (see [23], for more details).
where the tensor product is the graded one: (A ⊗ B)(C ⊗ D) = (−1) p(B)p(C) (AC ⊗ BD). We assume that every tensor product ⊗ in this paper is the graded one. We will also use an opposite co-product defined by (2.14) In addition to these, there are anti-poide and co-unit, which will not be used in this paper.
gives a shift automorphism of B + or B − . Here we omit the unit element multiplied by the above complex numbers. This automorphism played a role 6 in the construction of the Q-operators in [5,6,1].
There exists a unique element [25,26] R ∈ B + ⊗B − called the universal R-matrix which satisfies the following relations where 7 R 12 = R ⊗ 1, R 23 = 1 ⊗ R, R 13 = (σ ⊗ 1) R 23 . The (graded) Yang-Baxter equation is a corollary of these relations (2.16). The universal R-matrix can be written in the form where (d ij ) 1≤i,j≤M +N −1 is the inverse of the Cartan matrix (a ij ) 1≤i,j≤M +N −1 of sl(M|N). In case this Cartan matrix is degenerated (M = N), we have to consider an extended matrix 8 and take the inverse of it [27]. Here R is the reduced 6 When one takes a limit of the highest weight, one has to take a limit of these shift parameters at the same time to obtain a q-oscillator representation for the Q-operators. 7 We will use similar notations for the L-operators to indicate the space which they non-trivially act on. 8 This may be achieved by adding an extra Cartan element M+N j=1 (−1) p(j) k j to U q (ŝl(M |N )). Here k j are Cartan elements of U q (ĝl(M |N )), which we will introduce later.
universal R-matrix, which is a series in e j ⊗ 1 and 1 ⊗ f j and does not contain Cartan elements. Thus the reduced universal R-matrix is unchanged under the shift automorphism (2.15), while the prefactor of the universal R-matrix (2.18) is shifted as where we considered a shift on B + . There is a (finite) quantum superalgebra U q (gl(M|N)), which is generated by the elements {e ij } M +N i,j=1 . We assign the parity of these generators as p(e ij ) = p(i) + p(j) mod 2. Let us introduce the notations: e α i = e i,i+1 , e −α i = e i+1,i for i ∈ {1, 2, . . . , M + N − 1}. Then the defining relations of U q (gl(M|N)) (for the distinguished simple root system) are (cf. [27] The other elements are defined by The other relations can also be obtain by (3.23)-(3.25) and (A1)-(A16). Let E ij be a (M + N) × (M + N) matrix unit whose (k, l)-element is δ i,k δ j,l . π(e ij ) = E ij gives the fundamental representation of U q (gl(M|N)). There is an evaluation map 10 ev x : U q (ŝl(M|N)) → U q (gl(M|N)): where x ∈ C is a spectral parameter. Let π λ be an irreducible representation of U q (gl(M|N)) with the highest weight λ = (λ 1 , λ 2 , . . . , λ M +N ) and the highest weight vector |λ defined by e ii |λ = λ i |λ , e jk |λ = 0 for j < k, i, j, k ∈ {1, 2, . . . , M + N}. (2.23) Then the composition π λ (x) = π λ • ev x gives an evaluation representation of U q (ŝl(M|N)). For the fundamental representation, we will use a notation π(x) = π (1,0,...,0) (x). We also use a notation π + λ (x) for the evaluation representation based on the Verma module defined by the free action of the generators on the highest weight vector (2.23). In this case, the representation is not necessary irreducible. Our main task is basically to evaluate the universal R-matrix for various representations of U q (ŝl(M|N)) (or U q (ĝl(M|N))). Namely, to find matrices of the form (2.18) which satisfy (2.16) for various representations of B + and B − . The simplest example is the R-matrix for the Perk-Schultz model [28] (see [29] for N = 0 case), which is a multi-component generalization of the six-vertex model. Namely, the image of the universal R-matrix for π(x 1 ) ⊗ π(x 2 ) gives (up to an overall factor N(x 1 , x 