From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group $\mathrm{SL}(2,\mathbb{C})$ or Faddeev's modular double. Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces. In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches.


Introduction
The Yang-Baxter equation (YBE) is a major tool in building the quantum integrable systems [1,[22][23][24]31]. It has found numerous applications in mathematical physics and purely mathematical questions. At the dawn of quantum inverse scattering method the finite-dimensional solutions of the YBE (when the operators R ij (u) are given by ordinary matrices with numerical entries depending on the spectral parameter u) attracted much attention in view of their relevance for physical spin systems on the lattices admitting a successful treatment of their thermodynamical behavior [1,22].
Solutions of the YBE for infinite-dimensional representations revealed their importance in the integrability phenomena emerging in quantum field theories. An integrable spin chain with underlying SL(2, C) symmetry group and its noncompact representations naturally arises in the high-energy behavior of quantum chromodynamics. Corresponding model was discovered in [25] together with an additional integral of motion. Later, in [26] and [18] it was identified with the noncompact XXX spin chain which revealed its complete integrability (for further investigations of this model, see [10,14]).
There are three increasing levels of complexity of finite-dimensional solutions of YBE described by matrices with matrix coefficients expressed in terms of the rational, trigonometric, and elliptic functions. In the infinite-dimensional setting the latter hierarchy is replaced by solutions of YBE defined as integral operators with the integrands described by plain hypergeometric, q-hypergeometric and elliptic hypergeometric functions [29].
The notion of the modular double was introduced by Faddeev in [16] and its noncompact representations arise naturally in the study of the Liouville model [17,27]. The quantum dilogarithm function [15] plays an important role in the representation theory of this modular double and in the Faddeev-Volkov solution of YBE [34] as well as in its generalization found in [30]. The elliptic modular double was introduced in [28].
The general elliptic solution of YBE with the rank 1 symmetry algebra was found in [12]. It is based on the properties of an integral operator with an elliptic hypergeometric kernel, the key identity for which (given by the Bailey lemma, see e.g. [29]) coincides with the star-triangle In Sect. 2.4 we derive an analogous reduction for the general sℓ 2 -algebra R-operator to the space of polynomials or the Verma module.
Then we proceed to the fusion. In Sect. 2.5 we formulate the fusion for the sℓ 2 algebra case in a rather nonstandard fashion. We construct projectors to the highest spin representation by means of some auxiliary spinor variables that results in the Jordan-Schwinger realization of the "fused" representation. We describe also how the fusion procedure reproduces the L-operator as well. After that, in Sect. 2.6 we get back to the SL(2, C) group and carry out the fusion in this case.
In the second part of the paper we consider similar questions for the modular double. There the presentation closely follows the rational case in order to emphasize the striking similarity between these two cases. In Sects. 3.1 and 3.2 we outline general structure of the modular double and the general R-operator for it. Corresponding reduced R-matrix which is finite-dimensional in one of the quantum spaces (or both) is derived in Sect. 3.3. Finally, in Sects. 3.4 and 3.5 we derive finite-dimensional R-matrices in the q-deformed cases using the fusion procedure.

Representations of the group and the intertwining operator
We start with a short review of some basic well-known facts about representations of the group SL(2, C). They are formulated in a form that will be natural for dealing with R-operators. We outline how finite-dimensional representations decouple from infinite-dimensional ones emphasizing the role of the intertwining operator.
Consider representations of the group SL(2, C) realized on the space of single-valued functions Φ(z,z) on the complex plane. The principal series representation [20] is parametrized by a pair of generic complex numbers (s,s) subject to the constraint to 2 (s −s) ∈ Z. We refer to them as spins in what follows. In order to avoid misunderstanding we emphasize that s ands are not complex conjugated in general.
The method of induced representations is a robust tool that enables one to construct a number of interesting representations of the group (see for example [19]). In the case of interest the representation T (s,s) is given explicitly as follows [20] T (s,s) (g) Φ (z,z) = (d − bz) 2s d −bz Representations of the group SL(2, C) yield representations of the Lie algebra sℓ(2, C) in a standard way. Assuming that g lies in a vicinity of the identity g = 1 + ǫ · E ik , where E ik are traceless 2 × 2 matrices: (E ik ) jl = δ ij δ kl − 1 2 δ ik δ jl , one extracts generators E ik andĒ ik of the Lie algebra, The generators E ik ,Ē ik are first-order differential operators. We arrange them in 2 × 2 matrices E (s) andĒ (s) , which will be useful for consideration of the integrable structures, The substitution z →z , ∂ →∂ and s →s in this formula results in the matrixĒ (s) for the generators E ik .
There exists an integral operator W which intertwines a pair of principal series representations T (s,s) and T (−1−s,−1−s) at generic complex s ands, W(s,s) T (s,s) (g) = T (−1−s,−1−s) (g) W(s,s) . (3) We will refer to this pair as the equivalent representations. The described intertwining relation can be equally reformulated as a set of intertwining relations for the Lie algebra generators W(s,s) E (s) = E (−1−s) W(s,s) ; W(s,s)Ē (s) =Ē (−1−s) W(s,s) .
The operator W is defined up to an overall normalization and it has the following explicit expression [20] [ W(s,s)Φ ] (z,z) = const Obviously the integral operator (5) is well-defined at generic s,s and the problems emerge for the discrete set of points 2s = n , 2s =n at n,n ∈ Z ≥0 . These special values of the spins correspond to finite-dimensional representations which we are aiming at. That is why we would like to have a meaningful intertwining operator on this discrete set. In order to obtain it we note that the expression (5), considered as an analytical function of s,s, has simple poles exactly on this discrete set of (half)-integer points. Consequently we need to choose properly the normalization constant in (5) to suppress the poles at 2s = n , 2s =n. Further, pursuing this strategy we find the normalization constant as an appropriate combination of Euler gamma-functions such that the intertwining operator (5) becomes well-defined in the case of finite-dimensional representations as well. In order to implement the outlined program we resort to the text-book formula for the following Fourier transformation [20] A(α,ᾱ) where Γ(x) is the Euler gamma function. We replace p andp by the differential operators, p → i∂ x andp → i∂x, use the shift operator e a∂x f (x) = f (x + a) and come to the definition In order to avoid cumbersome expressions we prefer to recast this formula to a concise form Here and in the following we profit from the shorthand notation which unifies the holomorphic and antiholomorphic sectors. Let us remind one more time that α and α are not assumed to be complex conjugated. The constraint on the exponents α,ᾱ in (9) ensures that the function [z] α is single-valued, whereas at generic α the holomorphic and anti-holomorphic factors of [z] α taken separately have branch cuts. Bearing in mind that the holomorphic sector is always accompanied by the antiholomorphic one we omit theᾱ-dependence in the A-factor: Thus if the normalization in (5) is chosen properly the intertwining operator can be represented in two equivalent forms, either as a formal complex power of the differentiation operator W(s,s) = [i∂ z ] 2s+1 or as a well defined integral operator At special points 2s = n , 2s =n, n,n ∈ Z ≥0 , the integral operator turns to the differential operator of a finite order (i∂ z ) n+1 (i∂z)n +1 . Let us note that at generic s the holomorphic ∂ 2s+1 z and anti-holomorphic ∂ 2s+1 z parts (see (9)) of the intertwiner [i∂ z ] 2s+1 taken separately are ill-defined (working with the contour integrals with the kernel (z − x) α one cannot find a translationally invariant measure). However, being taken together they form a well-defined integral operator.
