Twisted rational r -matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models

In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl( 2 ) ⊗ gl( 2 ) -valued Cartan-non-invariant classical r -matrices with spectral parameters. We consider the two families of these r -matrices, namely, the two non-standard rational r -matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r -matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations.


Classical r-matrices and quantum integrable systems
Quantum integrable systems are of utmost importance for the quantum mathematical physics. In some sense, the most interesting quantum solvable systems are the ones admitting Lax representation, i.e. the so-called Lax-integrable models. These systems can be further divided into the two classes: the ones for which the relevant Lax algebra is linear and the ones for which the Lax algebra is quadratic. The commutation relations of linear and quadratic Lax algebras are determined by classical r-matrices and quantum R-matrices, respectively.
For a long while it was commonly believed that integrable systems associated with linear Lax algebras are just the artefacts of the integrable systems associated with quadratic algebras i.e. quantum groups. That is to say that for quantum integrability are pertinent only the skewsymmetric classical r-matrices r 12 (u 1 , u 2 ) ∈ g ⊗ g, such that r 12 (u 1 , u 2 ) = −r 21 (u 2 , u 1 ), and the corresponding linear Lax algebras. Here g is (semi)simple Lie algebra or reductive Lie algebra gl(n).

Aims of the paper
The aim of this paper is two-fold. The first aim is to continue the study of the generalized Gaudin models based on the non-skew-symmetric classical r-matrices, their generalized Richardson's counterparts and to specify physically interesting ones. This study was initiated by the first author in [16,17] for the case of the Gaudin-type models and in [22] for the case of the Richardson-type models. Moreover, the cases of r-matrices which are diagonal in some natural basis, have been studied in the papers [22][23][24]27], including their Richardson-type systems and corresponding spectral problem. With the present paper we begin the study of the general nondiagonal, non-Cartan-invariant non-skew-symmetric r-matrices, their Richardson-type models and their spectra. For this purpose we consider two non-standard rational skew-symmetric rmatrices written as follows [4,10]: where c is an arbitrary parameter. Furthermore, we construct their twisted counterparts (1.2) having, up to an equivalence, the following explicit form: here, again, c is an arbitrary parameter. These r-matrices may also be viewed as two oneparametric families of non-skew-symmetric, non-Cartan-invariant deformations of standard skew-symmetric rational and non-skew-symmetric trigonometric r-matrix [24]. The corresponding Richardson-type models will be one-parametric deformations of the Richardson model of the s-type [2] and p x + ip y -type [23,24], [28]. In particular, the r-matrix (1.6b) yields the following integrable Richardson-type fermion hamiltonian: constituting a one-parametric family of deformations of the p x + ip y reduced BCS hamiltonian [23,24]. Here c † n, , c m, , m, n ∈ 1, N, , ∈ {+, −} are standard fermion creation-anihilation operators, n is a free energy of the n-th fermion and g nm = g √ n m is a pairing interaction strength.
Notice that non-standard rational skew-symmetric r-matrices (1.5a)-(1.5b) have no diagonal shift elements satisfying the equation (1.3). Therefore one can not associate with them Richardson-type hamiltonians possessing the kinetic terms. This confirms the observation of the first author [27] that non-skew-symmetric classical r-matrices are more pertinent to the constructions of the integrable Richardson-type hamiltonians than the skew-symmetric ones. Now we come to the second aim of our paper, which is to the develop methods for complete integrability of the quantum systems related to the classical r-matrices.

