Quantum groups and functional relations for arbitrary rank

The quantum integrable systems associated with the quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{\, l + 1}))$ are considered. The factorized form of the transfer operators related to the infinite dimensional evaluation representations is found and the determinant form of the transfer operators related to the finite dimensional evaluation representations is obtained. The master $TQ$- and $TT$-relations are derived. The operatorial $T$- and $Q$-systems are found. The nested Bethe equations are obtained.

Functional relations satisfied by commuting integrability objects are known to be a powerful tool for solving quantum integrable models. 1 In this paper, using the quantum group approach, we construct and investigate functional relations for quantum integrable systems associated with the quantum loop algebras U q (L(sl l+1 )). The central object of the quantum group approach is the universal R-matrix being an element of the completed tensor product of two copies of the quantum loop algebra. The integrability objects are constructed by choosing representations for the factors of that tensor product. It is traditional to call the representation used for the first factor of this tensor product the auxiliary space, while treating the representation space of the second one as the quantum space. These conventions can also be interchanged. By choosing a representation of the quantum group in the auxiliary space, one fixes an integrability object, while subsequently fixing a representation in the quantum space one defines a physical model. The consistent application of the method based on the quantum group theory was initiated by Bazhanov, Lukyanov and Zamolodchikov [2][3][4]. They considered the quantum version of KdV theory. Later on this method proved to be efficient for studying other quantum integrable models. Within the framework of this approach, R-operators [5][6][7][8][9][10][11], monodromy operators and L-operators were constructed [1,[10][11][12][13][14]. The corresponding families of functional relations were found [1,12,[15][16][17]27]. Recently the quantum group approach was used to derive and investigate equations satisfied by the reduced density operators of the quantum chains related to an arbitrary loop algebra [18,19].
In fact, the most important functional relation is the factorized representation of the transfer operator. All other relations appears to be its consequences. Usually, the functional relations are obtained by using the appropriate fusion rules for representations of the quantum loop algebra [20][21][22][23][24][25][26]. In the papers [3,4,15,17,27] to prove the factorization relations a direct operator approach was used for l = 1 and l = 2. For the higher ranks, the computational difficulties that arise seem to be almost insurmountable. In this paper, we use a different approach based on the analysis of the ℓ-weights of the representations. The effectiveness of the method was demonstrated in the paper [28] for l = 1 and l = 2. The present paper is devoted to the case of general l.
In section 2 we define the quantum group U q (gl l+1 ) and discuss its Verma modules. These modules are used later in section 4 to define the evaluation representations of 1 For the terminology used we refer to the paper [1] and section 5 of the present paper. the quantum loop algebra U q (L(sl l+1 )). In section 3 we define the quantum loop algebras U q (L(g)). We give two equivalent definitions, the first in terms of the Drinfeld-Jimbo generators, and the second is the second Drinfeld's realization. The need for two definitions is that the first is convenient for defining evaluation representations of U q (L(sl l+1 )), and the generators used in the second definition contain an infinite commutative subalgebra used to define ℓ-weights of the quantum loop algebras representations. In the same section we describe the construction of the Poincaré-Birkhoff-Witt basis of U q (L(sl l+1 )) used in the present paper. In section 4 we describe the category O of U q (L(g))-modules, introduce the concept of ℓ-characters and define the Grothendieck ring of O. In the same section we define the evaluation representations of the quantum groups U q (L(sl l+1 )), and the q-oscillator representations of their Borel subalgebras U q (L(b + )). We use the q-oscillator representations to construct the Q-operators. In the paper [29] the prefundamental representations, introduced in the paper [30], are used for this purpose. We use the q-oscillator representations because of their explicit form. In section 5 we define various types of integrability objects and discuss their properties. Here we also obtain the explicit form of L-operators related to q-oscillator representations. The functional relations for U q (L(sl l+1 )) are studied in section 6. We obtain the factorization representation for transfer operators in terms of Q-operators. This allows us construct TQ-relations, TT-relations and give the operator form of T-systems and Q-systems.
It is worth noting here that the functional relations for the integrable systems associated with the Yangians Y(gl n ) were found and investigated in the paper [31,32]. The authors of these papers used fusion of the elementary L-operators with a subsequent appropriate factorization. This method was also used for the case of the quantum loop algebra U q (L(sl 2 )) in the paper [33]. It seems that the generalization to the higher ranks is very cumbersome. Therefore, we use a completely different approach.
In our paper l is a fixed positive integer. We also fix the deformation parameterh in such a way that q = exp(h) is not a root of unity and assume that q ν = exp(hν) for any ν ∈ C. We define q-numbers by the equation and use the notation An algebra, if it is not a Lie algebra, is understood as a unital associative algebra. All algebras and vector spaces are assumed to be complex. By a tuple (s i ) i∈I we mean a mapping from a finite ordered set I to some set of objects S. When a tuple has only one component or is used as a multi-index we omit the parentheses in the notation.
2. QUANTUM GROUP U q (gl l+1 ) 2.1. Definition. We start with a brief reminder of some basic facts on the Lie algebras gl l+1 and sl l+1 . The Lie algebra gl l+1 is formed by the square matrices of order l + 1. The standard basis of the standard Cartan subalgebra k of the general linear Lie algebra gl l+1 consists of the matrices K i defined as 2 K i = E ii , i = 1, . . . , l + 1.
Let (ǫ i ) l+1 i=1 be the dual basis of k * . Below we often identify an element µ ∈ k * with the (l + 1)-tuple formed by the components µ i = µ, K i of µ with respect to this basis. There are l simple roots α i = ǫ i − ǫ i+1 , i = 1, . . . , l, and the full system ∆ + of positive roots is formed by the roots The system of negative roots is ∆ − = −∆ + , and the full root system is The special Lie algebra sl l+1 is a subalgebra of gl l+1 formed by the traceless matrices. The standard basis of the standard Cartan subalgebra h of sl l+1 is formed by the matrices As the positive and negative roots one takes the restriction of the roots of gl l+1 to h. We have The matrix A = (a ij ) l i, j=1 is the Cartan matrix of the Lie algebra sl l+1 . We define the quantum group U q (gl l+1 ) as an algebra generated by the elements satisfying the following defining relations In addition, there are the Serre relations, whose explicit form is not used in the present paper.