The formula (1) implies that at special values of spins 2s = n , 2s =n discussed above an (n + 1)(n + 1)-dimensional representation decouples from the general infinite-dimensional case [20]. Indeed, the space of polynomials spanned by (n + 1)(n + 1) basis vectors z kzk , where k = 0, 1, · · · , n andk = 0, 1, · · · ,n, is invariant with respect to the action of the operators T (s,s) (g). Instead of working with the separate basis vectors we prefer to deal with a single generating function which contains all of them. The generating function for basis vectors of this finite-dimensional representation can be chosen in the following form where x,x are some auxiliary parameters. Indeed, expanding (11) with respect to x andx we recover all (n + 1)(n + 1) vectors z kzk , where k = 0, 1, · · · , n andk = 0, 1, · · · ,n. The decoupling of a finite-dimensional representation and the explicit expression for the generating function (11) allow us to give a very natural interpretation to the situation from the point of view of the intertwining operator. Indeed, an immediate consequence of the definition (3) is that the null-space of W(s,s) -the space annihilated by the operator -is invariant under the action of the operators T (s,s) (g). Therefore, if the intertwining operator has a nontrivial null-space then a subrepresentation decouples and the corresponding invariant subspace appears. In the case at hand, when 2s = n and 2s =n, the intertwining operator turns into the differential operator ∂ n+1∂n+1 .
Of course this operator annihilates all (n+1)(n+1) basis vectors z kzk , where k = 0, 1, · · · , n and k = 0, 1, · · · ,n, but the whole null-space of this operator is too big (it includes all harmonic functions) and we need some additional characterization for the considered finite-dimensional subspace. Relation (3) shows that the image of the intertwining operator W(−1 − s, −1 −s) is also invariant under the action of the operators T (s,s) (g). Moreover, formula (10) in the considered situation clearly shows that for special values of the spins 2s = n and 2s =n discussed above the integral in the right-hand side is polynomial with respect to z andz and the image of the operator W(−1 − s, −1−s) (after dropping the numerical factor Γ (−s −s + |s −s|) which diverges at these points) is exactly the needed finite-dimensional subspace. After all we obtain a characterization of our finitedimensional subspace: it is the intersection of the null-space of the intertwining operator W(s,s) and image of the operator W(−1 − s, −1 −s) both being properly normalized for special values of the spins 2s = n and 2s =n.
Further we demonstrate that the intertwining operator annihilates the generating function of the finite-dimensional representation (11) using solely the basic properties of the intertwining operator.
Moreover, the following calculation suggests us the generating function. The differential form of the intertwining operators formally indicates that W(−1 − s) and W(s) are inverses to each other, However, this inversion relation is broken for special values of the spins. Let us rewrite the identity (13) taking into account the explicit expression for kernels of the integral operators W(−1 − s) (12) and 1l, which is given by the Dirac delta-function. In this way we find the relation At special points 2s = n , 2s =n the gamma-function Γ (−s −s + |s −s|) has poles, and therefore the right-hand side of the latter formula vanishes. So, one obtains i.e. the generating function of the finite-dimensional representation coincides with the kernel of the intertwining operator W(−1 − n/2) after a proper normalization. Our calculation may seem superfluous since the relation (14) is evident per se. However, we presented it here because all its basic steps remain valid after the trigonometric (see Sect. 3.1) and elliptic deformations (see [12,13]) of the symmetry algebra. The deformations complicate significantly the intertwining operator and the generating function of finite-dimensional representations such that the deformed analogues of (14) are far from being obvious and in the elliptic case they are much more involved [8,13].

The general SL(2, C)-invariant R-operator
Emergence of the periodic integrable spin chain with SL(2, C) symmetry in the high energy asymptotics of quantum chromodynamics was discovered in [18,25,26]. The detailed consideration of the corresponding formalism was performed in [10,14]. In these papers the quantum-mechanical model of interest has been solved, i.e. the relevant Baxter Q-operators has been constructed and the separation of variables has been implemented. The general R-operator for the SL(2, C) group has been extensively studied in the first part of [11] as a simplest nontrivial example of the general SL(N, C)-construction.
Here we briefly outline the main steps in the construction of this R-operator before proceeding to its finite-dimensional reductions.
Firstly we tailor a pair L-operators out of the Lie algebra generators E (s) ,Ē (s) (2) and the spectral parameters u andū which are assumed to be restricted like the representation parameters, u −ū ∈ Z [10,11], Here we apply the shorthand notation that turns out to be extremely useful in what follows Each of the L-operators (15), (16) respects the RLL-relation with the Yang's 4 × 4 R-matrix, where a, b, · · · = 1, 2, and the summation over repeated indices is assumed, R ab,cd (u) = u · δ ac δ bd + δ ad δ bc (cf. (49)). The described relations supplemented by the commutativity condition [ L(u) ,L(v) ] = 0 are equivalent to the set of commutation relations for the Lie algebra generators of SL(2, C).
The L-operators (15), (16) respect simultaneously another RLL-relation with some general Roperator which intertwines the co-product of L-operators in the pair of quantum spaces where parameters u 1 and u 2 are defined in (17), and v 1 , v 2 are analogous linear combinations of v and ℓ, The lower indices of R 12 and L 1 , L 2 denote quantum spaces on which the operators act nontrivially. The L-operators are multiplied as conventional 2 × 2 matrices and the R-operator acts as an identity operator on the auxiliary 2-dimensional spaces of L-operators, but it acts non-trivially on the tensor product of two infinite-dimensional representations: the first representation is specified by spins s,s and it is realized on the functions of variables z 1 ,z 1 , the second representation is specified by spins ℓ,l and it is realized on the functions of variables z 2 ,z 2 . In (19), (20) we skip the dependence of the R-operator on the representation parameters. The full-fledged notation would be R(u − v,ū −v|s,s, ℓ,l). Let us note that the R-operator serves for both L-operators, i.e. it is not just the holomorphic or anti-holomorphic as opposed to the L-operators (15), (16). In the following we frequently omit the dependence of the R-operator (and other intertwining operators) on the anti-holomorphic parameters denoting it R(u). The R-operator is invariant with respect to the SL(2, C) group, i.e. it commutes with the co-product of sℓ(2, C) generators which follows immediately from the RLL-relations (19), (20).
Apart from the RLL-relations (19), (20) the general R-operator satisfies the YBE where both sides are endomorphisms on the tensor product of three infinite-dimensional spaces realizing arbitrary principal series representations of SL(2, C).
In [10,11] the general R-operator, i.e. an integral operator solution of the equation (21), was found solving a pair of intertwining relations (19), (20). The construction implies that the Roperator naturally takes a factorized form. In [11] a number of factorized forms of the R-operator has been derived which were related with an operator representation of the symmetric group S 4 . Here we do not go into details of the construction and just indicate the factorized form which is appropriate for our current purposes. The R-operator factorizes to the product of four elementary intertwining operators [11] where we assume the shorthand notation z ij = z i − z j and (9). Taking into account (8) one can rewrite (22) explicitly as an integral operator. The notation (9) implies that the R-operator consists of the holomorphic and anti-holomorphic parts which, being taken separately, are ill-defined at generic spectral and representation parameters. The merge of holomorphic and antiholomorphic parts yields a well-defined integral operator. The formula (22) will play a crucial role in the subsequent discussion. It admits deformations [9] leading to R-operators for the modular double [5] and the elliptic modular double [12].