Algebraic Bethe ansatz
The exact solvability of the quantum integrable models, i.e. the diagonalization of the corresponding Hamiltonian and related integrals of motion, can be obtained by several methods. For the Lax integrable models the most important method is the algebraic Bethe ansatz. For the quadratic Lax algebras and Cartan-invariant quantum R-matrices the Bethe ansatz technique was proposed and developed in the papers of Leningrad school of mathematical physics (see [14,15] for the reviews), for the linear Lax algebra case and the Cartan-invariant skew-symmetric classical r-matrices it has been proposed in [1] and developed in [5,6].
In the papers of the first author the algebraic Bethe ansatz approach was applied on the Laxintegrable systems governed by Cartan-invariant non-skew-symmetric classical r-matrices: for the case of Lie algebras g = sl (2), gl (2) in [19,23] and for the case of Lie algebras g = gl(n) in [25,26]. The Cartan-non-invariant case is investigated much less [32,34]. Thus the second aim of this paper is to develop the algebraic Bethe ansatz for the Lax-integrable systems with linear Lax algebras governed by Cartan-non-invariant non-skew-symmetric classical r-matrices r 12 (u, v).
For the Cartan-invariant case and Lie algebras g = sl (2), gl (2) the algebraic Bethe ansatz is implemented to the non-skew-symmetric case just in the same manner as in the skew-symmetric case [5]. In more details, the eigenvectors of the quantum hamiltonians are given by: where |0 is the vacuum vector such that ij (u)X ij is the Lax matrix and rapidities v i satisfy a set of Bethe-type equations.
In order to apply the algebraic Bethe ansatz for the considered Cartan-non-invariant r-matrices (1.6a)-(1.6b) we use the approach of the second author to the skew-symmetric classical r-matrix (1.5a)-(1.5b) in [29,30] with the assumption that non-skew-symmetric case is organized analogously. Namely, for the Bethe states we take the vectors of the following from where the vacuum vector satisfies the conditions (1.8) and the operators B k (v k ) are defined as follows: here f (v k ) = c in the case of the r-matrix (1.6a) and f (v k ) = cv k in the case of the r-matrix (1.6b).
Moreover, the form of the Bethe equations that guarantee the diagaonalization of the generating functions of the quantum Hamiltonians for the r-matrices (1.6a)-(1.6b) -as the for the skew-symmetric r-matrices (1.5a)-(1.5b) -is the same as in the undeformed c = 0 case. Thus we have that the one-parametric families of the obtained deformed BCS-Richardson's Hamiltonians of sand p x + ip y -type have the same spectrum as the non-deformed Hamiltonians of the same type, but their eigenvectors are different and depend on the deformation parameter c.

The structure of the paper
The structure of the paper is the following: in the Section 2 we outline some general facts about the quantum integrable systems and non-skew-symmetric classical r-matrices. In the Sections 3 and 4 we consider the non-skew-symmetric r-matrices (1.6a)-(1.6b) correspondingly, their Gaudin-type and Richardson-type models and the implementation of the algebraic Bethe ansatz for them.

Quantum integrable systems and classical r-matrices
In this section we will briefly review the relation of the theory of general non-skew-symmetric classical r-matrices with spectral parameters with the theory of quantum integrable systems [16][17][18][19]. Although the constructions presented in this section hold true for any simple (reductive) Lie algebra, we will state them in the case of the reductive Lie algebra g = gl(2).

Definition and notations
Let g = gl(2) be the Lie algebra of the general linear group over the field of complex numbers. Let X ij , i, j = 1, 2 be a standard basis in gl (2) with the commutation relations: (2.1) Definition 1. A function of two complex variables r(u 1 , u 2 ) with values in the tensor square of the algebra g = gl (2) is called a classical r-matrix if it satisfies the following generalized classical Yang-Baxter equation [7,9,8] [r 12 (u 1 , u 2 ), r 13 1 ⊗ X kl , etc. and r ij,kl (u, v) are matrix elements of the r-matrix r(u, v).

Remark 1.
In the case of skew-symmetric r-matrices, i.e. when r 12 (u 1 , u 2 ) = −r 21 (u 2 , u 1 ) the generalized classical Yang-Baxter equation reduces to the proper classical Yang-Baxter equation [3,4]: In the present paper we are interested in the meromorphic r-matrices that possess the decomposition: Casimir.
For the subsequent we will also need the following definition: c ij (u)X ij of one complex variable is called generalized shift element if it satisfies the following equation:

The σ -twisted classical r-matrices with spectral parameters
Let r 12 (u 1 , u 2 ) be a skew-symmetric r-matrix i.e. a non-degenerate meromorphic solution of the proper classical Yang-Baxter equation (2.3). Let σ be an automorphism of g = gl(2) of the second order: σ 2 = 1.
Let us assume that the following anti-invariance condition is satisfied: where σ is an extension of σ onto the algebra of meromorphic functions, i.e. σ (X(u)) = σ (X(u σ ))), σ : u → u σ is a certain involution in C, e.g. u σ = −u or u σ = u −1 . The index i ∈ 1, 2 in σ i refers to the component of the tensor product in which the automorphism σ acts. Taking into account the definitions above, it is possible to show [18,19] that the following tensor is a solution of the generalized classical Yang-Baxter equation (2.2), i.e. is a non-skew-symmetric classical r-matrix with spectral parameters. A particular case of such a non-skew-symmetric classical r-matrices will be the main interest of the present article.