2 ); This obeys the graded Yang-Baxter equation which is an image of (2.17) for π( of U q (ĝl(M|N)), which is related to the generators of U q (ŝl(M|N)) under (2.10) as where the indices i, j should be interpreted modulo M + N. These are even elements p(k i ) = 0. It is sometimes convenient to define and rewrite (2.2) as Later on, we will renormalize the generators and consider the case where k i differs from −k i (cf. Appendix B). Moreover, this difference can be infinite in some limit. Now the Borel subalgebras B + and B − are generated by the pre-factor of the universal R-matrix (2.18) can be rewritten as satisfies (2.16) for U q (ĝl(M|N)) generators. Then we regard 11 this renormalized universal R-matrixR as a universal R-matrix for U q (ĝl(M|N)) (under the condition (2.10)). For M = N,R is related to R via an overall central element:R = Rq 1 M −N C⊗C . However,R itself is well-defined for M = N case as well. For any c i ∈ C (multiplied by a unit element), the following transformation gives the shift automorphism of the Borel subalgebra. This keeps the level zero condition (2.10) for any c i . The prefactor of the universal R-matrix (2.33) is shift by the shift automorphism (2.34) for B + as The evaluation map for the Cartan elements is defined by The evaluation representations are defined via this map in the same way as U q (ŝl(M|N)) case (the same symbols will be used). In particular π(x)(C) is a (M + N) × (M + N) unit matrix. In the subsequent sections, the contribution of the difference between R andR to each formula will be absorbed into a (representation dependent) overall factor of it. For example, the factor (π( 3 L-operators from FRT realization of the quantum affine superalgebra U q (ĝl(M |N )) The quantum affine superalgebra U q (ĝl(M|N)) (and its subalgebra U q (gl(M|N))) has another realization, called FRT realization [12] (see also, [30]), based on the Yang-Baxter equation (RLL = LLR relation). In this section we use this realization. The (centerless) quantum affine superalgebra U q ( gl(M|N)) is defined by where x, y ∈ C and and The above relations came from the graded Yang-Baxter equation (2.17) for the universal R-matrix under the specialization (2.24) and and N(x) are overall factors. In order to obtain the defining relations for U q ( sl(M|N)), we will have to impose a condition that the quantum super-determinants of the above L-operators are 1. But we do not impose this explicitly here. Let us introduce a function: θ(True) = 1, θ(False) = 0.
One can rewrite (3.3) as and (3.4) as .9) and (3.5) as gives an automorphism of U q (ĝl(M|N)) since R(cx 1 , cx 2 ) = R(x 1 , x 2 ). The restriction of the relations (3.1)-(3.5) to the relation for L(x) defines a sort of Borel subalgebra of U q (ĝl(M|N)) called q-super-Yangian. Note that the following transformation (multiplication of diagonal matrices in the second space) 12 This is related to the parameters c i in the shift automorphism (2.34)-(2.35) via H (i) R = q ci . This also came from the first relation for the Cartan elements of U q (ĝl(M |N )) in (2.16). If we restrict these Cartan elements to the ones for U q (ŝl(M |N )), we will obtain a restriction In this case, (3.12) (for H (i) L = 1) should correspond to the shift automorphism (2.15) and (2.19). Here we assumed that these parameters are not 0 at first. However, we will have to consider limits that some of these go to ∞ or 0.
Then the inverse of L We will meet a situation where some of L R ) −2 remain finite in some limit. The restriction of this transformation to the q-superYangian gives an automorphism of it. In addition, if we consider a 'bigger' algebra (a kind of an asymptotic algebra [15]) which does not assume (3.2), it can be an automorphism of such algebra.