The form (22) of the R-operator might seem rather unusual. In [11] it was shown that the holomorphic part of the R-operator (22) restricted to the space of polynomials coincides with a familiar R-operator constructed in [23,31] in the form of the beta-function of the "square root" of the Casimir operator. Note that the form (22) does not demand extra information about the structure of tensor products and corresponding Clebsch-Gordan coefficients. Further we will show that the integral R-operator (22) contains a number of finite-dimensional solutions of the Yang-Baxter relation (21).
The factors appearing in (22), which we call the elementary intertwining operators, fulfill the following operator relations Using (23) one can easily prove that the R-operator (22) respects the Yang-Baxter relation (21). The previous formulae have a remarkable connection with representations of the symmetric group S 4 and Coxeter relations (see [9,11]). The elementary intertwining operators satisfy the equations and similar ones with L substituted byL , which justify the chosen terminology. Moreover, the previous relations uniquely fix ( (24) and (25) one can easily check that the composite R-operator (22) obeys the RLL-relations (19) and (20). The identities (23) are equivalent to the famous star-triangle relation which can be represented in the following three equivalent forms: 1) as an integral identity [10,32] provided that the exponents respect the uniqueness conditions α + β + γ =ᾱ +β +γ = 2 ; 2) as a particular point in the image of the operator [i∂ z ] α−1 (with the same restriction on the exponents as before) 3) or as an operator identity [21] [

Finite-dimensional reductions of the general R-operator
Now we reduce the R-operator (22) to finite-dimensional representations in its first space. The principal possibility of this reduction is based on the following relation for the R-operator (22) which can be proved using the identity (23). Here we use the R-operator R 12 = P 12 R 12 , where P 12 is a permutation operator: P 12 Ψ(z 1 , z 2 ) = Ψ(z 2 , z 1 )P 12 . This relation shows that both, the null-space of the intertwining operator [i∂ 1 ] 2s+1 and the image of the intertwining operator [i∂ 1 ] −2s−1 , are mapped onto themselves by our R-matrix R 12 . Therefore, if we find invariant finite-dimensional subspaces of the latter spaces they will be invariant with respect to the action of R-operator itself.
We consider the function [z 13 ] 2s (with 2s = u 2 − u 1 − 1) and act upon it by the R-operator. We break down the calculation in several steps according to the factorized form (22) of the R-operator. At the end of calculation we choose 2s = n, 2s =n with n,n ∈ Z ≥0 such that [z 13 ] 2s turns into the generating function of the finite-dimensional representation in the first space (11) with an auxiliary parameter z 3 . However, for a while we assume the spin s to be generic.
Using the formula (27) we implement the first step. We act by the first two factors In order to apply the third factor [i∂ 2 ] u 1 −v 1 of the R-operator (22) we resort to the relation which follows immediately from the integral representation (8) for [i∂ z ] α . A merit of the previous formula is that we traded the integral operator [i∂ 2 ] u 1 −v 1 for [i∂ 1 ] 2s , which becomes just a differential operator for 2s = n and 2s =n. Incorporating into the latter formula the inert factors from (30) and the last factor [z 12 ] u 2 −v 1 of the R-operator (22), we find In order to polish the latter formula we denote z 3 = x like in (11) and rewrite (31) in terms of the representation parameters. Also we prefer to replace the R-operator by R 12 = P 12 R 12 .
Thus the general R-operator for the SL(2, C) group acting in the tensor product of two infinitedimensional representation spaces with spins s,s and ℓ,l can be reduced to a finite-dimensional subspace in the first factor if 2s = n, 2s =n (n,n ∈ Z ≥0 ). More precisely the following formula takes place where the normalization factor is .
The latter formula gives a number of solutions of the YBE (21) which are endomorphisms on the tensor product of an (n + 1)(n + 1)-dimensional and an infinite-dimensional spaces. We consider the formula (32) as one of the main results of this paper. It gives a concise expression for the higher spin R-operators. They are "mixed" objects in a sense that they are defined on the tensor product of finite-dimensional and infinite-dimensional representations. In addition they can be considered as generalizations of the L-operators from the fundamental to arbitrary finitedimensional representations. Moreover, the formula (32) produces all such solutions of the YBE related to the principal series representation. Its analogue for the modular double is derived in Sect. 3.3 and the elliptic modular double case is considered in [8].
In order to get accustomed to the reduction formula (32) let us consider a simple example. One can easily recover the holomorphic L-operator (15) substituting (n,n) = (1, 0) in (32) and choosing the basis in the space C 2 of the fundamental representation as e 1 = −z 1 , e 2 = 1. Then Consequently the restriction of R 12 (u − 1 2 | 1 2 , ℓ) to C 2 in the first factor takes the matrix form and coincides with the holomorphic L-operator (15). Analogously taking (n,n) = (0, 1) we recover the anti-holomorphicL-operator (16). Besides the L-operator, the formula (32) reproduces all its higher-spin generalizations. Simultaneously, it produces a number of finite-dimensional (in both spaces) R-operators. Indeed, substituting in (32) the generating function (11) of the finite-dimensional (m + 1)(m + 1)-dimensional representation in the second space, we find a solution of the YBE (21) for the spins n 2 ,n 2 and m 2 ,m 2 in the first and second spaces, respectively, Expanding both sides of this relation in auxiliary parameters x,x, y,ȳ one can rewrite it in a form of a square matrix with (n + 1)(n + 1)(m + 1)(m + 1) rows (or columns). The concise formula (36) produces all its entries. In particular taking the fundamental representation in both spaces n = m = 1,n =m = 0 we reproduce Yang's R-matrix (cf. (49)).

Verma-module reduction: reduction of the general R-operator to the space of polynomials
In this section we slightly digress from the discussion of the group SL(2, C) and outline how sℓ 2symmetric finite-dimensional solutions of the YBE arise from the infinite-dimensional ones. Similar to the previous considerations this approach yields a concise expression for finite-dimensional solutions that may find various applications. Since the corresponding calculations are essentially based on ideas explained above we will limit ourselves to the statement of the results.
Although the sℓ 2 algebra is "a half" of the Lie algebra of the group SL(2, C), it demands a special treatment. We deal with a functional representation of the sℓ 2 -algebra in the space of polynomials of one complex variable C[z]. Fixing a generic complex number s ∈ C and representing the algebra generators by the first order differential operators given in (2) we provide C[z] with a structure of the Verma module. At generic spin s the module is an infinite-dimensional space with the basis {1, z, z 2 , · · · } and there is no invariant subspaces, i.e. the representation is irreducible.
Invariant subspaces arise for the discrete set of spin values 2s = n, n ∈ Z ≥0 . The corresponding (n + 1)-dimensional representation is irreducible and it is realized on the submodule with the basis {1, z, · · · , z n }.