Poincaré-Birkhoff-Witt basis. The abelian group
is called the root lattice of gl l+1 . Assuming that we endow the algebra U q (gl l+1 ) with a Q-gradation. An element x of U q (gl l+1 ) is called a root vector corresponding to a root γ of gl l+1 if x ∈ U q (gl l+1 ) γ . In particular, E i and F i are root vectors corresponding to the roots α i and −α i , respectively. Following Jimbo [34], we introduce the elements E γ and F γ , corresponding to all roots γ ∈ ∆, with the help of the relations and It is clear that the elements E α ij and F α ij are root vectors corresponding to the roots α ij and −α ij , respectively. They are linearly independent and together with the elements q X , X ∈ k are called the Cartan-Weyl generators of U q (gl l+1 ). One can demonstrate that the ordered monomials constructed from the Cartan-Weyl generators form a Poincaré-Birkhoff-Witt basis of the quantum group U q (gl l+1 ). In this paper we choose the following ordering of monomials. First endow ∆ + with the colexicographical order. It means that α ij α mn if j < n, or if j = n and i < m. This is a normal ordering of roots in the sense of [35,36], see also [37]. This normal ordering is also a normal ordering of the roots of sl l+1 . Now we say that a monomial is ordered if it has the form where α i 1 j 1 · · · α i r j r , α m 1 n 1 · · · α m s n s and X is an arbitrary element of k. It is demonstrated in the paper [38] that such monomials really form a basis of U q (gl l+1 ), see also the paper [39] for the case of an alternative ordering.

Verma modules.
We use the standard terminology of the representation theory. In particular, we say that a U q (gl l+1 )-module V is a weight module if A weight U q (gl l+1 )-module V is called a highest weight module of highest weight µ if there exists a weight vector v ∈ V of weight µ such that The vector with the above properties is unique up to a scalar factor. We call it the highest weight vector of V.

The vectors
v m = F m 12 12 F m 13 13 F m 23 23 . . . F where for consistency we denote the highest weight vector by v 0 , form a basis of V µ . The explicit relations describing the action of the generators E i , F i and q X of the quantum group U q (gl l+1 ) on a general basis vector v m are obtained in the paper [38]. Note that V µ is an infinite-dimensional U q (gl l+1 )-module. However, if for all i = 1, . . . , l, there is a maximal submodule of V µ , such that the respective quotient module is simple and finite-dimensional. We denote this U q (gl l+1 )-module and the corresponding representation by V µ and π µ , respectively. Note that any finite-dimensional U q (gl l+1 )-module can be constructed in this way.
The weights ω i ∈ k * , i = 1, . . . , l + 1, defined as correspond to finite-dimensional representations called fundamental. The restriction of ω i , i = 1, . . . , l, to h are the fundamental weights of sl l+1 so that 3. QUANTUM LOOP ALGEBRA U q (L(g)) 3.1. Definition in terms of Drinfeld-Jimbo generators. Let g be a complex finite-dimensional simple Lie algebra of rank l, h a Cartan subalgebra of g, and ∆ the root system of g relative to h, see, for example, the books [40,41]. Fix a system of simple roots α i , i = 1, . . . , l. The corresponding coroots h i form a basis of h, so that The Cartan matrix A = (a ij ) l i, j=1 of g is given by the equation Note that the Cartan matrix A is always symmetrizable. It means that there exists a diagonal matrix D = diag(d 1 , . . . , d l ) such that the matrix DA is symmetric and d i , i = 1, . . . , l, are positive integers. It is evident that D is defined up to a nonzero scalar factor. We fix its normalization assuming that the integers d i are relatively prime. Following Kac [42], we denote by L(g) the loop algebra of g, by L(g) its standard central extension by the one-dimensional center C c, and by L(g) the Lie algebra obtained from L(g) by adding a natural derivation d. By definition, and the Cartan subalgebra of L(g) is the space The Lie algebra L(g) is isomorphic to the affine (Kac-Moody) algebra associated with the extended Cartan matrix (Dynkin diagram) A (1) of g, see, for example, the book [43, p. 166]. Here the basis coroots are h i , i = 1, . . . , l, and The integers aǐ, i = 1, . . . , l, together with aǐ = 1 are the dual Kac labels of the Dynkin diagram associated with the Cartan matrix A (1) . Thus, we have To introduce the corresponding simple roots, we identify the space h * with the subspace of h * defined as {λ ∈ h * | λ, c = 0, λ, d = 0}, and denote by δ the element of h * defined by the equations Then the simple roots are α i , i = 1, . . . , l, and is the highest root of ∆. The integers a i , i = 1, . . . , l, together with a 0 = 1 are the Kac labels of the Dynkin diagram associated with the Cartan matrix A (1) . One can demonstrate that the equation gives the entries of the Cartan matrix A (1) . Complementing the numbers d i , i = 1, . . . , l, with a suitable number d 0 , one can demonstrate that the matrix A (1) is symmetrizable. Note that for g = sl l+1 , d i = 1 for all i = 0, 1, . . . , l.
The system of positive roots of the affine algebra L(g) is where ∆ + is the system of positive roots of the Lie algebra g. The system of negative roots ∆ − of L(g) is ∆ − = − ∆ + , and the full system of roots is The roots mδ are imaginary, and all other roots are real, see the book [42] for definition. It is convenient for our purposes to denote It is easy to show that for any λ ∈ h * there is a unique element λ ∈ h * such that We define the quantum group U q (L(g)) as an algebra generated by the elements e i , f i , i = 0, 1, . . . , l, and q x , x ∈ h, satisfying the relations for all i = 0, 1, . . . , l. Here, for each i = 0, 1, . . . , l we set There are also the Serre relations, whose explicit form is not used in the present paper. The quantum loop algebra U q (L(g)) is a Hopf algebra. The comultiplication ∆, the antipode S, and the counit ε are given by the relations We do not use these relations in the present paper. They are given only to fix the conventions used.
To distinguish from the tensor product of mappings, we denote the tensor product of any two representations of U q (L(g)), say ϕ and ψ, as and, similarly, the tensor product of the corresponding U q (L(g))-modules V and W as V ⊗ ∆ W. More generally, given two homomorphisms ϕ and ψ from U q (L(g)) to algebras A and B, respectively, equation (3.1) defines a homomorphism of U q (L(g)) into A ⊗ B.
The abelian group Zα i is called the root lattice of L(g). Assuming that for any i = 0, 1, . . . , l and x ∈ h, we endow the algebra U q (L(g)) with a Q-gradation.
An element x of U q (L(g)) is called a root vector corresponding to a root γ ∈ ∆ if x ∈ U q (L(g)) γ . One can construct linearly independent root vectors corresponding to all roots from ∆, see, for example, the papers [5,[44][45][46] or the papers [47,48] for a different approach. We use the approach by Khoroshkin and Tolstoy [5,[44][45][46]. The linearly independent root vectors together with the elements q x , x ∈ h are called the Cartan-Weyl generators of U q (L(g)). If some ordering of roots is chosen, then appropriately ordered monomials constructed from the Cartan-Weyl generators form a Poincaré-Birkhoff-Witt basis of U q (L(g)). In fact, in applications to the theory of quantum integrable systems one uses the so-called normal orderings. An example of such an ordering for g = sl l+1 is described in section 3.4. We denote the root vector corresponding to a real positive root γ ∈ ∆ by e γ , and the root vector corresponding to a real negative root γ ∈ ∆ by f −γ . In fact, the root vectors corresponding to the imaginary roots mδ and −mδ, m > 0, are indexed by the simple roots of g and denoted as e ′ mδ; α i and f ′ mδ; α i . The prime in the notation is explained by the fact that one also uses another set of root vectors corresponding to imaginary roots. They are introduced by the functional equations where the generating functions are defined as formal power series.
3.2. Spectral parameters. In applications to the theory of quantum integrable systems, one usually considers families of representations of a quantum loop algebra and its subalgebras, parameterized by a complex parameter called a spectral parameter. We introduce a spectral parameter in the following way. Assume that a quantum loop algebra U q (L(g)) is Z-graded, so that any element x ∈ U q (L(g)) can be uniquely represented as Given ζ ∈ C × , we define the grading automorphism Γ ζ by the equation Now, for any representation ϕ of U q (L(g)) we define the family ϕ ζ of representations as If V is the U q (L(g))-module corresponding to the representation ϕ, we denote by V ζ the U q (L(g))-module corresponding to the representation ϕ ζ .
In the present paper we endow U q (L(g)) with a Z-gradation assuming that where s i are arbitrary integers. We denote where a i are the Kac labels of the Dynkin diagram associated with the extended Cartan matrix A (1) .