The quantum affine superalgebra U q (ĝl(M|N)) has a finite subalgebra U q (gl(M|N)) defined by Then the relation (3.16) leads the relation (3.17) leads (3.21) and the relation (3.18) leads For convenience, we list a more explicit form of these relations in Appendix A. These generators are related to the generators {e ij } in section 2 as Then the action of generators on the highest weight vector corresponding to (2.23) is There is an evaluation map from U q (ĝl(M|N)) to U q (gl(M|N)) such that Apparently, the difference between L(x) and L(x) are not very important under the evaluation map. Let us consider an irreducible representation of U q (ĝl(M|N)) with the highest weight (ν(x), ν(x)) and the highest weight vector |ν, ν defined by For the finite dimensional representations, there exist monic polynomials in x, called Drinfeld polynomials 13 P i (x), such that For the evaluation modules whose highest weights are given by (3.31) and (3.32), Here we define these so that these become monic polynomials of the spectral parameter from B + . We can also define them so that they are monic polynomials of the spectral parameter from B − . In this case, q in (3.34) will be replaced by q −1 .  (gl(M|N)). Namely, let us consider an algebra whose condition (3.15) is replaced by Then one can obtain 2 M +N kind of algebraic solutions of the graded Yang-Baxter equation via the map (3.27). In addition to the contraction (3.36), we consider the following subsidiary contraction. Suppose the set I has the form I = {k + 1, k + 2, . . . , k + n} for some k ≥ 0, n > 0, then we assume One may consider different contractions than (3.37),(3.38). Here we consider a contraction so that the location of the zeros becomes cyclic with respect to the shift of the suffixes by an operation: a → a+ 1 for a < M + N and M + N → 1. Namely, the contraction for k > 0 can be given by applying this operation k-times for the case k = 0. What is important here is to respect the commutation relations for the generators (3.20)- (3.22). In this way, we define a contracted algebra U q (gl(M|N; I)) by (3.14), (3.16)-(3.19) and (3.35)-(3.38). There is an option to exclude the conditions (3.37)-(3.38) from U q (gl(M|N; I)) and interpret them as relations on the level of the representation. We remark that these contractions on the L-operator for U q (ĝl (3)) (written in terms of the generators e ij and substituted into (3.27)) was previously considered in [13]. We also reported these contractions for U q (ĝl(2|1)) in conferences [14]. The next task is to consider representations of these contracted algebras. We are interested in q-oscillator representations. The q-oscillator (super)algebra (see for example, [31]) is generated by the generators c ai , c † ia , n ia for i ∈ I, a ∈ I, whose parities are defined by p(c ai ) = p(c † ia ) = p(a) + p(i) mod 2, p(n ia ) = 0. They obey the following defining relations: where i, j ∈ I, a, b ∈ I. From (3.39), we can derive the relations: . Note that the following transformation gives a |I||I|(|I||I| + 3)/2 parameter continuous automorphism of the q-oscillator algebra (3.39). We also remark that the following transformation gives a discrete automorphism of the q-oscillator algebra (3.39) for any i ∈ I and a ∈ I. For the diagonal part, we consider the following 15

49)
L aa = q (−1) p(a) n i,a for a ∈ I, (3.50) 51) 14 We consider these generators on the Fock space fixed by the vacuum (3.75). Then for the fermionic case p(i) + p(a) = 1 mod 2, these relation effectively becomes c ai c † ia = 1 − n ia , c † ia c ai = n ia . 15 For L, this satisfies a U q (sl(M |N ))-type relation i∈I L (−1) p(i) ii a∈I L (−1) p(a) aa = 1, but for L, it does not. 16 We used relations in Appendix A for the direct calculations.   