Generating Series for Nested Bethe Vectors

We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincar\'e 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra $U_q(\widehat{\mathfrak{gl}}_2)$.


Introduction
Hierarchical (nested) Bethe ansatz (NBA) was designed [12] to solve quantum integrable models with gl N symmetries. The cornerstone of NBA is a procedure which relates Bethe vectors for the model with gl N symmetry to analogous objects with gl N −1 symmetry. This hierarchical procedure is implicit, it allows to obtain Bethe equations for the parameters of the Bethe vectors while the explicit construction of these vectors itself remains rather non-trivial problem. Authors of the papers [16,17] proposed a closed expression for off-shell Bethe vectors as matrix elements of the monodromy operator built of products of fundamental L-operators. However a calculation of this expression in every particular representation is still a nontrivial problem. Such a calculations was done in [18] on the level of the evaluation homomorphism of U q ( gl N ) → U q (gl N ).
The construction of [16,17] yields the off-shell Bethe vectors in terms of matrix elements of the monodromy matrix which satisfies the corresponding quantum Yang-Baxter equation and generate a Borel subalgebra of quantum affine algebra U q ( gl N ) [15] for trigonometric R-matrix or the Yangian Y (gl N ) for the rational R-matrix. Those quantum affine algebras as well as doubles of Yangians possess another "new" realization introduced in [2]. In this realization the corresponding algebra is described in terms of generating series (currents) and an isomorphism between different realizations of these infinite-dimensional algebras was observed in [3]. Using this isomorphism one may try to look for the expressions for the universal off-shell Bethe vectors in terms of the modes of the currents. This program was realized in [9,10], where explicit formulas for the off-shell Bethe vectors in terms of the currents were found. A significant part of this approach to the construction of Bethe vectors is a method of projection introduced in [7] and developed in the recent paper [6]. This method operates with projections of Borel subalgebra to its intersections with Borel subalgebras of a different type. It was proved in [6] that universal off-shell Bethe vectors can be identified with the projections of products of the Drinfeld currents to the intersections of Borel subalgebras of different types. In the latter paper it was checked in a rather general setting that the Bethe vectors obtained from the projections of the currents satisfy the same comultiplication rule as the Bethe vectors constructed in terms of the fundamental L-operators. This approach was used in [11,14] for further generalization of the results obtained in [18].
A deduction of general expressions for off-shell Bethe vectors [11,14] is based on hierarchical relations between projections of products of the currents (see Proposition 4.2 in [11]). However these relations do not help in the investigation of the action of integrals of motions on off-shell Bethe vectors. The situation is quite different in classical approach of nested Bethe ansatz. The corresponding hierarchical relations allow to compute the action of integrals of motion and derive the Bethe equations [12,13], but can be hardly used in the computation of the explicit expressions of Bethe vectors. This signifies an existence of two types of hierarchical relations for off-shell Bethe vectors and thus of two different presentations of them. The goal of this paper is to observe these two type of the hierarchical relations within the approach of the method of projections.
The new important objects which appear in the application of the method of projections to the investigation of the off-shell Bethe vectors are so called strings and their projections. Strings are special ordered products of composed current introduced in [4]. Our hierarchical relations express U q ( gl N ) Bethe vectors via products of special strings and U q ( gl N −1 ) Bethe vectors. On the other hand the basic point of application of the method of projection is the ordered decomposition of the product of the currents. The factors of this decomposition are projections of the products of the currents (3.12). The crucial observation is that this decomposition and hierarchical relations for the opposite projections of the products of currents have a similar structure. Combinations of relations of these two different type can be used to obtain a new type hierarchical relations for off-shell Bethe vectors.
In order to solve the latter problem we collect all off-shell Bethe vectors into multi-variable generating series. We also introduce the generating series for the products of the currents and for the strings and their projections. We rewrite the hierarchical relations for U q ( gl N ) Bethe vectors as a simple relation on the product of the generating series of the projections of the strings and the generating series of the U q ( gl N −1 ) Bethe vectors. A similar construction can be performed for off-shell Bethe vectors related to opposite Borel subalgebra. However the product entering into these relations is not usual. It contains a q-symmetrization with a special functional weight. This ⋆-product is associative and the generating series are invertible with respect to this product. Finally the new type of the hierarchical relations reduces to inversion of the generating series of opposite projections of the strings. This inversion is effectively performed. To do this we introduce some combinatorial language of tableaux filled by the Bethe parameters (see Section 4.5).