Drinfeld's second realization.
The quantum loop algebra U q (L(g)) can be realized in a different way [49][50][51] as an algebra with generators ξ ± i, m with i = 1, . . . , l and m ∈ Z, q x with x ∈ h, and χ i, m with i = 1, . . . , l and m ∈ Z \ {0}. They satisfy the defining relations , and the Serre relations whose explicit form is not important for our consideration. The quantities φ ± i, ±m , i = 1, . . . , l, m ∈ Z, are defined by the equation and by the conditions Stress that we use the definition of φ ± i, m slightly different from the commonly used. The generators of Drinfeld's second realization are related to the Cartan-Weyl generators in the following way [45,46]. The generators q x of the quantum loop algebra in the Drinfeld-Jimbo's and Drinfeld's second realizations are the same, except that in the first case x ∈ h, while in the second case x ∈ h ⊂ h. For the generators ξ ± i, m and χ i, m of the Drinfeld's second realization we have where for each i = 1, . . . , l the number o i is either +1 or −1, such that o i = −o j whenever a ij < 0. It follows from (3.2), (3.3) and (3.6) that we also obtain 3.4. Poincaré-Birkhoff-Witt basis for U q (L(sl l+1 )). In this article, we often give general definitions for general g, however, concrete results are obtained for the case of the loop algebra U q (L(sl l+1 )). In this case d i = 1, a i = 1, and aǐ = 1, for all 0 ≤ i ≤ l, and we assume that o i = (−1) i . 3 To define an appropriate ordering of positive roots of U q (L(sl l+1 )), we start with the normal ordering of the positive roots of sl l+1 defined in subsection 2.2. Then we assume that α + mδ β + nδ, with α, β ∈ ∆ + and m, n ∈ Z ≥0 , if α β, or α = β and m < n. Further, (δ − α) + mδ (δ − β) + nδ, with α, β ∈ ∆ + , if α β, or α = β and m > n. Finally, we assume that the relation α + kδ mδ (δ − β) + nδ is valid. As the result we come to a normal ordering of positive roots of U q (L(sl l+1 )). For more details see the paper [54].
Following [5,[44][45][46], we define the root vectors inductively . In the case under consideration the procedure looks as follows. We start with the root vectors corresponding to the roots α i and −α i for 1 ≤ i ≤ l, which we identify with the generators e i and f i so that The next step is to construct root vectors e α ij and f α ij for all roots α ij ∈ ∆ + . We assume that for i > 1, and then for j < l + 1. The root vectors corresponding to the roots δ and −δ are indexed by the elements of ∆ + and defined by the relations It can be seen that we label the root vectors corresponding to the imaginary roots by all roots of sl l+1 and not only by the simple ones. This is, of course, redundant for building a Poincaré-Birkhoff-Witt basis, but useful as a technical tool.
The meaning of the element λ ∈ h * for λ ∈ h * is explained in section 3.1. The space V λ is called the weight space of weight λ, and a nonzero element of V λ is called a weight vector of weight λ. We say that is a weight module all of whose weight spaces are finite-dimensional; (ii) there exists a finite number of elements µ 1 , . . . , µ s ∈ h * such that every weight of V belongs to the set s i=1 D(µ i ) , where D(µ) = {λ ∈ h * | λ ≤ µ} with ≤ being the usual partial order in h * , see, for example, the book [41]. Let a U q (L(g))-module V be in category O. The algebra U q (L(g)) contains an infinitedimensional commutative subalgebra generated by the elements φ ± i, ±m , i = 1, . . . , l, m ∈ Z >0 and q h , h ∈ h. We can refine the weight decomposition (4.1) in the following way. Let λ be a weight of V. By definition, the space V λ is finite-dimensional. The restriction of the action of the elements φ ± i, ±m to V λ constitutes a countable set of pairwise commuting linear operators on V λ . Hence, there is a basis of V λ which consists of eigenvectors and generalized eigenvectors of all those operators, see, for example, the book [55]. This leads to the following definitions.
An ℓ-weight is a triple We denote the set of ℓ- . . , l and m ∈ Z >0 is said to be an ℓ-weight vector of ℓ-weight Λ. Every nontrivial ℓ-weight space contains an ℓ-weight vector.
For any two ℓ- is an associative operation with respect to which h * ℓ is an abelian group. Here . and the role of the unity is played by the ℓ-weight (0, (1, . . . , 1 l ), (1, . . . , 1 l )). Note that The vector with the above properties is unique up to a scalar factor. We call it the highest ℓ-weight vector of V.
Let V and W be highest ℓ-weight U q (L(g))-modules in category O of highest ℓ-weights Λ and Ξ, respectively. The submodule of V ⊗ ∆ W generated by the tensor product of the highest ℓ-weight vectors is a highest ℓ-weight module of highest ℓ-weight Λ Ξ.