Since the sℓ 2 generators are holomorphic, we have a single holomorphic L-operator given in (15). Now only the holomorphic spectral parameter u is present. The general R-operator R(u|s, ℓ) is defined on the tensor product of two Verma modules with spins s and ℓ. It has to satisfy holomorphic analogues of the RLL-relation (19) and of the YBE (21).
The general R-operator (22) for SL(2, C) group is well defined due to its non-analyticity, in other words, due to the presence of holomorphic and antiholomorphic parts. Thus we cannot get the general R-operator for sℓ 2 (which has to be holomorphic) by crossing out the anti-holomorphic part of (22). Anyway, the holomorphic RLL-relation (19) can be solved [11] resulting in a well- where ratios of the operator-valued gamma functions are defined with the help of the integral representation for Euler's beta-function This R-operator satisfies the YBE (21) as well. As we remarked above in Sect. 2.2 the operator (37) coincides with the one found in [23,31] in the early days of the quantum inverse scattering method in spite of the fact that they look completely different.
At 2s = n, n ∈ Z ≥0 , the general R-operator (37) can be restricted to an (n + 1)-dimensional representation in the first space. Taking into account permutation P 12 of the pair of tensor factors, R 12 = P 12 R 12 , one can show that the restricted R-operator acquires a concise form where the normalization factor is .
Formula (38) is completely analogous to the SL(2, C) reduction formula (32). Expanding both sides of (38) with respect to an auxiliary parameter x one recovers an (n + 1) × (n + 1)-matrix whose entries are differential operators of the order n with polynomial coefficients in spectral parameter u of degree n (or lower). In [6] the Lax operator has been recovered from the general R-operator by means of a quite bulky calculation. The formula (38) provides considerable simplification of that result generalizing it to the higher-spin analogues of the rational Lax operator.
In order to illustrate the power of the formula (38) we present below the R-operator for the spin 1 representation in the first space. In the basis e 1 = 1, e 2 = z 1 , e 3 = z 2 1 of the 3-dimensional space R(u|1, ℓ) takes the matrix form (we change notation Conventional methods demand laborious calculations to reproduce this complicated matrix. In our case the result follows immediately from the formula (38).

Fusion, symbols and the Jordan-Schwinger representation
The standard procedure for constructing finite-dimensional higher-spin R-operators out of the fundamental one is the fusion procedure [23,24]. Firstly, we remind how it works in the case of the symmetry algebra sℓ 2 using a formulation convenient for us. Then in the next section we straightforwardly extend it to the case of the SL(2, C) group and show that the reduction formula (32) is in line with the fusion construction.
For the the rank one symmetry algebras underlying an integrable system the recipe of [23,24] looks as follows. One forms an inhomogeneous monodromy matrix T j 1 ···jn i 1 ···in out of L-operators L j i multiplying them as operators in quantum space and taking tensor products of the auxiliary space C 2 , and then symmetrizes the monodromy matrix over the spinor indices. The parameters of inhomogeneity have to be adjusted in a proper way. The result T (j 1 ···jn) is an R-operator which has a higher-spin auxiliary space and solves the YBE. Thus constructing higher-spin R-operators one has to deal with Sym C 2 ⊗ n which is a space of symmetric tensors with a number of spinor indices Ψ (i 1 ···in) . The usual matrix-like action of operators has the form where the summation over repeated indices is assumed. We prefer not to deal with a multitude of spinor indices. Instead we introduce auxiliary spinors λ, µ and contract them with the tensors Thus the symmetization over spinor indices is taken into account automatically. Henceforth, in place of the tensors we work with the corresponding generating functions which are homogeneous polynomials of degree n of two variables T(λ|µ) is usually called the symbol of the operator. In this way the formula (39) acquires a rather concise form Note that, in fact, we do not need to take µ = 0 in (42). The µ variable disappears automatically since T(λ|µ) and Ψ(µ) have equal homogeneity degrees. In order to illustrate the merits of auxiliary spinors let us apply them to the text-book example of the quantum-mechanical system of spin n 2 , i.e. consider the symmetry group SU(2) and the generators J of the Lie algebra su 2 in the representation of spin n 2 . In the spin 1 2 representation the generators act on the space C 2 and they are given by the Pauli matrices σ 2 , so that Here the lower indices enumerate the rows and the upper indices -the columns. Taking the tensor product of n spin 1 2 representations we obtain the generators on the space C 2 ⊗n , In order to single out in the tensor product an irreducible maximal spin representation we symmetrize over spinor indices yielding the representation of spin n 2 , Further we introduce a pair of auxiliary spinors and find the symbol J(λ, µ) of the operator J (43) converting the formula (43) into where λ|µ = λ k µ k and λ| σ|µ = λ i σ j i µ j are symbols of the identity operator and Pauli matrices, respectively. In view of (42), (45), formula (44) acquires the indexless form Consequently, instead of tensors and finite-dimensional operators we deal with their symbols and generating functions. Note that due to the homogeneity of Ψ (41) λ|µ n is a symbol of the identity operator defined on the tensor product of n spaces Then taking into account that n 2 λ|µ n−1 λ| σ|µ = 1 2 λ| σ|∂ λ λ|µ n , we obtain an alternative expression for J, Thus we have realized the Lie algebra generators J as differential operators on the space of homogeneous polynomials of two variables (forming a projective space) or, more explicitly, This realization of the generators is known as the Jordan-Schwinger representation. We can choose the homogeneous function (λ 1 + xλ 2 ) n (see (41)) as a generating function of the (n + 1)-dimensional representation with an auxiliary parameter x. Instead of working with the projective space one can easily proceed to the space of polynomials of one complex variable. Indeed, due to the homogeneity all information about Ψ(λ 1 , λ 2 ) is encoded in a function of one complex variable -the ratio λ 1 λ 2 .
In order to make contact with the holomorphic set of the sℓ(2, C) generators (2) we choose λ 1 = −z , λ 2 = 1 and rewrite the generators (47) in terms of the variable z Furthermore, the generating function of the Jordan-Schwinger representation turns into the generating function of one variable (x − z) n (cf. (11); recall (38)).