One may also apply the transformations (3.41) or (3.42) to these solutions to get many parameter solutions. The q-oscillator solutions of the graded Yang-Baxter equation are given by substituting the above q-oscillator realizations of the Loperators into the map (3.27). We denote the corresponding solutions as We remark that the following renormalized L-operators reduce to L-operators similar to the ones in [17] in the rational limit q → 1. Now (3.73) defines an evaluation map from the q-superYangian to the contracted algebra. Let us calculate the actions of generators on the vacuum defined by n ia |0 = c ai |0 = 0 for all i ∈ I, a ∈ I. (3.75) They lead for a ∈ I. Thus the corresponding representation is the highest weight representations of the q-superYangian with the highest weight vector |0 and the highest weight given by (3.76). In addition, the ratio of the eigenvalues ν i (x) of L ii (x) on |0 is This is a kind of Drinfeld rational fraction 17 introduced in [15]. The finite dimensional representations of the quantum affine algebras are characterized by the Drinfeld polynomials. In contrast, q-oscillator representations given as limits of the Kirillov-Reshetikhin modules 18 of the Borel subalgebra of the quantum affine algebras are characterized by the Drinfeld rational fractions. For the other sets I, the highest weight condition (3.77) will have to be changed since they should be interpreted as representations permuted by automorphisms of U a (ĝl(M|N)). Let us consider a renormalized L-operator for the q-superYangian shifted by the automorphisms (3.11) and (3.12). The latter corresponds to c i = −m for i ∈ I, c i = 0 for i ∈ I (3.81) In addition, the map (2.36) becomes: We also define Due to the relation (3.15), (3.82) and (3.83) are consistent with (2.29). Let us substitute L ij given by (3.43)-(3.72) (for a fixed I) into the right hand side of (3.81)-(3.82). This gives evaluation map from B + or B − to the q-oscillator superalgebra. We denote this map as ρ I (x). Similar maps from (restricted to) B + to the q-oscillator (super)algebra were considered for U q (ŝl(2)) [5], U q (ŝl(3)) [6], U q (ŝl(M)) [9] and U q (ŝl(2|1)) [1]. Here we used L −1 ii in (3.81) instead of L ii since L −1 ii (for ∈ I) do not coincide with L ii for the contracted algebras U q (gl (M|N; I)). We remark that ρ I (x) is not an evaluation map from U q (ĝl(M|N)) to the q-oscillator superalgebra but rather a map from a contracted algebra U q (ĝl(M|N; I)) on U q (ĝl(M|N)). In fact, the following contracted commutation relations hold true under the map.
[e 2 , [e 0 , [e 2 , e 1 ] q −1 ]] = 0 for 1 ∈ I, 2, 3 ∈ I,  [6]. Some of the Serre-type relations in section 2 automatically hold true under these relations. For example, we found the following relations 23 : . This is because our L-operators are image of the universal R-matrix (up to an overall factor N I (x, y)): L I (y/x) = N I (x, y)(ρ I (x)⊗ π(y))(R) (see also discussions on the universal R-matrix in [34]). Note that the relation for f i , namely has the standard form only for the case i, i + 1 ∈ I (0 ≡ M + N) since we are considering a contracted algebra U q (ĝl (M|N; I)). In particular, this can be 0 = 0 for i, i + 1 / ∈ I case. We have observed the relations  (M|N; I)). Then we can consider evaluation representations of U q (ĝl(M|N; I)) based on the representations of U q (gl (M|N; I)). The co-product ∆ : U q (ĝl(M|N)) → U q (ĝl(M|N; I)) ⊗ U q (ĝl(M|N)) for e i and k i is the same as the one in section 2, while the one for f i is contracted as This may be rewritten as The co-product ∆(k i ) = k i ⊗ 1 + 1 ⊗ k i is well defined only for i ∈ I since k i ∈ U q (ĝl(M|N; I)) diverges for i ∈ I. However ∆(q k i ) = q k i ⊗ q k i is still well defined even for i ∈ I (it just becomes 0). We can define contracted universal R-matrices by the contracted co-products for the contracted algebras and (2.16). They are the universal R-matrices for the Q-operators. Furthermore, it will be important to evaluate a contracted universal 24 Here (2.29) is not always true since the generators are renormalized.
R-matrix in U q (ĝl(M|N; I)) ⊗ U q (ĝl (M|N; J)). For this, we will have to repeat similar calculations discussed in the appendix B for B + as well as B − . The original universal R-matrix (under a certain condition) will be factorized with respect to contracted universal R-matrices. This is our step toward the construction of the Q-operators for the generic representations on the quantum space.