Applications of the new type hierarchical relations to the investigation of the properties of quantum integrals of motion are given in [8]. Here we demonstrate how they work in the simplest case of the universal U q ( gl 2 ) Bethe vectors. As a result we get universal Bethe equations of the analytical Bethe ansatz [1]. In contrast to the usual Bethe equations these equations refer to Cartan currents instead of the eigenvalues of the diagonal elements of monodromy matrix on highest weight vectors. In the Appendix we collect the basic defining relations for the U q ( gl N ) composed currents.
2 Generating series and ⋆-product 2.1 A q-symmetrization Lett = {t 1 , . . . , t n } be a set of formal variables. Let G(t) be a Laurent series taking values in U q ( gl N ). Consider the permutation group S n and its action on the formal series of n variables t = {t 1 , . . . , t n } defined for the elementary transpositions σ i,i+1 as follows where the rational series 1 1−x is understood as a series n≥0 x n and q is a deformation parameter of U q ( gl N ). Summing the action over the group of permutations we obtain the operator Symt = σ∈Sn π(σ) acting as follows: The product is taken over all pairs (ℓ, ℓ ′ ), such that conditions ℓ < ℓ ′ and σ(ℓ) > σ(ℓ ′ ) are satisfied simultaneously. We call operator Sym t a q-symmetrization. One can check that the operation given by (2.2) is a projector Symt Symt (·) = Symt (·). Denote by [l,r] a set of segments which contain positive integers {l a + 1, l a + 2, . . . , r a − 1, r a } including r a and excluding l a . The length of each segment is equal to r a − l a . For a given set [l,r] of segments we denote byt [l,r] the sets of variables . We also name for a short collections [l,r] of segments as a segment.
Denote by Sl ,r = S l 1 ,r 1 × · · · × S l N−1 ,r N−1 a direct product of the groups S la,ra permuting integers l a + 1, . . . , r a . Let G(t [l,r] ) be a series depending on the ratios t a i /t b j for a < b and t a i /t a j for i < j. The q-symmetrization over the whole set of variablest [l,r] of the series G(t [l,r] ) is defined by the formula where the set σt [l,r] is defined as We say that the series G(t [l,r] ) is q-symmetric, if it is invariant under the action π of each group S la,ra with respect to the permutations of the variables t a la+1 , . . . , t ra for a = 1, . . . , N − 1: Due to (2.3) the q-symmetrization G(t [l,r] ) = Symt [l,r] Q(t [l,r] ) of any series Q(t [l,r] ) is a q-symmetric series.
The rational function which is understood as a series with respect to t a i /t a j is an example of a q-symmetric series.

Generating series
Let u i , i = 1, . . . , N − 1 be formal parameters. We denote the set of these parameters as u = {u 1 , . . . , u N −1 }. Define a generating series where the coefficients A(t [n] ) are arbitrary q-symmetric series (Symt We call generating series of this type q-symmetric generating series. Note that the multi-indexn of the coefficients of a q-symmetric generating series is uniquely defined by the set of formal variablest [n] , thus the coefficients A(t [n] ) are used without any additional index. However once it will be convenient for us to use instead the notation An ≡ A(t [n] ) (see proof of Proposition 5). For two generating series A(ū) and B(ū) we define ⋆-product as a generating series with coefficients where and (2.15) Proposition 1. The ⋆-product is associative, namely, for three arbitrary q-symmetric series A(ū), B(ū) and C(ū) of the form (2.10) Proof . We check an equality (2.16) first in the simplest case of the generating series depending on one generating parameter u. Equating the coefficients at the n-th power of this parameter we obtain from (2.16) an equality Symt   n≥m≥0 n≥s≥m A(t s+1 , . . . , t n ) · B(t m+1 , . . . , t s ) · C(t 1 , . . . , t m ) where the property (2.8) of the q-symmetric generating series was used andt is a set {t 1 , . . . , t n }. An equality (2.17) is an obvious identity if one replaces the ordering of the summations. It is clear that in the general case the arguments remain the same and the appearing of the series (2.15) does not change these arguments.