Evaluation representations.
To construct representations of the quantum loop algebra U q (L(sl l+1 )), it is common to use the Jimbo's homomorphism ε from U q (L(sl l+1 )) to the quantum group U q (gl l+1 ) defined by the equations [34] ) called an evaluation representation. Given µ ∈ k * , we denote the representation π µ • ε, where the representation π µ is described in section 2.3, by ϕ µ . Since the representation space of π µ and ϕ µ is the same, slightly abusing notation, we denote both corresponding modules as V µ . The same convention is used for the respective finitedimensional counterparts, if any. It is common to call V µ an evaluation module. It is a highest weight U q (L(sl l+1 ))-module in category O of highest weight being the restriction of µ to h. For any µ in h * the module V µ is a highest ℓ-weight module in category O with highest ℓ-weight vector v µ 0 . The ℓ-weight spaces and the corresponding ℓ-weights for l = 1 and l = 2 was found in the paper [52]. It appears that in this case all ℓ-weight spaces are one-dimensional and, therefore, generated by ℓ-weight vectors. Although the ℓ-weight vectors do not coincide with the basic vectors v µ m , defined by equation (2.1), they can also be labeled by the (l + 1)l/2-tuple m. It is natural to assume that this is the case for an arbitrary l, and we use for the ℓ-weights the notation It was found in the paper [38] that for an arbitrary l, In particular, for the highest weight we have Using formulas of the paper [52], one can demonstrate that for the components of Λ µ+ 0 (ζ) and Λ µ− see appendix A.
Below we use the (l + 1)-dimensional representation ϕ ω 1 ζ on the vector space C l+1 . For this representation we have and, further, Let v m , m = 1, . . . , l + 1, be the elements of the standard basis of C l+1 . It is clear that for any m = 2, . . . , l + 1. We see that the vector v m is a weight vector of weight λ m being the restriction of the element µ m ∈ k * defined as to the Cartan subalgebra h of sl l+1 . Explicitly, we have 4.3. ℓ-weights of U q (L(b + ))-modules. There are two standard Borel subalgebras of the quantum loop algebra U q (L(g)). In terms of the Drinfeld-Jumbo generators they are defined as follows. The Borel subalgebra U q (L(b + )) is defined as the subalgebra of U q (L(g)) generated by e i with 0 ≤ i ≤ l and q x with x ∈ h, and the Borel subalgebra U q (L(b − )) is the subalgebra generated by f i with 0 ≤ i ≤ l and q x with x ∈ h. It is clear that these subalgebras are Hopf subalgebras of U q (L(g)). The description of U q (L(b + )) and U q (L(b − )) in terms of the Drinfeld generators is more intricate. The category O for U q (L(b + )) or U q (L(b − )) is defined literally by the same words as it is defined in section 4.1 for the case of U q (L(g)). By definition, any module V in category O allows for the weight decomposition which can be refined again by considering ℓ-weights. Consider the Borel subalgebra U q (L(b + )). It does not contain the elements φ − i, −m , i = 1, . . . , l, m ∈ Z >0 , and its infinite-dimensional commutative subalgebra is generated only by the elements φ + i, m i = 1, . . . , l, m ∈ Z >0 , and q h , h ∈ h. Respectively, now an ℓ-weight Λ is a pair Λ = (λ, Λ + ), (4.12) where λ ∈ h * and Λ + is an l-tuple We denote the set of ℓ-weights (4.12) by h * + ℓ .
For any two ℓ-weights Λ = (ψ, Λ + ) and Ξ = (ξ, Ξ + ) we define the ℓ-weight Λ Ξ as The product (4.13) is an associative operation with respect to which h * + ℓ is an abelian group. Here , and the role of the unity is played by the ℓ-weight (0, (1, . . . , 1 is an invertible power series. Define a surjective homomorphism ̟ : . . , l and m ∈ Z >0 is said to be an ℓ-weight vector of ℓ-weight Λ. Every nontrivial ℓ-weight space contains an ℓ-weight vector. The vector with the above properties is unique up to a scalar factor. We call it the highest ℓ-weight vector of V. Let V and W be highest ℓ-weight U q (L(b + ))-modules in category O of highest ℓ-weights Λ and Ξ respectively. The submodule of V ⊗ ∆ W generated by the tensor product of the highest ℓ-weight vectors is a highest ℓ-weight module of highest ℓ-weight Λ Ξ.
For any U q (L(b + ))-module V in category O and an element δ ∈ h * , we define the shifted U q (L(b + ))-module V[δ] shifting the action of the generators q h . Namely, if ϕ is the representation of U q (L(b + )) corresponding to the module V and ϕ[ξ] is the representation corresponding to the module V[ξ], then (4.14) For an arbitrary U q (L(g))-module V we define the shifted module V[δ], δ ∈ h * , as a U q (L(b + ))-module obtained first by restricting V to U q (L(b + )) and then shifting the 4.4. Oscillator representations. One can get representations of U q (L(b + )) by restricting to it the representations of U q (L(sl l+1 )). However, the most interesting for theory of integrable systems are the representations which cannot be obtained by such a procedure.
In the present paper we use a class of such representations called the q-oscillator representations. We define the q-oscillator representations in two steps. First, we introduce a homomorphism of U q (L(b + )) into a tensor power of the q-oscillator algebra Osc q , then for each factor of the tensor product we choose a certain representation of Osc q and come to the required representations.
The q-oscillator algebra Osc q is an algebra defined by the generators b † , b, q νN , ν ∈ C, and the relations We are particularly interested in the two basic representations of Osc q . First, let W + be the free vector space with a basis (w n ) n∈Z ≥0 . One can show that the relations where we assume that w −1 = 0, endow W + with the structure of an Osc q -module. We denote the corresponding representation of the algebra Osc q by χ + . Further, let W − be again a free vector space with a basis (w n ) n∈Z ≥0 . The relations where we again assume that w −1 = 0, endow W − with the structure of an Osc q -module. We denote the corresponding representation of Osc q by χ − . Denoting for k ∈ Z >0 and ν ∈ C, and that for |q ν | < 1. For |q ν | > 1 we define the trace tr χ + by analytic continuation. Since the monomials (b † ) k q νN , b k q νN and q νN for k ∈ Z >0 and ν ∈ C form a basis of Osc q , the above relations are enough to determine the trace of any element of Osc q . It appears that Consider the tensor product of l copies of the q-oscillator algebra, and denote In the paper [38] a homomorphism o of U q (L(b + )) into (Osc q ) ⊗ l described by the equations where i = 1, . . . , l, was obtained by some limiting procedure starting from the evaluation representations ϕ λ of U q (L(sl l+1 )). Define the representation The basis of this representation is formed by the vectors where n i ∈ Z ≥0 for all 1 ≤ i ≤ l, and we use the notation n = (n i ) l i=1 . Here the vector w 0 is the vacuum vector, satisfying the equations The representation θ is in category O. It is a highest ℓ-weight representation with highest ℓ-weight vector w 0 .
There is an automorphism of U q (L(sl l+1 )) defined by the equations where it is assumed that q νh l+1 = q νh 0 , e l+1 = e 0 and f l+1 = f 0 . One can restrict σ to an automorphism of U q (L(b + )). It is useful to have in mind that σ l+1 = 1. We define a collection of homomorphisms from U q (L(b + )) into Osc ⊗l q as and a family of representations (4.23) Note that θ = θ l+1 . The corresponding basis vectors are The vacuum vector w a, 0 satisfies the equations All the representations θ a are highest ℓ-weight representations in category O. The vectors w a, n are the ℓ-weight vectors, and w a, 0 is the highest ℓ-weight vector of the representation θ a . We denote the U q (L(b + ))-module corresponding to the representation θ a by W a . The explicit form of the ℓ-weights Ψ a, n (ζ) = (ψ a, n , Ψ + a, n (ζ)) for the modules (W a ) ζ was found in the paper [53]. For the future usage, we collect below the relevant expressions. 4.5. ℓ-weights of the oscillator representations. In fact, below we need the ℓ-weights for the modules (W a ) ζ . 4.5.1. a = 1.
j=i−a n j −l+i−1 ζ s u 4.6. q-characters and Grothendieck ring. Let V be a U q (L(g))-module in category O.
We define the character of V as a formal sum By the definition of the category O, dimV λ = 0 for λ outside the union of a finite number of sets of the form D(µ), µ ∈ h * . For any two U q (L(g))-modules V and U in category O we have ch(V ⊕ U) = ch(V) + ch(U). More generally, if U q (L(g))-modules V, W and U in category O can be included in a short exact sequence 0 → V → W → U → 0, (4.24) then ch(W) = ch(V) + ch(U). It can be also shown that ch(V ⊗ ∆ U) = ch(V) ch(U) for any U q (L(g))-modules V and U in category O. Here, to multiply characters we assume that The Grothendieck group of the category O of U q (L(g))-modules is defined as the quotient of the free abelian group on the set of all isomorphism classes of objects in O modulo the relations V = U + W if the objects V, W and U can be included in a short exact sequence (4.24). Here for any object V, V denotes the isomorphism class of V. Defining we come to the Grothendieck ring of O. It is a commutative unital ring, for which the role of the unit is played by the trivial U q (L(g))-module. We see that the character can be considered as a mapping from the Grothendieck ring of O. However, it is not injective, and, therefore, does not uniquely distinguish between its elements. This role is played by the q-character.
We define the q-character of a U q (L(g))-module V in category O as a formal sum One can easily demonstrate that ̟(ch q (V)) = ch(V).
Here we assume that ̟([Λ]) = [̟(Λ)], and extend this rule by linearity. The q-character has the same properties as the usual character. Namely, if U q (L(g))-modules V, W and U in category O can be included in a short exact sequence (4.24), then ch q (W) = ch q (V) + ch q (U), and for any two U q (L(g))-modules V and U in category O, we have (4.25) see the paper [56]. To define the product of q-characters, we assume that