Lie algebra of Lax operators
In the space of certain gl(2) valued functions of the complex parameter u, using the classical r-matrix r(u 1 , u 2 ), it is possible define the tensor Lie bracket [7,9,8] [ 12 , and P 12 is the permutation operator which interchanges the first and second spaces in the tensor product. Tensor bracket (2.8) between the Lax matrices L 1 (u 1 ) and L 2 (u 2 ) is a symbolic notation for the Lie brackets between their matrix entries. Explicitly, in terms of matrix entries the bracket (2.8) takes the following form Remark 2. From the explicit form of the Lie bracket (2.9) it follows that the operators (L 11 (u) − L 22 (u)), L 12 (u), L 21 (u) span a subalgebra and an ideal in the Lax algebra spanned over the elements L 11 (u), L 22 (u), L 12 (u), L 21 (u), independently on the form of the classical r-matrix.

Generating functions of quantum integrals
Let us now consider the following linear function on the Lax algebrâ (2.10) From the tensor form of the Poisson brackets (2.8) it follows that it generates commutative subalgebra. Moreover, the following Proposition is valid Proof. The statement follows from the brackets (2.9).
By the direct calculation one can also prove the following Proposition Proposition 2.2. The gl(2) ⊗ gl(2)-valued r-matrices r 12 (u, v) that satisfy the conditions (2.11) can be brought to the sl(2) ⊗ sl(2)-valued form by the equivalence transformation 1 : where X(u, v) takes the value in gl (2).
Let us now consider the following quadratic function on the Lax algebrâ We can state the following Theorem

Theorem 2.1. Let the classical r-matrix r(u, v) satisfy the conditions (2.4) and (2.11). Then
Proof. In order to prove the theorem, we note that in the new notations corresponding to the sl(2) basis the generating function τ (2) (u) is re-written in the following waŷ τ (2) Now we note that under the conditions of the Theorem τ (1) (u) is a central element, hence it commutes with everything and its presence or absence does not influence the relation (2.14). Moreover, under the conditions of the Theorem by the virtue of the Proposition 2.1 and the arguments of the previous subsection, the Lax algebra (2.9) is a direct sum of sl(2)-valued Lax algebra with the generating functions of the basis being Ĥ (u), B (u) and Ĉ (u) and a center with the generating function being τ (1) (u). Finally, by the virtue of the Proposition 2.2 the r-matrix That is why the proof of the commutativity of the generating functions τ (2) (u) and τ (2) (v) is reduced to the proof of the commutativity of the generating functions τ(u) and τ (v), wherê is a quadratic generating function on sl(2)-valued Lax algebra with the sl(2) ⊗ sl(2)-valued rmatrix. Finally, to prove the theorem we recall that the proof of the commutativity of τ(u) and τ (v) under the condition (2.4) was done in the paper [19].

Generalized Gaudin models in external magnetic field
.., N be linear operators in some Hilbert space that span Lie algebra isomorphic to gl(2) ⊕N with the commutation relations Let us fix N distinct points of the complex plain ν m , m ∈ 1, N. It is possible to introduce the following quantum Lax operator [16,17,21] This quantum Lax operator corresponds to the generalized gl(2)-Gaudin system in the external magnetic field. As our next step, we apply the results of the previous subsection to the Lax operators of the generalized Gaudin systems. A direct calculation yields the explicit form of the corresponding generating functions: Under the conditions (2.11) on the r-matrix the generating function (2.17) is further reduced to the following form: i.e. it becomes the linear combinations of the linear Casimirs of gl (2). The second order generating function is given bŷ Maybe somewhat more transparent are the residues of τ (2) (u) at u = ν n : They have the following form where r ij,kl 0 (ν n , ν n ) are the matrix elements of the regular part of the classical r-matrix r(u, v) at the point u = v = ν n . The Hamiltonians (2.18) are the generalized Gaudin Hamiltonians corresponding to the gl(2) ⊗ gl(2)-valued r-matrix [16,17]. By the virtue of the results of the previous subsection as well as of the general results of [17,21] they mutually commute [Ĥ m ,Ĥ n ] = 0, ∀m, n ∈ 1, N.