Proposition 2.
For any generating series A(ū) there exist an unique q-symmetric series B(ū) such that Proof . Since A(ū) has the form of a Taylor series with the free term equal to 1, we can always reconstruct uniquely the inverse series solving recursively the equations for the coefficients of the series B(ū). By the construction the coefficients of this series will be also q-symmetric.
3 Universal nested Bethe vectors for U q ( gl N ) Quantum affine algebras in the current realization [2] provide examples of the q-symmetric generating series. We will construct these generating series for the quantum affine algebra U q ( gl N ) and show that ⋆-products of these generating series provide hierarchical relations for NBA. We now recall the current realization of the algebra U q ( gl N ). The quantum affine algebra U q ( gl N ) is generated by the modes of the currents where i = 1, . . . , N − 1 and j = 1, . . . , N subject to the commutation relations given in the Appendix A. The generating series F i (z), E i (z) and k ± (z) are called total and Cartan currents respectively. We consider two types of Borel subalgebras of the algebra U q ( gl N ). Generators of the standard Borel subalgebras U q (b ± ) ⊂ U q ( gl N ) can be expressed in terms of the modes of the currents (3.1). To do this one has to introduce the composed currents E a,b (z) and F b,a (z) for a < b − 1 and 1 ≤ a < b ≤ N (see Appendix A for the definition of the currents F b,a (z)). The Borel subalgebra U q (b + ) is generated by the modes of the currents: . . , N . The reader can find description of the standard Borel subalgebras in terms of the modes of the U q ( gl 3 ) currents in the paper [9]. This decomposition of the algebra U q ( gl N ) is related to the standard realization of this algebra in terms of pair of the dual L-operators, where generators of the standard Borel subalgebras serve as the modes of the Gauss coordinates of the corresponding L-operators.
Another type of Borel subalgebras is related to the current realization of U q ( gl N ) and was introduced in [ . . , N , n ∈ Z and m > 0. We call these subalgebras of U q ( gl N ) the current Borel subalgebras. Further, we will be interested in the intersections, and will describe properties of projections to these intersections. The current Borel subalgebras are Hopf subalgebras of U q ( gl N ) with respect to the current Hopf structure for the algebra U q ( gl N ) defined in [2]: The quantum affine algebra U q ( gl N ) with ommited central charge and gradation operator can be identified with the quantum double of its current Borel subalgebra constructed using the comultiplication (3.3). One may check that the intersections U − f and U + F are subalgebras. It was proved in [10] that these subalgebras are coideals with respect to Drinfeld coproduct (3.3) and the multiplication m in U q ( gl N ) induces an isomorphism of vector spaces According to the general theory presented in [6] we define projection operators P + : where ε is the counit map: ε : U q ( gl N ) → C. Denote by U F an extension of the algebra U F formed by linear combinations of series, given as infinite sums of monomials a i 1 [n 1 ] · · · a i k [n k ] with n 1 ≤ · · · ≤ n k , and n 1 + · · · + n k fixed, (1) the projections (3.4) can be extended to the algebra U F ;

Generating series for universal Bethe vectors
It was proved in the papers [10,11] that the projection of the product of the U q ( gl N ) currents can be identified with universal Bethe vectors (UBV). In this paper we show that the hierarchical relations for UBV can be presented in a compact form using ⋆-product of certain q-symmetric generating series. Then the formal inversion of generating series allows to obtain another form of hierarchical relations and to investigate further (see [8]) special properties of UBV when their parameters satisfy the universal Bethe equations appeared in the framework of the analytical Bethe ansatz [1]. In this paper we will demonstrate these properties for the U q ( gl 2 ) universal Bethe vectors. Products of the U q ( gl N ) currents yield examples of the q-symmetric generating series. We consider the generating series of the product of the currents ) means the following normalized product of the currents We set F N 0 ≡ 1. More generally, following the convention (2.14), for a segment [l,r] and the related collectiont [l,r] of variables, see (2.5), we set The superscript N in the notation F N (t [n] ) of the coefficients of the q-symmetric generating series signifies that these coefficients belong to the subalgebra U F ⊂ U q ( gl N ). Further on we will consider smaller algebras U q ( gl j+1 ), j = 1, . . . , N −1 embedded into U q ( gl N ) in two different way.