[Λ][Ξ] = [ΛΞ]
for any Λ, Ξ ∈ h * ℓ . Thus, q-character can also be considered as a mapping from the Grothendieck ring of O, and now one can demonstrate that this mapping is injective. In other words, different elements of the Grothendieck ring of O have different q-characters.
It follows from equation (4.25) that It means that the U q (L(g))-modules V ⊗ ∆ W and W ⊗ ∆ V belong to the same equivalence class in the Grothendieck ring of O. All above can be naturally extended to the categories O of the modules over U q (L(b + )) and U q (L(b − )). 5.1.1. Universal R-matrix. As any Hopf algebra the quantum loop algebra U q (L(g)) has another comultiplication called the opposite comultiplication. It is defined by the equation

UNIVERSAL R-MATRIX
There are several definitions of quantum groups. In particular,h can be treated not only as a complex number [34,48,57], but also as an indeterminate. In this case the quantum group U q (L(g)) is a C[[h]]-algebra [5,[44][45][46]49]. Let us assume temporally that it is the case. Herewith U q (L(g)) is a quasitriangular Hopf algebra. It means that there exists an element R of the completed tensor product U q (L(g)) ⊗ U q (L(g)), called the universal for all x ∈ U q (L(g)), and 4 The latter equations are considered as equalities in the completed tensor product of three copies of U q (L(g)). In fact, it follows from the explicit expression for the universal Rmatrix [5,[44][45][46] that it is an element of a completed tensor product of two Borel subalgebras U q (L(b + )) and U q (L(b − )).

Integrability objects.
Let ϕ be a homomorphism from U q (L(b + )) to an algebra A, and ψ a homomorphism from U q (L(b − )) to another algebra B. The corresponding integrability object X ϕ|ψ is defined by the equation where ρ ϕ|ψ is a scalar normalization factor. It is evident that this object is an element of A ⊗ B. Certainly, as ϕ and ψ one can use the restrictions to U q (L(b + )) and U q (L(b − )) of homomorphisms from U q (L(g)). However, this is not always the case. It follows from equation (5.1) that and equation (5.2) gives where in the first equation ϕ 1 , ϕ 2 are homomorphisms from U q (L(b + )) to algebras A 1 , A 2 , and ψ is a homomorphism from U q (L(b − )) to an algebra B, while in the second one ϕ is a homomorphism from U q (L(b + )) to an algebra A, and ψ 1 , ψ 2 are homomorphisms from U q (L(b − )) to algebras B 1 , B 2 . In fact, hereinafter, we assume that Each integrability object X ϕ|ψ ∈ A ⊗ B is accompanied by an integrability object of a different type, defined as follows. Let χ be a representation of the algebra A on a vector space V. Define a trace on A by the equation where tr End(V) is the usual trace on the endomorphism algebra of the vector space V.
When ϕ is already a representation of U q (L(b + )) on a vector space V we use as χ the identity mapping id V and omit χ in the notation. The integrability object in question is and t i are complex parameters. The element q t ∈ U q (L(b + )) ∩ U q (L(b − )) is a group like element 5 called a twisting element. It is necessary for convergence of the trace in the case when ϕ is an infinite dimensional representation. We see that Y χ ϕ|ψ is an element of the algebra B.