The σ -twisted case
The non-skew-symmetric classical r-matrices we will consider in the present paper are the σ -twisted classical r-matrices (2.7). The corresponding Lax matrix of the Gaudin-type models in the external magnetic field can be written as followŝ In this case, the Gaudin-type Hamiltonians (2.18) have the following form where r ij,kl (ν m , u) in the formula (2.20) are the components of the initial skew-symmetric classical r-matrix from which the r-matrix (2.7) is obtained. By the virtue of all we have stated above as well as the results of [17,18,21], the Hamiltonians (2.20) mutually commute Remark 3. In all the applications below we will consider only the case where the involution is given by ν σ n = −ν n .

Integrable fermion models
Base on integrable quantum spin chains it is possible to derive integrable fermion systems. To this end it is necessary to consider the realization of the corresponding spin operators in terms of fermion creation-anihilation operators. In other words, it is necessary to obtain the fermionization of the underlying Lie algebra gl(2) ⊕N , where N is the length of the chain.

Fermionization
Here we will consider only the simplest fermionization of the Lie algebra gl(2) ⊕N corresponding to the case when all the Lie algebras gl (2) in the direct sum have the representation with the lowest weight λ = (0, 1).
More explicitly, let c j, , c † i, , i, j ∈ 1, N, , ∈ {+, −} be fermion creation-annihilation operators, then we have By direct calculation it is possible to show that the following formulaê provide realization of the Lie algebra gl(2) ⊕N with the lowest weight λ j = (0, 1), j ∈ 1, N. Here operators c j, are chosen to annihilate, and operators c † j, are chosen to create fermion in the state j, .

Remark 4.
Note, that after the restriction to the subalgebra sl (2) and after the identification 22 we obtain the well-known fermionization of the Lie algebra sl(2) [11,12].

BCS-type Hamiltonians
In order to construct some interesting integrable fermion Hamiltonian Ĥ of the BCS-type it is necessary to apply the above fermionization formulae to the certain combination of the generalized Gaudin Hamiltonians, Casimir functions and, possibly, some linear integrals. The concrete form of the Hamiltonian Ĥ will depend on the underlying classical r-matrix. We will consider some explicit examples in the next sections.

Twisted non-standard rational r-matrix
Let us consider skew-symmetric rational r-matrix of the following form [10] Furthermore, let σ be a trivial automorphism of gl(2): σ (X) = X and let us also consider the following involution in C: u σ = −u.
Due to the fact that r 12 (−u, −v) = −r 12 (u, v), it immediately follows that Thus, we can define the following twisted non-skew-symmetric classical r-matrix of the type (2.7) Making the equivalence transformation, namely, dividing this r-matrix by 2v, and changing the parametrization: u 2 → u, v 2 → v we come to the following shifted non-skew-symmetric rational r-matrix which we will consider in this section. In order to simplify our notation in the following we will denote it simply by r 12 (u, v).

Remark 5.
Observe that the r-matrix (3.1) can be viewed as a shifted standard rational r-matrix 12 , with the shift tensor being constant and non-skew-symmetric: The r-matrix (3.1) evidently satisfies the condition (2.4) as well as the conditions (2.11).
It is straightforward to show that the r-matrix (3.1) possess the following constant shift element (3.2)

Linear Lax algebra and generating functions of the integrals of motion
With the help of the classical r-matrix (3.1) one can define the linear Lax algebra (2.8) As a consequence of the fact that the r-matrix (3.1) satisfies the conditions (2.11) the function generates a center of the linear Lax algebra (2.8). Also, it follows from the Theorem 2.1 the function is a generating function of the commuting quantum integrals [τ (2) (u),τ (2) (v)] = 0.
The main task of the subsequent subsections will be to diagonalize this operator function. For the subsequent it will be convenient to introduce the following notations: In terms of these operators the commutation relations of the considered Lax algebra (2.9) become Also, the generating functions τ (1) (u), τ (u) ≡τ (2) (u) are given bŷ

Vacuum vector and Bethe vectors
where the operators B k (u) are defined by the following formulâ From the commutation relation (3.3d) we have that: Therefore the vector |v 1 , v 2 , ..., v M is a symmetric function of its arguments.

The spectrum of the generating functions
In this subsection we seek the spectrum of the generating functions τ (1) (u) and τ (2) (u) corresponding to the Bethe vectors |v 1 , v 2 , ..., v M (3.5).
Notice that due to the fact that τ (1) (u) is a Casimir function, the spectrum of τ (1) (u) iŝ Below we will simplify our notation by τ(u) ≡τ (2) (u). To calculate the spectrum of τ (u) we will need the following Proposition.