Using these embeddings, for any segment [l,r] such that and for any segment [l,r] such that l i = r i for all i = 1, 2, . . . , j − 1 a series These series are gathered into q-symmetric generating functions F N −j+1 (u 1 , . . . , u N −j ) and F N −j+1 (u j , . . . , u N −1 ). Subscripts a = 1, . . . , N − 1 of the formal parameters u a in the definition of these generating series denote the indices of the simple roots of the algebra gl N . For example, the notation F N −1 (u 1 , . . . , u N −2 ) means the generating series which coefficients take values in the subalgebra On the other hand, the notationF N −1 (u 2 , . . . , u N −1 ) means the generating series taking value in the subalgebra U q ( gl N −1 ) embedded into U q ( gl N ) by means of the mapτ N −2 . Further on we use the special notation for the ordered product of the noncommutative entries.
A a will mean ordered products of noncommutative entries A a , such that A a is on the right (resp., on the left) from A b for b > a: Using these notations we write then-th term of the generating series (3.6) as follows Besides of the generating series of products of the total currents we consider also the generating series of the projections of the products of the currents In the same manner we define series and The series P + F N (ū) is the generating series of all possible universal off-shell Bethe vectors. Our goal is to show that the hierarchical relations of the nested Bethe vectors imply the factorization property of this generating series with respect to the ⋆-product of certain q-symmetric generating series. An associativity of this product allows to obtain a new presentation for the universal Bethe vectors.
We call any expression i f The q-symmetric generating series (3.6) can be written using a ⋆-product in a normal ordered form where the q-symmetric generating series P ± F N (ū) are defined by (3.9).
Proof . Using the property of the projections (3.5) an equality of the series was proved in [11] (see, Proposition 4.1 therein). That proof was based on the comultiplication property (3.3) and the commutation relation between currents. Using a definition of the ⋆-product and considering the coefficients in front of monomial u n 1 1 · · · u n N−1 N −1 in the both sides of the equality (3.11) we obtain the formal series equality (3.12).
4 Generating series of strings and nested Bethe ansatz 4
Define an ordered normalized product of the composed currents, which we call a string of the type j: 1 taking values in the subalgebra U F ⊂ U q ( gl j+1 ) embedded into U q ( gl N ) by the map τ j . The composed currents V (t a+1 m a+1 +sa , . . . , t a+1 m a+1 +1 ; t a na , . . . , t a ma+1 ), We define the q-symmetric generating series of the strings of the type j by the formula Here superscript j+1 signifies that this generating series takes values in the subalgebra U q ( gl j+1 ) embedded into U q ( gl N ) by the map τ j . The subscripts of the parameters u 1 , . . . , u j signify that this subalgebra is generated by the U q ( gl N ) currents corresponding to the simple roots with indices 1, . . . , j. Let P + F N (u 1 , . . . , u N −1 ) be the generating series of the universal off-shell Bethe vectors for the algebra U q ( gl N ) and P + F N −1 (u 1 , . . . , u N −2 ) be the analogous series for the smaller algebra U q ( gl N −1 ) embedded into U q ( gl N ) by the map τ N −2 . We have the following Proposition 4. Hierarchical relations between universal weight functions for algebras U q ( gl N ) and U q ( gl N −1 ) can be written as the following equality on generating series: Proof . Taking the coefficients in front of the monomial u n 1 1 u n 2 2 · · · u n N−1 N −1 we obtain an equality of the formal series (4.7) An equality (4.7) coincides with the statement of the Proposition 4.2 of the paper [11] up to renormalization of the universal weight function by the combinatorial factor.
Recall that the generating series P + S j (u 1 , . . . , u j−1 ) belongs to subalgebra U F ⊂ U q ( gl j ) embedded into U q ( gl N ) by the map τ j−1 which removes the currents F a (t), E a (t) and k ± a+1 (t) with a = j, j + 1, . . . , N − 1.