We may also interpret U q (ĝl (M|N; I)) as a subalgebra of an asymptotic algebra (cf. [15]) associated with U q (ĝl(M|N)). In terms of the asymptotic algebra, the vanishing of the action of the Cartan generator q k i for i ∈ I in (2.30) occurs on the level of the representation. Here we regarded this as a phenomenon on the level of the algebra and defined the contracted algebra U q (ĝl(M|N; I)). As for the FRT formulation of U q (ĝl(M|N; I)), we must replace the condition (3.2) with On the other hand, in the context of the asymptotic algebra, we just forget about (3.2) and interpret that (3.108) occurs on the level of the representation. There is an option to exclude (3.85) or (3.86)-(3.97) form the definition of U q (ĝl(M|N; I)) and interpret them as relations on the level of the representation. However, it will be useful to look for smaller algebras to factorize the universal R-matrix as the product of elemental ones.
In this paper, we consider contractions defined by (3.35)-(3.36). Instead of (3.36), one can consider the following: (3.109) The L-operators based on this contraction have one to one correspondence to the ones proposed in this paper. They seem to be the image of the Cartan anti-involution for our L-operators. One may also consider more general contractions than (3.36) and (3.109): L ii = 0 for i ∈ I 1 , L ii = 0 for i ∈ I 2 , I 1 , I 2 ⊂ I.

T-and Q-operators
In this section, we define Q-operators based on the q-oscillator representations introduced in the previous section and sketch an idea how to write the T-operators in terms of them. This gives a cue for operator realization of the formulas in our previous papers [2,3].
We introduce the universal boundary operator where ϕ i ∈ C. This boundary operator is a Cartan element of U q (ĝl(M|N)). Due to the first relation in (2.16), its co-product commutates with the universal R-matrix The images of the evaluation map (2.22) and ρ I (x) are given as 3) We define the universal T-operator by Note that T λ (x) is an element of B − and this definition of the T-operator does not depend on the particular representation of the quantum space. It is convenient to introduce operators where 1 ≤ k ≤ M + N. Then the T-operator (4.5) can be rewritten as where whereK is introduced in (2.32). Here we have renormalized the boundary operator (4.1) by the prefactor of the universal R-matrix (2.18) as in [1]. In the U q (ŝl(M|N))-picture, we may define (4.1), (4.6) and (4.8) respectively as where d k0 = d M +N,j = 0, and the parameter ϕ M +N is defined by the relation In this case, the following relation holds: If there is no reduced universal R-matrix in (4.7), the following quantity Z(λ) = (Str π λ (x) ⊗ 1) D , (4.11) gives the supercharacter. For finite dimensional modules, it is a supersymmetric Schur function on the variables (4.6). In particular for the Verma module, it leads . (4.13) In the above formulas, the reduced universal R-matrix plays a role to put the spectral parameter into the supercharacters, or to change the supercharacters to the q-supercharacters. This induces sort of shits on the parameters (4.6) in the supercharacters. Let F I be the Fock space defined by the action of the generators {c ai , c † ia , n ia } (i ∈ I, a ∈ I) of the q-oscillator superalgebras on the vacuum (3.75). We define the universal Q-operator by where the normalization function reads . (4.17) As expected, this coincides with a limit of a normalized character of the Kirillov-Reshetikhin module at least for the case 26 N = 0 (cf. [15]):.