Universal integrability objects.
For any homomorphism ϕ from U q (L(b + )) to an algebra A, it is productive to define universal integrability object X ϕ = (ϕ ⊗ id U q (L(b − )) )(R), (5.8) being an element of A ⊗ U q (L(b − )). It is clear that for any homomorphism ψ from U q (L(b − )) the integrability object X ϕ|ψ is related to X ϕ by the equation Similarly, each universal integrability object X ϕ is accompanied by universal integrability object being an element of U q (L(b − )). By definition, for any homomorphism ψ from U q (L(b − )) to an algebra B we have ψ(Y χ ϕ ) = ρ ϕ|ψ Y χ ϕ|ψ . Furthermore, let ϕ 1 , . . ., ϕ m be homomorphisms from U q (L(b + )) to algebras A 1 , . . ., A m , and χ 1 , . . ., χ m are representations of A 1 , . . ., A m , then 5.2. Quantum group as a C-algebra. 5 An element a of a Hopf algebra is called group like if ∆(x) = x ⊗ x. Note that any group like element is invertible. 5.2.1. Integrability objects. The expression for the universal R-matrix of a quantum loop algebra U q (L(g)) considered as a C[[h]]-algebra can be constructed using the procedure proposed by Khoroshkin and Tolstoy [44][45][46]. However, in this paper, we define the quantum loop algebra U q (L(g)) as a C-algebra. In fact, one can use the expression for the universal R-matrix from the papers [44][45][46] to construct the integrability objects also in this case, having in mind that U q (L(g)) is quasitriangular only in some restricted sense, see the paper [58], the book [59, p. 327], and the discussion below. We restrict ourselves to the following case. Let ϕ be a homomorphism from U q (L(b + )) to an associative algebra A and ψ a representation of U q (L(b − )) on a vector space U. Define an integrability object X ϕ|ψ as an element of A ⊗ End(U) by the equation Here R ≺δ , R ∼δ and R ≻δ are elements of U q (L(b + )) ⊗ U q (L(b − )), K ϕ|ψ is an element of A ⊗ End(U), and ρ ϕ|ψ a scalar normalization factor. The element K ϕ|ψ ∈ A ⊗ End(U) is given by the equation where Π λ ∈ End(U) is the projector on the component U It can be shown that with an appropriate choice of R ≺δ , R ∼δ and R ≻δ the integrability objects defined by equation (5.12) satisfy equations (5.4) and (5.5), see the paper [58] and next section.
The Y type companion of X ϕ|ψ is defined by the equation Y χ ϕ|ψ = (tr χ ⊗ id End(U) )(X ϕ|ψ (ϕ(q t ) ⊗ 1 End(U) )). (5.14) It is not difficult to demonstrate that It is clear that the integrability object Y χ ϕ|ψ depend on χ and ϕ only through the equivalence class in the Grothendieck ring of the category O to which the representation χ • ϕ belongs.

5.2.2.
The case g = sl l+1 . We give explicit expressions for the elements R ≺δ , R ∼δ R ≻δ and K ϕ|ψ only for the most important for this paper case g = sl l+1 . The element R ≺δ is the product over the set of roots α ij + nδ, 1 ≤ i < j ≤ l + 1, n ∈ Z ≥0 , of the q-exponentials Hereinafter, the q-exponential is defined as The order of the factors in R ≺δ coincides with the normal order of the roots α ij + nδ, described in section 3.4.
The element R ∼δ is defined as where for each n ∈ Z >0 the quantities u n, ij are given by the equations The definition of the element R ≻δ is similar to the definition of the element R ≺δ . It is the product over the set of roots (δ − α ij ) + nδ of the q-exponentials The order of the factors in R ≻δ coincides with the normal order of the roots (δ − α ij ) + nδ, described in section 3.4.
The matrix A is tridiagonal, and the matrix elements of the matrix C entering the expression (5.13) for K ϕ|ψ can be found using the results of the paper [60]. We obtain

Universal integrability objects. When U q (L(g)) is defined as a C[[h]
]-algebra, we use equations (5.8) and (5.10) to define universal integrability objects. This is not possible if U q (L(g)) is defined as a C-algebra. Here the universal integrability objects are defined only as formal objects with specific rules of use. If ϕ is a homomorphism from U q (L(b + )) to an algebra A, the universal integrability object X ϕ behaves as an element of A ⊗ U q (L(b − )). To obtain usual integrability object we use the rule The universal object Y χ ϕ , where ϕ is a homomorphism from U q (L(b + )) to an algebra A and χ a representation of A, behaves as an element of U q (L(b − )). It obeys the following rule. Let ϕ 1 , . . . , ϕ m be homomorphisms from U q (L(b + )) to algebras A 1 , . . . , A m , and χ 1 , . . . , χ m representations of A 1 , . . . , A m , respectively, then for any representation ψ of U q (L(b − )). Formally, the two above relations coincide with equations (5.9) and (5.11). However, (5.9) and (5.11) are consequences of the definitions, while (5.17) and (5.18) is a part of the definitions. Using (5.15), we obtain and come to the following equation for formal objects It is clear that the integrability object Y χ ϕ depend on χ and ϕ only through the equivalence class in the Grothendieck ring of the category O to which the representation χ • ϕ belongs.

Commutation with group like elements.
Let ϕ be a homomorphism from U q (L(b + )) to an algebras A, and ψ a representation of U q (L(b − )) on a vector space U. If x ∈ U q (L(b + )) ∩ U q (L(b − )) is a group like element, then using equation (5.4), we obtain Assuming that a commutes with the twisting element q t , we come to the equation Applying to both sides of this equation the mapping tr χ , where χ is a representation of A, we come to the equation For the universal integrability object Y for any group like element a commuting with the twisting element q t .

5.2.6.
Behavior under action of automorphism σ. Return again to the case g = sl l+1 , and find the connection between the integrability objects X ϕ•σ|ψ•σ and X ϕ|ψ , where σ is the automorphism of U q (L(sl l+1 )) defined by equation (4.21). Adapting the reasoning given in the paper [5] to our situation, we can demonstrate that Using the fact that q ν ∑ l k=0 h k = 1, we find where S ik are the matrix entries of the square matrix S of size l defined as One can get convinced that S t C = C S −1 , and obtain σ q . Then, using equation (4.14), we obtain Since for any x ∈ h, we come to the equation For the integrability objects Y ϕ|ψ and Y ϕ we obtain the equations Adding spectral parameters. To define integrability objects depending on spectral parameters, we use as a homomorphism ϕ the tensor product of m homomorphisms ϕ ζ 1 , . . ., ϕ ζ m , and as ψ the nth tensor power of a representation ψ. 6 The corresponding universal integrability objects are denoted as while for usual integrability objects we use the notation When n = 1 we usually write just X ϕ|ψ (ζ 1 , . . . , ζ m ) and Y χ ϕ|ψ (ζ 1 , . . . , ζ m ). In accordance with our conventions, we assume that If m = n = 1, an integrability object X ϕ|ψ (ζ) = (1) X ϕ|ψ (ζ) defined by equation (5.29) is said to be a basic integrability object. With the help of equations (5.5), all other integrability objects of type X can be expressed through the basic ones.
All integrability objects that we use in this paper, are constructed as described above. However, depending on the role they play in the integration procedure, they are given different names. Below we describe the main classes of integrability objects. It is worth to have in mind that the proposed classification is rather conditional, although it is widespread.
The most famous integrability objects are the R -operators. They form a special class of integrability objects of type X used to permute integrability objects of type Y. However, there is a more general method for demonstrating commutativity of integrability objects of type Y, described in the previous section, and we do not define and use R-operators in the present paper.