Hierarchical relations for the negative projections
One of the results of the papers [11,14] is the hierarchical relation for the positive projections of the product of the currents. Let us recall shortly the main idea of this calculation. In order to calculate the projection we separate all factors F a (t a ℓ ) with a < N − 1 and apply to this product the ordering procedure based on the property (3.12). We obtain under total projection the q-symmetrization of terms x i P − (y i )P + (z i ), where x i are expressed via modes of F N −1 (t) and y i , z i via modes of F a (t) with a < N − 1. Then we used the property of the projection that and reorder the product of x i and P − (y i ) under positive projection to obtain the string build from the composed currents (cf. equation (4.7)).
We will use an analogous strategy to calculate the negative projection of the same product of the currents Now we separate all factors F a (t a ℓ ) with a > 1 and apply to this product the ordering rule (3.12). Again, we obtain under total negative projection the q-symmetrization of terms P − (x i )P + (y i )z i , where z i are expressed via modes of F 1 (t) and y i , z i via modes of F a (t) with a > 1. Using now the property of the projections and reordering the product of P + (y i ) and z i under negative projection we obtain the desired hierarchical relations for the negative projection of the currents product.
We will not repeat these calculations since they are analogous to the ones presented in [11], but will formulate the final answer of these hierarchical relations. For two sets of variables {t 1 1 , . . . , t 1 l } and {t 2 1 , . . . , t 2 l } we introduce the series When k = N − 1 we setX(·) = 1.
Define an ordered normalized product of the composed currents, which we call a dual string of the type N − k: Note that the notion of the dual string is different from the notion of the inverse string used in [11]. More generally, letm = {m 1 , . . . , m N −1 } andn = {n 1 , . . . , n N −1 } be a pair of collections of nonnegative integers such that n a − m a = 0 for a = 1, . . . , k − 1 and n a − m a = s a for any a = k, . . . , N − 1. Then for the collection of variables (t a+1 n a+1 , . . . , t a+1 m a+1 +1 ; t a na , . . . , t a na−s a+1 +1 ), Doing the calculations described above we obtain the recurrence relations for the negative projections We define the generating series of the dual strings of the type N − j by the formulã taking values in the subalgebra U F ⊂ U q ( gl N −k+1 ) embedded into U q ( gl N ) by the mapτ N −k which removes the currents F a (t), E a (t) and k ± a (t) for a = 1, . . . , k − 1. The recurrence relations (4.14) can be written as the ⋆-product of the generating series Generating series of the negative projections of the product of the currents can be written using ordering ⋆-product of the generating series of the dual strings

Other type of the hierarchical relations
A special ordering property of the universal Bethe vectors when their parameters t a ℓ satisfy the universal Bethe equations [1] was investigated in [8]. This property leads to the fact that the ordering of the product of the universal transfer matrix and the universal nested Bethe vectors is proportional to the same Bethe vector modulo the terms which belong to some ideal in the algebra if the parameters of this vector satisfy the universal Bethe equations. We will demonstrate this property for the U q ( gl 2 ) universal Bethe vectors in the Section 4.7.
A cornerstone of this ordering property lies in a new hierarchical relations for the universal Bethe vectors, which can be proved using the technique of the generating series. Here we give the detailed proof of the relation which particular form was used in the paper [8].
Using normal ordering relation (3.11) and (4.15) we may write the generating series of the product of the currents in the form On the other hand these generating series may be presented as the factorized product Applying the ordering relation (3.11) to the seriesF N −1 (u 2 , . . . , u N −1 ) again we obtain an alternative to (4.17) expression for the generating series F N (u 1 , . . . , u N −1 ): Equating the right hand sides of (4.17) and (4.19) we obtain the identity The identity (4.21) relates the universal off-shell Bethe vectors for the algebra U q ( gl N ) and for the smaller algebra U q ( gl N −1 ). They can be considered as an universal formulation of the relation used in the pioneer paper [12] for the obtaining the nested Bethe equations. The equality (4.21) between generating series contains many hierarchical relations between UBV. In order to get some particular identities between these UBV one has to invert explicitly the generating series P − S N (u 1 , . . . , u N −1 ) . This will be done in the next subsection.  Its solution can be written in the form

Inverting generating series of the strings
After this we consider the relation (4.23) for arbitrary n 1 ≥ 0, n 2 = 1 and n 3 = · · · = n N −1 = 0. Avoiding writing the dependence on the 't' parameters, that is using notations Ek instead of E(t [k] ), the relation (4.23) takes the form The rational series Z m 1 ,0,0 disappears in the first sum of (4.26) by the same reason as in (4.24). Let us consider the relation (4.26) for n 1 = 0. The second sum is absent and the first sum contains only one terms D 0,1,0 · E 0,0,0 = 0 which is equal to zero. This proves that the coefficient D 0,1,0 = 0 vanishes identically. Now the first sum in the relation (4.26) is terminated at m 1 = n 1 − 1 and this relation allows to find all coefficients D n 1 ,1,0 starting from D 1,1,0 = −Z 1,1,0 E 1,1,0 . Considering the relation (4.23) for n 1 ≥ 0, n 2 = 2 and n 3 = · · · = n N −1 = 0 we prove first that D 0,2,0 = D 1,2,0 = 0 and then can find all coefficients D n 1 ,2,0 starting from D 2,2,0 . It is clear now that the coefficients D n 1 ,n 2 ,0 are non-zero only if n 1 ≥ n 2 . Continuing we prove the statement of the proposition.