In compact notation the previous formula looks like R(u|λ, µ) = λ|µ n u + 1 2 + 1 2 n σ u − 1 2 + 1 2 n σ · · · u − n + 3 2 + 1 2 n σ = = u(u − 1) · · · (u − n + 1) λ|µ n u + 1 − n 2 + n 2 n σ and it can be easily proven by induction using the identity ( n σ) 2 = 1l . Up to the inessential normalization factor r n (u) = u(u − 1) · · · (u − n + 1) and the shift of the spectral parameter u → u − 1 + n 2 , we obtain the following symbol (see (45)) L(u|λ, µ) = r −1 n (u) R(u − 1 + n 2 |λ, µ) = u λ|µ n + n 2 λ|µ n−1 λ| σ|µ σ = u λ|µ n + J(λ, µ) σ (54) for the higher-spin R-operator which acts on the tensor product of the spin n 2 and spin 1 2 representations. Such an R-operator is usually called the Lax operator with an (n + 1)-dimensional local quantum space. Let us emphasize once more that (54) is a symbol of the Lax operator solely with respect to the local quantum space, but it is a matrix in the 2-dimensional auxiliary space. In order to avoid misunderstandings we showed in (53) its explicit matrix form. The expression λ|µ n is a symbol of the unit operator and J(λ, µ) is a symbol of the Lie algebra generators. Hence the fusion procedure yields the familiar Lax operator, The auxiliary spinors enabled us to reproduce this well-known result in a remarkably simple and explicit way. They saved us from the need to construct projectors which single out irreducible representations and which are inevitable in the standard formulation. Now we are going to describe another way for deriving the L-operator (55) by means of the fusion procedure. The main reason to embark upon one more calculation is that it can be generalized easily to the case of q-deformation (see Sect. 3.4) and, more importantly, to the elliptic deformation [8]. As before we deal with the symbols of finite-dimensional operators. The new ingredient is a factorization of the L-operator (cf. (15)). Calculating the symbol R(u|λ, µ) of the "fused" R-matrices (50) R(u|λ, µ) = λ|R(u)|µ λ|R(u − 1)|µ · · · λ|R(u − n + 1)|µ (56) we choose the parametrization of the auxiliary spinor λ 1 = −z , λ 2 = 1 from the very beginning. Remind a realization of the spin 1 2 generators as differential operators (cf. (48)) which act in the two-dimensional space of linear functions ψ(z) = a 1 z + a 0 . In the basis e 1 = −z , e 2 = 1 of this space the matrices of the generators coincide with the Pauli-matrices J ± (e 1 , e 2 ) = (J ± e 1 , J ± e 2 ) = (e 1 , e 2 ) σ ± ; J 3 (e 1 , e 2 ) = (J 3 e 1 , J 3 e 2 ) = (e 1 , e 2 ) 1 2 σ 3 .
Next we use the fusion procedure and derive the Lax operator (55) together with a representation of the spin n 2 generators (48) acting in the (n + 1)-dimensional space of polynomials ψ(z) = a n z n + . . . + a 0 .
The symbol λ|R(u)|µ of the Yang's R-matrix has been already found above (52), but now we are going to rewrite it in a different form. We represent it as a differential operator in the spinor variables acting on the symbol of identity operator. Indeed, let us rewrite relations (57) in the equivalent form and use these formulae for calculating the symbol of Yang's R-matrix (49) We see that the Lax operator (35) at spin ℓ = 1 2 (with the shifted spectral parameter u → u + 1 2 ) acting on the symbol λ|µ = (µ 2 − µ 1 z) of the identity operator stands on the right-hand side of the previous formula. Then we observe that it can be casted in the factorized form which can be easily checked by a direct calculation. Let us draw attention that the factorization in (59) is slightly different from (15) (at ℓ = 1 2 ). It is easy to realize that (59) is the normal ordered in z and ∂ form of (15). In other words the normal ordering is compatible with the factorization of the Lax operator up to the shift of spectral parameter.
Then we consider the product of two consecutive symbols in (56) and profit a lot from the factorization (59) which provides cancellation of two adjacent matrix factors (which are underlined in the following formula) By now the generalization of the previous result to the product of n − 1 symbols (56) is evident λ|R(u)|µ λ|R(u − 1)|µ · · · λ|R(u − n + 1)|µ = r n (u) · Further we multiply all matrices on the right hand side of the previous formula and obtain where on the last step we use an obvious formula The final result for the symbol (56) of "fused" Yang's R-matrices is where the generators J ± , J 3 for the representation of spin n 2 are given by (48). The factorization of the L-operator plays an important role in the construction of the general R-operator for deformed [9,12] and non-deformed [9] rank 1 symmetry algebra, as well as in a higher rank case [11]. Here see that this factorization finds its natural place in the fusion construction as well (in the elliptic case [8] we will have another opportunity to strongly support this statement).

Fusion construction for SL(2, C)
The fusion procedure enables one to produce even more intricate sℓ 2 -symmetric solutions of the YBE. Starting with the Lax operator which acts in the tensor product of spin 1 2 and spin ℓ representations (we assume ℓ to be generic such that the corresponding representation is infinite-dimensional) one obtains the R-operator which acts in the tensor product of spin n 2 and spin ℓ representations.
Since we are mainly interested in the R-operators which are invariant with respect to the SL(2, C) group, we will thoroughly study how the fusion procedure applies in this case. As before we profit a lot from the auxiliary spinors notation. However, from now on the holomorphic and antiholomorphic sectors are present and we need to introduce a pair of auxiliary spinors λ i ,λī. They are independent variables not related by the complex conjugation. Thus we introduce a pair of scalar objects (without spinor indices) which are linear combinations of the L-operators' entries (15), (16). An easy calculation shows that 1 The factorized expression (61) looks much like formula (15). Indeed in both expressions the differential operators are sandwiched between some multiplication by a function operators. The analogous relation takes place forL (16). Then we multiply a number of Λ-operators with shifted spectral parameters to form a Λ-string Here we apply the formula which can be easily proven by induction, Then we take into account the anti-holomorphic sector and form the product of Λ-andΛ-strings resulting in the symbol for a higher-spin R-operator R fus (u,ū|λ,λ, µ,μ) = Λ(u)Λ(u − 1) · · · Λ(u − n + 1)Λ(ū)Λ(ū − 1) · · ·Λ(ū −n + 1) = This R-operator acts in the tensor product of the infinite-dimensional representation specified by spins ℓ,l and the finite-dimensional representation with spins n 2 ,n 2 . Let us remind that R fus is a symbol with respect to the finite-dimensional space only, but it is a differential operator in the infinite-dimensional space. We note that the right-hand side of (62) is evidently polynomial in λ and µ, as it should be. In order to reconstruct the operator itself from its symbol we resort to the rule (42). More precisely, we apply the corresponding relation to a function Φ(λ,λ|z,z), which is homogeneous in λ andλ of the homogeneity degree n andn, respectively, [ R fus (u,ū) Φ ] (λ,λ|z,z) = R fus (u,ū|λ,λ, ∂ µ , ∂μ) Φ(µ,μ|z,z) µ=μ=0 .
Then we act by R fus on the generating function according to (63) and choose the symbol in the form (64). At the same time we do not act by the R fus -operator on any function in its second space. At this point we take into account that and obtain Here we profited from the operator form of the star-triangle relation (28) at the last step. In order to compare the reduction of the general R-operator (32) with the expression (66) following from the fusion formula (62) we just need to pass from the Jordan-Schwinger representation to the standard representation of SL(2, C) (in the space of functions of one complex variable) described in Sect. 2.1.
Consequently we choose λ 1 = −z 1 ,λ 1 = −z 1 , λ 2 =λ 2 = 1 and denote z = z 2 ,z =z 2 . Finally, we see that both formulae are identical up to a numerical normalization. We conclude that both ways to construct the higher-spin finite-dimensional (in one of the spaces) R-operators give identical results, and the general R-operator (22) contains all solutions of the Yang-Baxter equation associated with the principal series representations of the SL(2, C) group.

The Faddeev modular double
Using the patterns of the previous sections, in the following we show that all described constructions for the group SL(2, C) can be straightforwardly adapted to the modular double. Additionally, we construct corresponding finite-dimensional solutions of the YBE using the fusion.