Proposition 3.1. The following commutation relations hold
Proof. The proof of the proposition is by the mathematical induction. To prove the first step we calculate the following commutator using the commutation relations (3.3) and the direct calculation. We obtain that Using the definition of the operator B 1 (v) (3.6) the M = 1 case follows directly from the formula above (3.9) Now let us assume that the formula (3.8) is valid for M. We have to prove that it is also valid for M + 1. Using the Leibnitz rule for the commutator we have Using further the formulae (3.8) and (3.9) we obtain In order to obtain the desired formula we have to pass the operator B M+1 (v M+1 ) to the left of the operators Â (u), D (u), Ĉ (u), Â (v k ), D (v k ) on the rhs of the formula (3.11). To this end we use 12c) The careful analysis shows that the additional terms obtained from the commutators (3.12) transform the right hand side of (3.11) to the form (3.8) with M → M + 1. To show this we have used the operator identitŷ the definition of B M+1 (v) and the following identity: Therefore the proposition is shown.

Now we can formulate the following Theorem
Theorem 3.1. Let the rapidities v k , k ∈ 1, M satisfy the following set of Bethe equations . (3.14) Proof. The statement of the theorem follows from the previous Proposition. Indeed, we havê On the other hand we have that where we have used the commutation relations (3.3h) in the limit v → u. Then, using the relations (3.4) we obtain Now, making use of the Proposition 3.1 we obtain that, by the virtue of the conditions (3.4), for the rapidities v i that satisfy Bethe equations (3.13) the following equality holds true: This completes the proof of the Theorem.

Remark 6.
It is of interest to notice that the spectrum and the Bethe equations of the models associated with the r-matrix (3.1) coincide with that in the standard rational case, when c = 0 [5]. But, as we have shown, when c = 0, the Bethe vectors have different form.

Lax matrix and Gaudin-type Hamiltonians
Let us now consider the Lax algebra of the Gaudin-type models. In the case when r-matrix is (3.1) the corresponding Lax matrix (2.16) has the following form The mutually commuting quantities it produces with the help of the generating functions τ (1) (u), τ (2) (u) are linear and quadratic Casimirs of the direct sum gl(2) ⊕N as well as the following Gaudin-type Hamiltonians (2.18) in an external magnetic field

Algebraic Bethe ansatz
Let us apply the construction of the previous subsection to the case of the Lax operators of the generalized Gaudin models. Let us consider a finite-dimensional irreducible representation of the algebra gl(2) ⊕N in some space V . Due to the fact that any irreducible representation of the direct sum of the Lie algebras is a tensor product of irreducible representations of their components, we will have V = V λ 1 ⊗ V λ 2 ⊗ · · · ⊗ V λ N , where V λ k is an irreducible finitedimensional representation of the k-th copy of gl (2) with the lowest weight λ k = (λ (k)

Let us consider the following vacuum vector in the space
From the definition of the Lax matrix (3.15) it follows that Thus, we have the following action of the entries of the Lax matrix on the vacuum vector where, due to (3.19) and (3.17a)-(3.17b), the eigenvalues are given by In this case, the creation operators B k (u) have the following form Due to the results of the previous subsection we know that the Bethe vectors are given by where the rapidities v i should satisfy the Bethe equations (3.13), which now read The spectrum of the generating function τ(u) is given by . (3.22) Also, the spectrum of the generalized Gaudin Hamiltonians is

The spectra and Bethe equations
In particular case when λ (k) 2 = 1, λ (k) 1 = 0, k ∈ 1, N the Bethe equations (3.21) have the following simple form The spectra of the generalized Gaudin Hamiltonians is also simple in this case where the rapidities v i satisfy the Bethe equations (3.27).
The spectrum of the BCS-like Hamiltonian (3.26), up to the constant, has the following form (3.29) here we have used the definition of Ĥ and the Bethe equations (3.27). It is important to notice that the spectrum of the Hamiltonian Ĥ (3.26) is the same as in the standard Richardon's case [2,11], but the Bethe vectors are different.

Shifted twisted trigonometric r-matrix
Let us consider another non-standard skew-symmetric rational r-matrix of the following form [4]: It is straightforward to check that where u σ = −u, v σ = −v and the automorphism σ on g = gl(2) is defined by Thus we can define the following non-skew-symmetric classical r-matrix By making the equivalence transformation, namely, multiplying this r-matrix by v 2 we come to the following non-skew-symmetric r-matrix of the type (2.7) In this section we will focus on this r-matrix and it will be denoted simply by r 12 (u, v).