Inversion and combinatorics
To invert explicitly the generating series of the projection of the strings we have to introduce certain combinatorial data. First of all, according to Proposition 5 we fix a sequence of nonnegative integers n 1 ≥ n 2 ≥ · · · ≥ n N −1 and the corresponding set of the variablest [n] . Choose any positive integer n and p = 1, . . . , n. A diagram χ of size |χ| = n and height p = h(χ) is an ordered decomposition of n into a sum of p nonnegative integers, Equivalently, a diagram χ consists of p rows and the i-th row contains χ i boxes, An example of such diagram is shown in the Fig. 1. The rows of the diagrams are numbered from the bottom to the top. If χ i = 0 for some i = 1, . . . , p then the diagram contains several disconnected pieces. We will call the diagram χ connected if all χ i = 0 for i = 1, . . . , p.
A tableauxχ with a given diagram χ is a filling of all boxes of χ by the indices {1, 2, . . . , N −1} of the positive roots of the algebra gl N with the condition of non-increasing from the left to the right along the rows. If an index a is associated to a box of the tableaux, we say that this box has a 'type' a. We will call the tableaux associated to the connected diagrams the connected for all a = 1, . . . N − 1. This formula demonstrates in particular that the weight of a tableaux always satisfies the admissibility conditions n 1 ≥ n 2 ≥ · · · ≥ n N −1 .
To each connected tableauxχ of the weightn =n(χ) we associate a decomposition of the set of the variablest [n] into the union of |χ| disjoint subsets, each corresponding to a box of the diagram χ of tableauxχ. To each box of the type a we associate one variable of the type 1, one variables of the type 2, etc., one variable of the type a, altogether a variables. We will number variables of the each type starting from the most bottom row and the most right box where variable of this type appear for the first time.
Let us give an example of the decomposition and of the ordering for the tableaux shown on the Fig. 3. The most bottom and the right box has the type 1. We associate to this box one variable 2 t 1 1 of the same type. Next to the left along the same row box has the type 2. We associate to this box two variables t 1 2 and t 2 1 of the types 1 and 2. Last box in the bottom row has type 3 and we associate to this box three variables t 1 3 , t 2 2 and t 3 1 . Next box is in the next row and also has the type 3. To this box we associate also three variables t 1 4 , t 2 3 and t 3 2 . Next two boxes in the third row both have the type 1 and we associate to the most right box in this row one variable t 1 5 and to the last box also one variable t 1 6 . In general, for each tableauxχ of the weightn(χ) and any segment [l,r], such thatr −l =n the set of the variablest The variable t a k belongs to the subsettχi if l a + h i−1 a < k ≤ l a + h i a . It is located in the (l a + h i a + 1 − k)th box of the rowχ i counting boxes in this row from the left edge of the tableaux.