) is the Schur function. Here we meant the equality in (4.18) by the substitution of elements of B − (4.6) for the complex numbers {z k } on the right hand side after the limit. The normalization factor in (4.18) came from the shift automorphism (2.34) on B + for the parameters in (3.79). We expect [2,3] that the T-operator is given by the Baxterizaiton of the supercharacter 27 where d k are differential operators which evaluate the degrees of the monomials on {z j } in the right of the dot ·. They effectively act as We assume d k act on the functions in the left of the dot · as just an identity, although {Q {k} } are also functions of {z k }. In particular for the Verma module 28 , we have [3] T We remark that the most of the T-operators can be written as summentions of the above formula (4.20) based on the Bernstein-Gelfand-Gelfand resolution and rewritten as Wronskian-like determinants (see [5] for U q (ŝl(2)), [6] for U q (ŝl(3)), [9] for finite dimensional representations of U q (ŝl(M)) (see also a Wronskian like determinant in [35]), [1] for U q (ŝl(2|1)); [2,3] for the Wronskian-like determinants for any U q (ĝl(M|N))). We expect our universal Q-operators obey functional relations 26 We have also checked that a normalized Sergeev-Pragacz formula produces (4.17) in the large Young diagram limit under a similar condition for the case M N = 0. 27 The shift of the spectral parameter of the Q-operators in [2,3] can be recovered by putting q → q −1 after the replacement Q I (x) → Q I (xq k∈I (−1) p(k) ). 28 This formula (4.20) was presented first as a poster at a conference 'Integrability in Gauge and String Theory 2010', Nordita, Sweden, 28 June 2010 -2 July. To fit the formula in [3], one has to make an overall shift of the spectral parameter x → zq 2(M−N ) after the manipulation in the footnote 27. of the form: for p(i) = p(j): (4.21) and for p(i) = p(j): At the moment, these functional relations are fully proven for U q (ŝl(2)) [5], for U q (ŝl (3)) [6] and for U q (ŝl(2|1)) [1]. Their proof is based on decompositions of qoscillator representations of B + and does not rely on the representation of B − on the quantum space. See also [20,17] for discussions on rational models (q = 1). On the level of the eigenvalues of Q-operators for rational models, (4.22) were discussed in details in relation to the Bäcklund transformations [36]. Here we used expressions based on the 2 M +N index sets on the Hasse diagram presented in [2]. Now that we have the universal T-and Q-operators (4.5), (4.14), our next task is to evaluate these for particular representations of B − on the quantum space of the model. For example, the T-operator for the lattice model whose quantum space is the fundamental representation on each site is given as λ (x) (π(ξ 1 ) ⊗ π(ξ 2 ) · · · ⊗ π(ξ L )) ∆ (L−1) T λ (x) (4.23) = Str π λ L 0L (ξ L /x) · · · L 02 (ξ 2 /x)L 01 (ξ 1 /x)(D ⊗ 1 ⊗L ) , (4.24) where L is the number of the lattice site; the complex parameters {ξ j } L j=1 are inhomogeneities on the spectral parameter; and N (L) λ (x) is a function for the normalization. In (4.24), the evaluation map (3.27) is used and the supertrace is taken over the auxiliary space denoted as '0'. The Q-operators for the same system are given by I (x) (π(ξ 1 ) ⊗ π(ξ 2 ) · · · ⊗ π(ξ L )) ∆ (L−1) Q I (x) (4.25) where Z I := (π(ξ 1 ) ⊗ π(ξ 2 ) · · · ⊗ π(ξ L )) ∆ (L−1) Z I and the normalization function is N (L) It is instructive to calculate the lattice T-operator (4.24) for the Verma module 29 and the lattice Q-operator (4.26) even for one site 29 We remark that a formula similar to the first equality in (4.27) (for the characters of finite dimensional representations of U q (gl(M ))) was previously derived by Anton Zabrodin in 2007 based on the trigonometric version of the co-derivative for L = 1 case. L = 1 case. Let us introduce a notation Z + (λ) := π(ξ 1 )(Z + (λ)). Then we obtain for i ∈ I,  2π 0 du e i ⊗ V i (u) , where V i (u) are q-vertex operators obeying V i (u)V j (v) = (−1) p(i)p(j) q a ij V j (v)V i (u) for u > v and e i are the generators of B + . Thus, if we substitute our q-oscillator realizations of B + through (3.81) into the formula and taking the supertrace over the Fock space for B + we will obtain Q-operators for the CFT. Examples of such Q-operators can be seen for (M, N) = (2, 0) in [5], (M, N) = (3, 0) in [6], N = 0 in [9], (M, N) = (2, 1) in [1] and for U q (C(2) (2) ) in [7]. See also a related recent paper [34].