Monodromy operators and transfer operators.
When ϕ is a homomorphism from the quantum loop algebra U q (L(g)) to an algebra A, ψ a representation of U q (L(b − )) on a vector space U, the integrability object where q t is a twisting element, which we define by equation (5.7), χ is a representation of the algebra A, and the trace tr χ is given by equation (5.6).
Let ϕ 1 and ϕ 2 be homomorphisms from U q (L(g)) to algebras A 1 and A 2 , respectively, ψ a representation of U q (L(b − )), and χ 1 and χ 2 representations of A 1 and A 2 . It follows from equation (5.20) that for any ζ 1 , ζ 2 ∈ C × . Similar commutativity takes place for the universal transfer operators, T . Emphasize that it is important for commutativity that the twist element is group like.
In this paper we construct the monodromy operators using as ϕ the evaluation representations ϕ λ and ϕ λ defined in section 4.2, and as ψ the finite-dimensional evaluation representation ϕ ω 1 . Here the following notation is used Similarly, for the corresponding universal transfer operators we use the notation One can demonstrate that the transfer operators for the evaluation representations of U q (L(b + )) depend on ζ via ζ s , see the papers [17,27] for l = 1 and l = 2.

5.4.1.
Definition. Now let ϕ be a homomorphism from U q (L(b + )) to an algebra A which cannot be extended to a homomorphism of U q (L(g)), and ψ a representation of the Borel subalgebra U q (L(b − )) on a vector space U. In this case the integrability object L ϕ|ψ (ζ). The companion of an L-operator L n ϕ|ψ (ζ) of type Y is called a Q-operator and is denoted where q t is a twisting element, which we define by equation (5.7), χ is a representation of the algebra A and the trace tr χ is given by equation (5.6). Let ϕ 1 and ϕ 2 be homomorphisms from U q (L(b + )) to algebras A 1 and A 2 which cannot be extended to homomorphisms of U q (L(g)), ψ a representation of U q (L(g)), and χ 1 and χ 2 representations of A 1 and A 2 . It follows from equation (5.20) that for any ζ 1 , ζ 2 ∈ C × , and similarly for the universal Q-operators, . One can demonstrate that the Q-operators for the oscillator representations of the Borel subalgebra U q (L(b + )) depend on ζ via ζ s , see the papers [17,27] for l = 1 and l = 2.

5.4.2.
Case of oscillator representations. Let us find the basic L-operator for the case when ϕ is the homomorphism o defined by equations (4.18)-(4.20), and ψ the representation ϕ ω 1 described by equations (4.6)-(4.9). We denote this L-operator simply as L(ζ). We have First find the expression for the factor K o|ϕ ω 1 . To this end we use equation (5.13). Recall that the representation ϕ ω 1 is (l + 1)-dimensional, the basic vectors v m are the weight vectors of weights λ m given by equation (4.10). It is clear that the projector on the weight vector v m is E mm . Thus, we have Using equation (5.16), we obtain Now, taking into account (4.19) and (4.20), we see that Thus, we have Here and below we use the notation N ij = ∑ j−1 k=i N k . It follows from equations (3.12)-(3.14) that Using the first equation of (3.2), we obtain o ζ (e nδ; α ij ) = ζ ns δ l+1, j κ −1 q (−1) in q −(i−l−1)n q 2nN i+1, l+1 for all 1 ≤ i < j ≤ l + 1. In turn, equations (3.15)- (3.17) give (5.34) and the second equation of (3.2) implies Now we find the expression and obtain Here the transcendental function F l+1 (ζ) is defined as Here we use the identities for all 1 ≤ i < j ≤ l + 1 and k > 1, and Assuming that the normalization factor is given by the equation we finally obtain 7 Using the set of the homomorphisms o a , given by equation (4.22), we define a family of L-operators L a (ζ) = L o a |ϕ ω 1 (ζ).
By definition, L a+l+1 (ζ) = L a , so that there are only l + 1 distinct L-operators of this type.
It is easy to demonstrate that for any x ∈ U q (L(sl l+1 )), where s → σ(s) stands for the set of substitutions Using equations (5.27) and (5.38), we obtain Taking this equation into account, we come to .
One can verify that for any x ∈ U q (L(sl l+1 )) where O is the square matrix of size l + 1 defined as Hence, we have To define the corresponding Q-operators we use the trace tr χ a , where χ a is the representation of Osc ⊗l defined by equation (4.23), and use the notation for the usual Q-operators. We use a prime to indicate that we redefine these operators below, see equation (6.8).
6. FUNCTIONAL RELATIONS 6.1. Factorization of transfer operators. In this section we generalize the consideration given in the paper [28] for l = 1 and l = 2 to the case of general l. Below we treat the U q (L(sl l+1 ))-modules V µ ζ and V µ ζ as U q (L(b + ))-modules. Let us consider the following tensor product of l + 1 oscillator U q (L(b + ))-modules is the half sum of positive roots of gl l+1 . Using the formulas of section 4.5, equations (4.4) and (4.5), we see that the product This means that the submodule of W µ (ζ) generated by the tensor product of the highest ℓ-weight vectors of the oscillator factors is isomorphic to ( V µ [δ]) ζ . Hence, the q-character of W(ζ) should contain a summand coinciding with the q-character of ( V µ [δ]) ζ . Let us show that the entire q-character of W(ζ) is the sum of such q-characters.
Introduce an independent labeling for the oscillators, using for it the tuple n = (n a, i ), where a = 1, . . . , l + 1 and i = 1, . . . , l. Denote the ℓ-weights of the module W µ (ζ) as Ξ µ n (ζ) = (ξ µ n , Ξ µ+ n (ζ)). Using expressions from section 4.5, we obtain that j=i−a n aj +i−2a+1 ζ s u) j=i−a+1 n aj +i−2a+1 ζ s u) j=i−a+2 n aj +i−2a+3 ζ s u) j=i−a+1 n aj +i−2a+3 ζ s u) . (6.4) One can easily get convinced that Ξ µ+ n (ζ) does not depend on n a, i with a + i > l + 1. For a + i ≤ l + 1 denote m a, i = n a, i−a (6.5) and rewrite (6.3) as Denote also the subtuple of n formed by n a, i with a + i > l + 1 by n ′ . Comparing the above equation with (4.3), we see that the weights ξ µ n for n ′ = 0, after the identification (6.5), coincide with the weights of the module V µ ζ [δ](ζ), where δ is defined by equation (6.2). It is natural to assume that Ξ µ+ n (ζ) for n ′ = 0, after the identification (6.5), coincides with the component Λ More generally, it is natural to assume that the ℓ-weight Ξ µ n (ζ) for any fixed n ′ , after the identification (6.5), coincides with the ℓ-weight Λ This very plausible assumption is supported at least by the fact that it is true for l = 1 and l = 2, and we will assume that it is true for arbitrary l. Similarly as above, this results in the relation satisfied by q-characters, which is equivalent to the relation in the Grothendieck ring Here the summation over n ′ means the summation over the components of n ′ from 0 to ∞. It follows that Having in mind that ), see equation (5.19), and taking into account equation (5.28), we come to the equation By a direct calculation we obtain Thus, we have the following factorization relation ). (6.6) Now introduce the notation and define new universal Q-operators In terms of these operators the factorization relation takes the form The advantage of this formula over (6.6) is that the factor C l does not depend on µ, and this is necessary to obtain the determinant representation of the universal transfer operator T λ (ζ).