In the same setting we define Zχi ,χ j (tχi;tχj ), (4.28) where is a rational series defined by the interchanging of the variables of the type a + 1 from the i-th row and variables of the type a from the jth row of the tableauxχ. In our example, the group of variables decomposes into three groups In this example the rational series Zχ(t [n] ) is equal to For a given tableauxχ we define the ordered product ) and Proof . For arbitrary non-empty setn =0 we substitute expression (4.33) into (4.23) to obtain the relation which has to be equal 0. After this substitution the product of the series Zχ′(t [m,n] ) · Eχ′(t [m,n] ) should be under q-symmetrization over the set of the variablest [m,n] . Since the series Zm(tn) is symmetric with respect to the set of these variables and the series E(tm) does not depend on the variablest [m,n] we may include these series under q-symmetrization over the variablest [m,n] . Then, since the variablest [m,n] forms a subset of the variablest [n] , the q-symmetrization over variablest [m,n] disappear due to the property (2.3). We will prove the cancellation of the terms in (4.34) in the sums over tableaux of the fixed height. Keep the terms in the summation of the first line of this relation which correspond to the connected tableauxχ such that n a (χ) = n a and h(χ) = p + 1. Keep the terms in the summation of the second line of (4.34) which correspond to all connected tableauxχ ′ such that n a (χ ′ ) = n a − m a and h(χ ′ ) = p for fixed p = 0, . . . , n 1 − 1.
Fix a term in the first sum of (4.34) corresponding to some connected tableauxχ with a weightn. Consider the first (bottom) line of the tableauxχ. Denote by m a = c 1 a + · · · + c 1 N −1 the nonnegative integers defined by this row. It is clear that this set of integers satisfies the admissibility condition m 1 ≥ m 2 ≥ · · · ≥ m N −1 and m a ≤ n a . In the second double sum of (4.34) choose the term corresponding to this setm and the tableauxχ ′ defined by the following rule.
If we glue from the bottom of the tableauxχ ′ the row of boxes such that it has a length m 1 and the number of boxes of the type a is equal to m a − m a+1 then for the obtained tableauxχ we require r i a (χ) = r i a (χ) and h(χ) = p + 1 for all possible values i and a. We claim that for each fixed tableauxχ there are unique setm such that m a ≤ n a and there are a single tableauxχ ′ which satisfies above conditions. The tableauxχ andχ coincide actually. The product of the coefficients Eχ′(t [m,n] ) · E(tm) will be equal to Eχ(t [n] ).
According to the definitions of the series Zm(tn) (2.15) and Zχ′(t [m,n] ) (4.28) their product will be equal to the series Zχ(t [n] ). The term corresponding to the fixed tableauxχ in the first line of (4.34) and the term from the second line given bym andχ ′ described above cancel each other since they will enter with different signs: h(χ) = h(χ ′ ) + 1 = p + 1.
For the example of the tableaux shown in the Fig. 3 The series Zχ′(t [m,n] ) is . (4.36) The product of (4.35) and (4.36) obviously coincides with (4.31).
4.6 Inversion of generating series for U q ( gl 2 ) Quantum affine algebra U q ( gl 2 ) in its current realization formed by the modes of the currents 3 E(t), F (t) and Cartan currents k ± 1 (t), k ± 2 (t). Let us invert explicitly the generating series P − (F(ū)) in the simplest case of one generating parametersū = u 1 ≡ u in (3.11) to show what kind of relations can be obtained for the generating series of the U q ( gl 2 ) off-shell Bethe vectors P + (F(u)): (4.37) For any positive n and non-negative p ≤ n we define the set of p + 1 positive integers {k p } = {k 1 , k 2 , . . . , k p , k p+1 } such that 0 = k 0 < k 1 < k 2 < · · · < k p < k p+1 = n. Using this data we define the ordered products It is clear that the inverse generating series P − (F(u)) can be written using q-symmetrization of these ordered products Fk p (t [n] ) as follows Then using the definition of the ⋆-product we can obtain from (4.37) a special presentation for the U q ( gl 2 ) universal weight function where summation over the set {k p } runs over all possible k i such that s = k 0 < k 1 < k 2 < · · · < k p < k p+1 = n. An extreme term in the sum when s = n and the sum over p is absent corresponds to the product of the total currents F (t n ) · · · F (t 1 ). Note that an equality (4.39) can be treated as generalization of Ding-Frenkel relation P + (F (t)) = F (t) − P − (F (t)) [3] when the positive projection is taken from the product of the currents.