Representations of the quantum algebra
The modular double of U q (sℓ 2 ) was introduced by Faddeev in [16]. This algebra is formed by two sets of generators E , K , F and E , F , K. The usual commutation relations for E , K , F which generate U q (sℓ 2 ) with q = e iπτ (τ ∈ C and it is not a rational number) are supplemented by similar relations for E, F, K with the deformation parameter q = e iπ/τ . The generators E and F commute with E and F. The generator K anti-commutes with E and F, K anti-commutes with E and F while K commutes with K.
For particular representations of the modular double see [2,3,16,17,27] and references therein. We use the parametrization τ = ω ′ ω , where ω and ω ′ are complex numbers with the positive imaginary parts, Im ω > 0, Im ω ′ > 0, satisfying the normalization condition ωω ′ = − 1 4 . Then and the change q ⇄ q is equivalent to ω ⇄ ω ′ . We denote also In the following we deal with a representation π s of the modular double when the generators K s = π s (K) , E s = π s (E) , F s = π s (F) are realized as finite-difference operators acting on the space of entire functions rapidly decaying at infinity along contours parallel to the real line. This representation is parameterized by one complex parameter s called the spin, and the generators have the following explicit form [3][4][5] wherep denotes a momentum operator in the coordinate representationp = 1 2πi ∂ x . The formulae for generators K s , E s , F s are obtained by a simple interchange ω ⇄ ω ′ in (69).
The modular double is associated with two basic special functions. The first one is the noncompact quantum dilogarithm which has the following integral representation where the contour goes above the singularity at t = 0. In the context of quantum integrable systems it has been found first in [15]. Some basic formulae for γ(z) can be found in [17,33]. This function respects a pair of finite-difference equations of the first order and the reflection relation One can interpret 2ω and 2ω ′ as some quasi-periods of the quantum dilogarithm. The second function we need is In fact it coincides with the Faddeev-Volkov R-matrix [34]. Some relations for this functions are presented in [4]. It naturally arises when one looks for the intertwining operator of equivalent representations of the modular double [27], and it serves as the main building block in the construction of a general R-matrix as an integral operator [4,5]. This general R-operator is a product of four Faddeev-Volkov's R-matrices. The function D a (z) obeys simple reflection relations and a pair of finite-difference equations of the first order Note that the functions γ(z) and D a (z) are symmetric with respect to ω and ω ′ . A generalization of the Faddeev-Volkov model was found in [30]. It is associated with the more general R-operator than we consider here, which can be obtained as a limit from the most complicated known R-operator derived in [12]. We do not consider in the present paper the model [30] postponing it to a later work. Now we proceed to finite-dimensional representations of the modular double. In order to fix the spin s 0 such that a finite-dimensional representation decouples from the infinite-dimensional representation π s 0 we resort to the intertwining operator of equivalent representations of the modular double. It is known that the representations π s and π −s are equivalent. The corresponding intertwining operator [27] is expressed in terms of the special function (72), such that wherep is the momentum operator. There are analogous relations for E, F, K, since the D-function is invariant with respect to permutation of ω and ω ′ . The latter relations can be easily checked using equations (74). Applying the Fourier transformation of the D-function (72) [4,17,33] A(a) we immediately represent the intertwiner D −s (p) as an integral operator (in analogy with (5)) Thus the intertwiner admits two forms at generic values of s: as a formal function of the momentum operator and a well-defined integral operator. A finite (n+1)(m+1)-dimensional representation decouples from the infinite-dimensional one for special values of the spin s 0 = −ω ′′ − nω − mω ′ , where the integers n, m ∈ Z ≥0 enumerate the points of a quarter-infinite lattice on a complex plane (or a line, for real ω/ω ′ ). Such two-index finitedimensional representations emerged first in the theory of elliptic hypergeometric functions [29]. They naturally lead to the two-index finite-dimensional representations of the elliptic modular double [12,13]. The cases we consider here in principle can be derived as certain limits from the corresponding constructions, but we give here an independent consideration and, moreover, describe the finite-dimensional R-matrices analogous to (38). Note that in the case of the group SL(2, C) the finite-dimensional representations were also parametrized by a pair of non-negative integer numbers n andn, but the integern has a different nature emerging from a discretization of the separate spin variables, which is absent in our case.
In order to find finite-dimensional representations of interest we investigate the null-space of the intertwiner. We take the formal operator identity which is a consequence of the reflection formula (73), and rewrite it in an equivalent form substituting D s (p) and 1l for their kernels (see (77)) Then we note that zeros of the quantum dilogarithm γ(z) = 0 are located at z = ω ′′ + 2nω + 2mω ′ , n, m ∈ Z ≥0 , which indicates that the relation (78) is broken down at the corresponding points.
Consequently at the values of spin specified above, s = s 0 , the right-hand side of (79) vanishes and a nontrivial null-space of D −s 0 (p) arises The latter formula is a deformed analogue of (14). From the intertwining relations (75) the nullspace is seen to be invariant under the action of the modular double generators. Corresponding representation is finite-dimensional as we will seen shortly. The generators are fixed by expressions (69) and their modular duals with the spin parameter s = s 0 . One can also show that the corresponding representation is irreducible.
In this way we have found that D nω+mω ′ (x − y) is a generating function of the finite-dimensional representation which contains in a concise form all its basis vectors. Here y is an auxiliary parameter, which is convenient to write in the exponential form Since we assume that the quasi-periods are incommensurate (i.e., that τ = ω ′ ω is not a rational number), the auxiliary variables Y and Y are algebraically independent for generic y (i.e., if Y k Y l = 1 for some integer k and l, then k = l = 0). Using the finite-difference equations (74) our generating function can be rewritten as a finite product where we use the shorthand notation Expanding the generating function (81) in integer powers of Y (y) and Y (y) we extract (n+1)(m+1) basis elements of the finite-dimensional representation, which are monomials X n−2k X m−2l with k = 0, 1, · · · , n , l = 0, 1, · · · , m .
Let us note that for s = s 0 the integral in (77) diverges. The divergence is compensated by the normalization factor, which turns to zero, A(s 0 − ω ′′ ) = 0 . The ambiguity can be resolved and for finite-dimensional representations the intertwiner D −s 0 (p) becomes a sum of finite-difference operators which follows from (74). One can directly check as well that the basis vectors (83) are annihilated by D −s 0 (p).

An infinite-dimensional R-operator for the modular double
Now we proceed to integrable structures for the modular double. The L-operator is constructed out of the modular double generators taken in the representation π s (69) [4], This L-operator respects the standard RLL-intertwining relation (cf. (18)) which is equivalent to the commutation relations (67). The second L-operator is obtained from L(u) by the interchange ω ⇄ ω ′ : L(u) = L(u)| ω⇄ω ′ . The same is true for the corresponding R-matrix. In the following we indicate formulae only for the L-operator (84), and all relations for the L-operator have the same form with ω ⇄ ω ′ . The L-operator (84) can be represented in the factorized form where we introduced the "light-cone" parameters u 1 and u 2 instead of u and s In the notation L(u) we omit for simplicity the dependence on the spin parameter s.
The factorization formula (86) is completely analogous to formula (15) for the SL(2, C) group.