Finally, we can define the universal master T-operator [22] by τ (x, t) = λ S λ (t)T λ (x), (4.30) where t = (t 1 , t 2 , . . . ) are time variables in the KP hierarchy and S λ (t) is the Schur function labeled by the Young diagram λ. This is a τ -function of the modified KP hierarchy and allows embedding of the quantum integrable system into the soliton theory. Basically, all the functional relations among T-and Q-operators in the Hirota form can be derived from this (see [22,20] for more details).

Concluding remarks
In this paper, we have developed our preliminary discussions on L-operators for the Baxter Q-operators for U q ( sl(2|1)) [14,1] and U q ( gl (3)) [16], and generalized them to the higher rank case U q ( gl(M|N)). The contraction of the algebra related to these L-operators was discussed. The model independent universal Q-operators are defined as supertrace of the universal R-matrix. This is a step toward our trial [1,2,3] (also [20,22]) to construct systematically Q-operators and Wronskian-like expressions of T-operators in terms of them. The L-operators given in this paper can be building blocks of them. Our next task [37] directly related to this paper will be mainly two fold: to generalize our q-oscillator realization of the L-operators for the Q-operators to all the intermediate ones labeled by any 2 M +N index set I introduced in [2], and to generalize these for more general representations on the quantum space. All these will be basically accomplished by evaluating the universal R-matrix in the light of asymptotic representations of the quantum affine algebra [15]. We find that a fusion method [17,19] on L-operators for Q-operators developed for rational models is also helpful for this. A generalization to the elliptic case is perhaps interesting. Although whether the contraction of the Sklyanin algebra works is not clear at the moment, elliptic L-operators may be given by twists 31 of our trigonometric L-operators since the elliptic algebras (for both vertex type models and face type models) can be obtained by twists on the quantum affine algebras [38].
The other obvious direction of further development will be a generalization to the other quantum affine superalgebras. For this, it will be helpful to characterize our L-operators as sort of Lax operators for the generalized Toda system [39] in terms of the asymptotic algebra [15] and investigate the system in the light of the soliton theory [20,22]. or b ≤ a < d ≤ c, h i = h i − (θ(i ∈ I) − θ(i + 1 ∈ I))m, (B3) where the right hand side of these should be understood under the evaluation map ev xq 2m (3.81)-(3.82); the effect of the renormalization is denoted by tilde; and the suffix i should be interpreted under modulo M + N. (B3) and (B4) came from the transformations for the shift automorphisms (2.15) and (2.34), respectively. Then the commutation relations become Let us consider the limit m → ∞ for |q| < 1 (or m → −∞ for |q| > 1). We assume the renormalized generators except for (B5) do not diverge in this limit at least for the evaluation representation π I (xq 2m ) in an appropriate basis, where π I is the highest weight representation of U q (gl(M|N)) with the highest weight (3.80). Then, in the limit, we obtain: q (−1) p(i)k i q (−1) p(i)k i = q 2m(1−θ(i∈I)) → θ(i ∈ I), and in particular q (−1) p(i)k i → 0 for i ∈ I.
The inverse of q (−1) p(i)k i , namely q −(−1) p(i)k i coincides with q (−1) p(i)k i only for i ∈ I in the limit. Then the commutation relations (B6) reduce to the contracted commutation relations (3.84) in the limit. Note that the limit of (B6) automatically hold true iff i = 0 for i, i + 1 ∈ I in the limit. Let us multiply (U q (ĝl(M|N)) case of) the first relation in (2.16) for f i by q (2−θ(i∈I)−θ(i+1∈I))m (1 ⊗ q −m j∈I k j ) from the right: whereR is defined in (2.33). One can see an effect of the shift automorphism for B + by the transformation (2.34) with the parameters (3.79). Then this relation for π I (xq 2m ) ⊗ π(y) suggests (3.102) in the limit.