Remark 9.
Notice that the r-matrix (4.1) may be also viewed as a shifted trigonometric r-matrix: with the shift tensor c 12 (v) defined as follows c 12 The trigonometric parametrization is obtained by the following substitution u = exp iφ 2 , v = exp iψ 2 . The r-matrix (4.1) satisfies the condition (2.4) in the trigonometric parametrization, since the r-matrix r trig 12 (u, v) satisfies it in this parametrization. Furthermore, is straightforward to show that the r-matrix (4.1) satisfies the conditions (2.11).

Linear Lax algebra and generating functions of the integrals of motion
In the standard way the classical r-matrix (4.1) defines the linear Lax algebra (2.8): Using the fact that the r-matrix (4.1) satisfies the conditions (2.11) we have that the function generates a center of the linear Lax algebra (2.8). Also, it follows from the Theorem 2.1, that the function τ (2) is a generating function of the commuting quantum integrals of the second order The spectral decomposition of this function will be the main topic of the subsequent subsections. As in the previous section, it will be convenient to use the following notationŝ In these terms the commutation relations of the Lax algebra (2.8) acquire the following form: (4.2h) In terms this notation, the generating functions τ (1) (u), τ (u) ≡τ (2) (u) have the following form:

Vacuum vector and Bethe vectors
As usual, let us assume that in the representation space V of the Lax algebra (4.2) there exists a vacuum vector |0 such that: Following the ideas of [29][30][31], we consider the following vectors |v where the operators B k (u) are defined bŷ From the commutation relation (4.2d) we have that: Therefore the vector |v 1 , v 2 , ..., v M is a symmetric function of its arguments.

The spectrum of the generating functions and Bethe equations
In this subsection we will study the spectra of the generating functions τ (1) (u) and τ (u) relative to the Bethe vectors |v 1 , v 2 , ..., v M obtained in the previous subsection.
Since τ (1) (u) is a Casimir function, its spectrum on the Bethe vectors (4.4) iŝ In order to calculate the spectrum of τ(u) we will need the following Proposition.
Proposition 4.1. The following commutation relation holds 3 : It can be seen that the additional terms obtained from the commutators (4.11) transform the right hand side of (4.10) to the form (4.7) with M → M + 1. In order to show this one also has to use thatB the definition of the creation operator B M+1 (v) (4.5) and the following identity This completes the proof of the proposition. Now we can state the following Theorem . (4.13) Proof. The statement of the theorem follows from the previous Proposition. Indeed, we have: On the other hand we have that: where we have used the commutation relations (4.2h) in the limit v → u. That is why using the relations (4.3) we obtain that: Now, making use of the Proposition 4.1 we obtain that, by the virtue of the conditions (4.3), and for the rapidities v i that satisfy Bethe equations (4.12), the following formula holds true This completes the proof of the theorem.

Remark 10.
The spectrum and Bethe equations of the systems associated with the r-matrix (4.1) coincide with the ones of the non-skew-symmetric trigonometric c = 0 case [24]. But the form of the Bethe states is different.

Lax matrix and Gaudin-type Hamiltonians
Let us consider the universal example of the Lax algebra which yields the Gaudin models. The relevant Lax matrix (2.16) has the form 14) The commutative quantities it produce with the help of the generating functions τ (1) (u), τ (u) are linear and quadratic Casimirs of the direct sum gl(2) ⊕N

Algebraic Bethe annsatz
Let us apply the construction of the previous subsection to the case of the Lax operators of the generalized Gaudin models. Let us consider a finite-dimensional irreducible representation of the algebra gl(2) ⊕N in some space V , just as we have done in the subsection 3.4.2. As we have seen already, any irreducible representation of the direct sum of the Lie algebras is a tensor product of irreducible representations of their components. That is to say that V = V λ 1 ⊗ V λ 2 ⊗ · · · ⊗ V λ N , where V λ k is an irreducible finite-dimensional representation of the k-th copy of gl (2) with the lowest weight λ k = (λ (k) 1 , λ (k) 2 ) with λ (k) 1 , λ (k) 2 ∈ N. Each representation V λ k contains the lowest weight vector v λ k such that S (k) acts on each vector v m λ k ∈ V λ k in the usual way Ĉ (2) k v m λ k = 1 2 (λ (1) k ) 2 + (λ (2) k ) 2 + (λ (2) k − λ (1) k ) v m λ k . Also, as we already know, the vacuum vector in the space V is given by  In this case, the creation operator B k (u) has the following form The spectrum of the generating function τ(u) is given by .

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.