The same can be said about spectral parameters u 1 , u 2 in (17) and (87). However, although the operators L(u), L(u) for the modular double look analogous to L(u),L(ū) for SL(2, C), in fact, they are different in their nature. At the level of R-operators an analogy with the rational case persists as well. The general Roperator acts in the tensor product of two infinite-dimensional representations π s 1 ⊗ π s 2 (69). It has been found first in [4], but the corresponding form of the R-operator is not suitable for our purposes. Here we profit from another construction implemented in [5], where it has been obtained solving a pair of RLL-relations (cf. (19), (20)), The spin parameters s 1 , s 2 and the spectral parameters u, v appearing in the RLL-relations (88), (89) are combined to four "light-cone" parameters u 1 , u 2 , v 1 , v 2 in accordance with (87), i.e.
The notation R 12 (u − v) is a shortened version of R 12 (u − v|s 1 , s 2 ) or R 12 (u 1 , u 2 |v 1 , v 2 ) which takes into account the spin parameters. The R-operator is invariant with respect to the modular double, i.e. it commutes with the co-product of the generators. More precisely, where we abbreviate the co-product taken in the tensor of representations with spins s 1 and s 2 , (π s 1 ⊗ π s 2 ) • ∆ , to ∆ bearing in mind the specified representations. The co-product is given by the formulae Analogous relations take place for E, F, K. The invariance (90) follows straightforwardly from the RLL-relations (88), (89) subject to the shift of spectral parameters u → u + w, v → v + w with arbitrary w.
The first construction of the general R-operator for the modular double from [5] Here we denote x ij = x i − x j . The latter formula has to be compared with (22) which has the same structure, only the building blocks are different. The change ω ⇄ ω ′ does not alter the Roperator (91) which satisfies both RLL-relations (88) and (89). Similar to the expression (22), the representation (91) for our infinite-dimensional R-operator plays a major role in what follows. In the next section we find its reductions to finite-dimensional invariant subspaces. According to (91) the general R-operator is a product of four Faddeev-Volkov's R-matrices [34]. Applying (77) one can rewrite it explicitly as an integral operator. Let us note that it is not the only possible form of the R-operator. Initially constructed in [4] the R-operator for the modular double was obtained there in the form which is not convenient enough to address the current problem. In [4] the R-operator appeared in disguise of D-function (72) and arcosh of the Casimir operator. Thus, dealing with such a form of the R-operator, one has to decompose tensor products to a sum of irreducible representations and to make use of the Clebsch-Gordan coefficients [27]. The form (91) of the R-operator has the virtue of not demanding any auxiliary information.
In order to justify the terms elementary intertwining operators chosen for the R-operator factors in (85), we indicate here the relations (cf. (24), (25)) which have a clear meaning in terms of the permutation group S 4 and which enable us to check the RLL-relations (88) and (89).
The elementary intertwining operators possess a number of peculiar properties. They satisfy the Coxeter relations (cf. (23)) Using these relations one can check [5] that the R-operator (91) satisfies the YBE Both sides in the latter relation are endomorphisms on the space π s 1 ⊗ π s 2 ⊗ π s 3 . For brevity we do not indicate dependence on the spin parameters. The Coxeter relations (94) are equivalent to the star-triangle relation [34] which has three disguises: 1) an integral identity [2,4,33] 2) a particular point in the image of the operator D −a−ω ′′ (p 1 ) (with the same restriction on the parameters as before) 3) an operator identity

Finite-dimensional reduction of the R-operator
Now we have all ingredients at hand to perform a reduction of the described R-operator for modular double to a finite-dimensional representation in one of its tensor factors. The calculation follows precisely the same pattern as in the SL(2, C) case (see Sect. 2.3). Again the principal possibility of this reduction is based on the following relation for the R-operator (91) which can be proved using the identity (94). Here, again, R 12 = P 12 R 12 , where P 12 is a permutation operator. This relation shows that both, the null-space of the intertwining operator D u 2 −u 1 (p 1 ) and the image of the intertwining operator D u 1 −u 2 (p 1 ), are mapped onto themselves by our R-matrix R 12 . Therefore the invariant finite-dimensional subspaces of the null-space are invariant with respect to the action of the R-operator as well.
We consider the R-operator R 12 (u|s 0 , s) acting on the tensor product π s 0 ⊗ π s and introduce the "light-cone" parameters (see (87)) We apply the R-operator to the function D −ω ′′ +u 2 −u 1 (x 13 ) in the first space. For s 0 = −ω ′′ − nω − mω ′ , n, m = 0, 1, 2, · · · , the latter function becomes a generating function of the finite-dimensional representation in the first space. However, for a moment s 0 is assumed to be generic. According to the structure of R-operator (91) we consider sequential action of its separate factors. On the first step, we apply D u 2 −v 2 (p 1 ) D u 1 −v 2 (x 12 ) to D −ω ′′ +u 2 −u 1 (x 13 ) Φ(x 2 ) and, using formula (96), obtain Further we apply the third factor D u 1 −v 1 (p 2 ) of the R-operator to both sides of this relation. On the right-hand sides we use the relation which can be easily checked taking into account the integral form of the intertwiner (77). In a full analogy with the SL(2, C) calculation we traded a complicated integral operator D u 1 −v 1 (p 2 ) for D u 2 −u 1 −ω ′′ (p 1 ) which turns to D nω+mω ′ (p 1 ) in the finite-dimensional setting. The latter operator is just a sum of the finite-difference operators which follows from equations (74). The substitution x−y →p 1 in (81) yields an explicit expression for D nω+mω ′ (p 1 ). The fourth factor of the R-operator is inert being the multiplication by a function operator. Thus the integral R-operator for the modular double (91) acting on the tensor product of two infinite-dimensional representations π s 0 ⊗ π s can be reduced to a finite-dimensional representation in the first space for s 0 = −ω ′′ − nω − mω ′ , n, m ∈ Z ≥0 . It acts on the generating function of finite-dimensional representation (81) according to the following explicit formula where the normalization factor is and x 3 is an auxiliary parameter. Both sides of the equality (101) can be expanded in integer powers of the variables X 3 (x 3 ) , X 3 (x 3 ) (see (80)). This yields simultaneously an expansion in integer powers of the variables X 1 (x 1 ) , X 1 (x 1 ) (see (82)), which form a basis of the finite-dimensional representation of interest. The resulting formula (101) is very helpful in applications. We use it as follows. Firstly, finitedifference operators in the sum D nω+mω ′ (p 2 ) act from the left on the D-functions and shift their arguments. After these shifts we trade all D-functions (72) in (101) for quantum dilogarithms (70) and apply the finite-difference equations (71). In this way we completely get rid off the quantum dilogarithms. The final result contains only trigonometric functions, i.e. a linear combination of the products of X 1 (x 1 ) , X 1 (x 1 ) , X 3 (x 3 ) , X 3 (x 3 ) , and Φ(x 2 ) with the shifted argument. Thus the restriction of the general R-operator can be represented as an (n + 1)(m + 1)-dimensional matrix whose entries are finite-difference operators with the trigonometric coefficients.
The formula (101) constitutes one of the main results of this paper. It gives a rich class of solutions of the YBE which are endomorphisms on a tensor product of finite-dimensional and infinite-dimensional representations belonging to the modular double series specified in (69). As mentioned earlier, we do not discuss in this work similar reductions of the R-operator for the model found in [30].