Determinant representation.
The Weyl group W of the root system of gl l+1 is generated by the reflections r i : (gl l+1 ) * → (gl l+1 ) * , i = 1, . . . , l, defined by the equation The minimal number of generators r i necessary to represent an element w ∈ W is said to be the length of w and is denoted by l(w). It is assumed that the identity element has the length equal to 0. It is easy to demonstrate that the reflection r i transposes the components µ i and µ i+1 of µ, and the whole Weyl group W can be identified with the symmetric group S l+1 . Here (−1) l(w) is evidently the sign of the permutation corresponding to the element w ∈ W. The order of W is equal to l!, and the length of the elements of W runs from 0 to l(l + 1)/2.
6.5. Q-system and nested Bethe ansatz equations. In this section we generalize the consideration given in the paper [32] for the integrable system associated with the Yangian Y(gl n ) to the case of the integrable system associated with the quantum loop algebra U q (L(sl l+1 )). Let a = (a 1 , . . . , a p ), 1 ≤ p ≤ l + 1, be a p-tuple of distinct elements of the set {1, . . . , l + 1}. Define the generalized universal Q-operator Q a (ζ) by the equation We use the symbol '∅' for the empty tuple and set Here and below we use the symbol '∪' for concatenation of tuples. By definition, we have and it follows from (6.14) that Q 1, ..., l+1 (ζ) = C l . (6.22) Let a = (a 1 , . . . , a p ) be a p-tuple of distinct elements of the set {1, . . . , l + 1}, while b and c are elements of {1, . . . , l + 1 } not included in a. One can easily demonstrate that the Jacobi identity with a 0 = b and a p+1 = c, implies the equation Note that the case a = ∅ is allowed if we set It is convenient, using the antisymmetry of generalized universal Q-operators under the permutations of the indices, to rewrite the above equation in the form Q a∪b∪c (ζ)Q a (ζ) = Q a∪b (q −1/s ζ)Q a∪c (q 1/s ζ) − Q a∪c (q −1/s ζ)Q a∪b (q 1/s ζ). Q a (ζ) = ψ(Q a (ζ)), see equations (5.5), (5.36) and (5.35). For the generalized Q-operator (Q a (ζ)).
In fact, we have As follows from the definition (5.35) of the function F l+1 (z), it satisfies the equation and, using (6.22), we obtain Note also that 26) The operators Q a (ζ) commute for any a and ζ, see equation (5.20). Although we do not know a rigorous proof of this fact, there are many indications that the operators Q a (ζ) are diagonalizable, and we assume that this is indeed the case. Therefore, all these operators can be diagonalized simultaneously, see, for example, the book [67, Theorem 6.8].
Let k = (k 1 , . . . , k l+1 ) be a tuple of non-negative integers such that k 1 + · · · + k l+1 = n. Denote by U k the linear span of the basis vectors containing k i vectors v i for any i = 1, . . . , l + 1. Clearly, it is a subspace of U. It is evident that for any v ∈ U k . Hence, the subspace U k is invariant with respect the action of the operators ψ(q νh i ). One can demonstrate that a vector v belongs to U k if an only if for any ν ∈ C and i = 1, . . . , l + 1. The operators (n) Q a (ζ) commute with the operators ψ(q νh i ), see equation (5.22). Therefore, U k is an invariant subspace for these operators as well.
Restricting ourselves to U k , we can consider (6.24) as an equation for the eigenvalues of the operators (n) Q a (ζ), which we denote by (n) Q a, k (ζ). Thus, we have the equation We call i in the above equation the level. Introduce one more notation where D a is defined by equation (6.7). It follows from equation (6.27) that the eigenvalues of the operators D a on U k are D a, k = s 2(l + 1) (ζ s − ζ a, k, j ), (6.32) where and c a, k are some complex coefficients. Here we set k ∅ = 0. Now, substitute (6.32) into (6.30), and consider three different cases. In all cases we need the equation which follows from the equation (6.31). Introducing new parameters τ a related to t a by the equation t a = τ a − τ a+1 , we obtain 2(D b, k − D c, k )/s = −k b + k c + τ b − τ c . (6.33) For the first level, equation (6.26) implies −1 = Q a 1 , k (q 2 ζ a 1 , k, m ) 1/s Q a 2 , k (q −1 ζ a 1 , k, m ) 1/s Q a 1 , k (q −2 ζ a 1 , k, m ) 1/s Q a 2 , k (q ζ a 1 , k, m ) 1/s . Using (6.32) and (6.33), we come to the equation q τ a 2 −τ a 1 = k a 1 ∏ j=1 j =m q 2 ζ a 1 , k, m − ζ a 1 , k, j ζ a 1 , k, m − q 2 ζ a 1 , k, j k a 2 ∏ j=1 ζ a 1 , k, m − qζ a 2 , k, j q ζ a 1 , k, m − ζ a 2 , k, j .
Thus, we come to the nested Bethe ansatz equations. We have l! variants of the equations corresponding to l! paths on the Hasse diagram, see the paper [32] for the Yangian case.

CONCLUSION
As a byproduct we have a conjectured formula for the components Λ µ+ m, i (ζ, u) of the ℓ-weight Λ µ m (ζ) of the evaluation U q (L(sl l+1 ))-module V λ . It can be obtained from the expression (6.4) for Ξ µ+ m, i (ζ, u) after the substitution n a, i = m a, a+i . The result is j=i m aj +i−2a+1 ζ s u) j=i+2 m aj +i−2a+3 ζ s u) j=i+1 m aj +i−2a+3 ζ s u) .
The highest ℓ-weight of the module V λ is rational, see equations(A.6) and (A.7). In this case the components Λ µ− m, i (ζ, u) are uniquely determined by the components Λ µ+ m, i (ζ, u), see, for example, the paper [78]. In our case we have .
Recall that the component λ µ m is given by equation (4.3). It would be interesting to generalize the consideration of the present paper to the supersymmetric case.

ACKNOWLEDGMENTS
This work was supported in part by the RFBR grant # 20-51-12005. The author is grateful to H. Boos, F. Göhmann, A. Klümper, and Kh. S. Nirov, in collaboration with whom some important results, used in this paper, were previously obtained, for useful discussions.
APPENDIX A. HIGHEST ℓ-WEIGHT OF EVALUATION U q (L(sl l+1 ))-MODULES We start with equation (5.8) of the paper [38]  Here and below m + kǫ ij means shifting by k the entry m ij in the l(l + 1)/2-tuple m. Using this relation, we first derive some auxiliary equations.