Universal Bethe ansatz for
, generated by all element of the form U q (b + ) · E[n], n > 0. As it was mentioned above the standard Borel subalgebra U q (b + ) in terms of the currents generators is formed by the modes F [n], k + 1,2 [n], n ≥ 0 and E[m], m > 0. Let us denote subalgebras generated by these modes as U + f , U + k and U + e , respectively. The multiplication in U q (b + ) implies an isomorphism of the vectors spaces We introduce ordering of the generators in the Borel subalgebra U q (b + ) (4.40) 3 Since algebra gl 2 has only one root we remove index of this simple root in the notation of the currents in case of the current realization of the algebra Uq( b gl 2 ).
induced by the circular ordering of the Cartan-Weyl generators in the whole algebra U q ( gl 2 ) [6]. We call any element w ∈ U q (b + ) normal ordered and denote it as : W : if it is presented as the linear combination of the elements of the form W 1 ·W 2 ·W 3 , where W 1 ∈ U + f , W 2 ∈ U + k , W 3 ∈ U + e . It is convenient to gather the generators of the subalgebras U + f and U + e into generating series which we call the half-currents.
A universal transfer matrix is the following combinations of the Cartan and half-currents Using the commutation relations in the algebra U q (b + ) ⊂ U q ( gl 2 ) one may check that these transfer matrices commute for the different values of the spectral parameters [T (t), T (t ′ )] = 0 and so generates the infinite set of commuting quantities 4 . We are interesting in the ordering relations between universal transfer matrix T (t) and the universal Bethe vector P + F(t [n] ) . Note that the universal transfer matrix is ordered according to the ordering (4.40).
becomes much more involved in the general case of the algebra U q ( gl N ). There is a simple way to avoid these difficulties using the relation (4.39). This relation allows to replace the projection of the product of the currents onto positive Borel subalgebra U q (b + ) by the linear combination of the terms A Current realization of U q ( gl N ) The commutation relations for the algebra U q ( gl N ) in the current realization are given by the following set of the relations and the Serre relations for the currents E i (z) and F i (z) Formulae (A.1) and (A.2) should be considered as formal series identities describing the infinite set of the relations between modes of the currents. The symbol δ(z) entering these relations is a formal series n∈Z z n . Following [4,9], we introduce composed currents F j,i (t) for i < j. The composed currents for nontwisted quantum affine algebras were defined in [4]. According to this paper, the coefficients of the series F j,i (t) belong to the completion U F of the algebra U F .
The completion U F determines analyticity properties of products of currents (and coincide with analytical properties of their matrix coefficients for highest weight representations [5]). One can show that for |i − j| > 1, the product F i (t)F j (w) is an expansion of a function analytic at t = 0, w = 0. The situation is more delicate for j = i, i ± 1. The products F i (t)F i (w) and F i (t)F i+1 (w) are expansions of analytic functions at |w| < |q 2 t|, while the product F i (t)F i−1 (w) is an expansion of an analytic function at |w| < |t|. Moreover, the only singularity of the corresponding functions in the whole region t = 0, w = 0, are simple poles at the respective hyperplanes, w = q 2 t for j = i, i + 1, and w = t for j = i − 1. Recall, that the deformation parameter q is a generic complex number, which is neither 0 nor a root of unity.
The definition of the composed currents may be written in analytical form for any a = i + 1, . . . , j − 1. It is equivalent to the relation F j,i (t) = F j,a (t)F a,i (w) dw w − q −1 − qt/w 1 − t/w F a,i (w)F j,a (t) dw w , F j,i (t) = F j,a (w)F a,i (t) dw w − q −1 − qw/t 1 − w/t F a,i (t)F j,a (w) dw w .
(A. 4) In (A.4) dw w g(w) = g 0 for any formal series g(w) = n∈Z g n z −n . Using the relations (A.1) on F i (t) we can calculate the residues in (A.3) and obtain the following expressions for F j,i (t), i < j: (A.5) For example, F i+1,i (t) = F i (t), and F i+2,i (t) = (q − q −1 )F i (t)F i+1 (t). The last product is welldefined according to the analyticity properties of the product F i (t)F i+1 (w), described above.
In a similar way, one can show inductively that the product in the right hand side of (A.5) makes sense for any i < j. Formulas (A.5) prove that the defining relations for the composed currents (A.3) or (A.4) yields the same answers for all possible values i < a < j.
Calculating formal integrals in (A.4) we obtain the following presentations for the composed currents: which are useful for the calculation of their projections.