In order to demonstrate how formula (101) works in practice we recover the L-operator (84) out of the R-operator (91). With this task in mind, we choose the spin s 0 = −ω ′ − ω ′′ , i.e. fix n = 0, m = 1. The generating function (81) of the 2-dimensional representation in the first space is Consequently e 1 = e iπ 2ω x 1 , e 2 = e − iπ 2ω x 1 form a basis of C 2 . The finite-difference operator in (101) is Up to a normalization factor the right-hand side of (101) takes the form cosh iπ 2ω Expanding this function in terms of X ±1 Thus we have reproduced the desired result (84). In a similar way one reproduces the L-operator at s 0 = −ω − ω ′′ . Implementing these reduced R-operators in the YBE (95) one recovers the RLL-relations (89). A reduction of the R-operator to finite-dimensional representations in both spaces can be constructed as well. One just should choose an appropriate discrete value of the spin s in (101) and substitute Φ(x 2 ) for the corresponding generating function. In this way one generates a number of finite-dimensional solutions of the YBE including the trigonometric R-matrix (85) among them. More precisely, let us fix the spin parameters as s 1 = −ω ′′ − n 1 ω − m 1 ω ′ , n 1 , m 1 ∈ Z ≥0 , and s 2 = −ω ′′ − n 2 ω − m 2 ω ′ , n 2 , m 2 ∈ Z ≥0 , in the first and second spaces, respectively. Then is a concise expression for the finite-dimensional (in both spaces) R-matrix. After expansion with respect to auxiliary parameters X 3 (x 3 ) , X 3 (x 3 ) , X 4 (x 4 ) , and X 4 (x 4 ) it can be rewritten explicitly is the form of an (n 1 + 1)(m 1 + 1)(n 2 + 1)(m 2 + 1)-dimensional matrix.
The general R-operator exists as well for U q (sℓ 2 )-algebra [9], which is a "one-half" of the modular double. The reduction of this R-operator leads to the trigonometric L-operator as was shown in [7]. Derivation of the corresponding higher-spin finite-dimensional solutions of YBE using the described reduction procedure will be presented elsewhere.

The fusion and symbols for U q (sℓ 2 ) algebra
Now we would like to show that the reduction result of the previous section can be derived with the help of the fusion procedure. In the present section we develop the fusion for the quantum algebra U q (sℓ 2 ) and in the next one we consider the modular double. Our approach is not that well known since we extensively use the symbols of operators.
Similar to the discussion in Sect. 2.5 we construct the Lax operator with a finite-dimensional local quantum space out of the q-deformed Yang's R-matrix 2 . The latter acts on the tensor product of two fundamental representations and is given by the matrix In a full analogy with the non-deformed case, the recipe of [23,24] suggests to form an inhomogeneous monodromy matrix out of the q-deformed Yang's R-matrices and to symmetrize it, i.e. R (j 1 ···jn) where Sym implies symmetrization with respect to (i 1 · · · i n ) and (j 1 · · · j n ) and the indices refer to the first space of the R-matrix (102) The standard way of treating (103) implies construction of the symmetrizer, i.e. a projector to the highest spin representation in the decomposition of the product of n fundamental representations. We implement the projection by means of the auxiliary spinors λ , µ that is equivalent to dealing with the symbols of R-matrices. The symbol of (51) (with respect to the local quantum space, not the auxiliary one) factorizes (see (51)) R(u|λ, µ) = λ i 1 · · · λ in R j 1 ···jn i 1 ···in (u) µ j 1 · · · µ jn = λ|R(u)|µ λ|R(u − 1)|µ · · · λ|R(u − n + 1)|µ (104) to a product of the symbols for q-Yang's R-matrices λ|R(u)|µ = λ i R j i (u) µ j , λ|R(u)|µ = [u + 1] q λ 1 µ 1 + [u] q λ 2 µ 2 λ 2 µ 1 λ 1 µ 2 [u] q λ 1 µ 1 + [u + 1] q λ 2 µ 2 .
Summation formulae in (107), (108) facilitate reconstruction of operators from the symbolic entries of the matrix (106). In analogy with the non-deformed case we again remove the inessential normalization factor r n (u) = [u] q [u − 1] q · · · [u − n + 1] q and shift the spectral parameter u → u − 1 + n

Fusion construction for the modular double
The fusion procedure for the modular double closely follows the construction from Sect. 2.6. One forms inhomogeneous monodromy matrix out of the L-operators and then symmetrizes it over spinor indices resulting in a finite-dimensional (in one of the spaces) higher-spin R-operator. Again, instead of working with the higher-rank tensors we introduce auxiliary spinors λ i , λ i , µ j , µ j and contract them with the monodromy matrix according to (40). The homogeneity (41) implies that there are redundant variables. We get rid off them by choosing the gauge λ 1 λ 2 = −1 , µ 1 µ 2 = −1 that is equivalent to the parametrization of the spinors by means of independent variables a, b as follows Analogous relations hold for spinors λ, µ obtained from (115) after the interchange ω ⇄ ω ′ with the same a, b. Since we assume that the ratio of quasi-periods τ is not rational, λ and λ are algebraically independent for generic a and the same is true for µ and µ for generic b. Further, we form symbols of the L-operators (84) (i.e., some scalar operators) contracting them in the matrix space with the auxiliary spinors 3 λ i L j i (u) µ j = Λ(u, λ, µ) , λ i L j i (u) µ j = Λ(u, λ, µ) .
In view of the reflection formula (73) and relation (117) this product can be recast to the form Finally, we reconstruct the operator of interest from its symbol using formula (42), which results in the representation [ R fus (u) Φ ] (λ, λ|x) = R fus (u|λ, λ, ∂ µ , ∂ µ ) Φ(µ, µ|x) where the symbol R fus is fixed in (118). Let us stress once more that the fusion formula (118) is completely analogous to the SL(2, C) group case formula (63). The higher-spin R-operator acts on a function Φ(λ, λ|x) having the homogeneity degrees m in λ and n in λ, respectively. In (118) one has differentiations over spinors µ , µ , but the operator R fus (118) formally depends on b and not on the exponential of b . In order to see that there is no contradiction, we note that according to the definitions (116) Λ and Λ are linear in spinors. Consequently R fus (118), being a product of them, has to be polynomial in spinors. This can be checked directly as well. Recalling the definition of µ , µ (115) and we conclude that R fus in (118) depends polynomially on µ and µ. Thus the fusion formulae (118) and (118) match to each other. The right-hand side of (118) explicitly depends on a , so its polynomiality in spinors λ , λ is not obvious at all. It is necessary to demonstrate it explicitly. Furthermore, we need to compare (118) with the reduction formula (101) since both give rise to a higher-spin R-operator. We will accomplish both tasks if we show that the R-operators do coincide. Thus we take the generating function D nω+mω ′ (a − y) of a finite-dimensional representation and act upon it by the "fused" R-operator in the first space according to the prescription (118). The generating function with auxiliary parameter y explicitly depends on λ , λ (see (81) and has the homogeneity degrees m in λ and n in λ, respectively. Now, using the relations (see (119), (120)) where at the last step we profited from the operator star-triangle relation (97). Identifying the variables a = x 1 , x = x 2 , y = x 3 , we find a nice agreement of the fusion formula (122) with the reduction formula (101). Thus both approaches are equivalent and yield identical results.