From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
Dmitry Chicherin, Sergey E. Derkachov, Vyacheslav P. Spiridonov
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2016), 028, 34 pages From Principal Series to Finite-DimensionalSolutions of the Yang–Baxter Equation
Dmitry CHICHERIN † , Sergey E. DERKACHOV ‡ and Vyacheslav P. SPIRIDONOV §† LAPTH, UMR 5108 du CNRS, associ´ee `a l’Universit´e de Savoie, Universit´e de Savoie, CNRS,B.P. 110, F-74941 Annecy-le-Vieux, France
E-mail: [email protected] ‡ St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,Fontanka 27, 191023 St. Petersburg, Russia
E-mail: [email protected] § Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, 141980, Russia
E-mail: [email protected]
Received November 17, 2015, in final form March 04, 2016; Published online March 11, 2016http://dx.doi.org/10.3842/SIGMA.2016.028
Abstract.
We start from known solutions of the Yang–Baxter equation with a spectral pa-rameter defined on the tensor product of two infinite-dimensional principal series representa-tions of the group SL(2 , C ) or Faddeev’s modular double. Then we describe its restriction toan irreducible finite-dimensional representation in one or both spaces. In this way we obtainvery simple explicit formulas embracing rational and trigonometric finite-dimensional solu-tions of the Yang–Baxter equation. Finally, we construct these finite-dimensional solutionsby means of the fusion procedure and find a nice agreement between two approaches. Key words:
Yang–Baxter equation; principal series; modular double; fusion
Contents
SL(2 , C ) group 4 , C )-invariant R-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Finite-dimensional reductions of the general R-operator . . . . . . . . . . . . . . . . . . . 102.4 Verma module reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Fusion, symbols and the Jordan–Schwinger representation . . . . . . . . . . . . . . . . . . 132.6 Fusion construction for SL(2 , C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 U q ( sl ) algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Fusion construction for the modular double . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References 33 a r X i v : . [ m a t h - ph ] M a r D. Chicherin, S.E. Derkachov and V.P. Spiridonov
The Yang–Baxter equation (YBE) R ( u − v ) R ( u ) R ( v ) = R ( v ) R ( u ) R ( u − v )is a major tool in building the quantum integrable systems [1, 26, 28, 29, 39]. It has foundnumerous applications in mathematical physics and purely mathematical questions. At thedawn of quantum inverse scattering method the finite-dimensional solutions of the YBE (whenthe operators R ij ( u ) are given by ordinary matrices with numerical entries depending on thespectral parameter u ) attracted much attention in view of their relevance for physical spinsystems on lattices admitting a successful treatment of their thermodynamical behavior [1, 26].Solutions of the YBE for infinite-dimensional representations revealed their importance inthe integrability phenomena emerging in quantum field theories. An integrable spin chain withunderlying SL(2 , C ) symmetry group and its noncompact representations naturally arises in thehigh-energy behavior of quantum chromodynamics. Corresponding model was discovered in [30]together with an additional integral of motion. Later, in [31] and [21] it was identified with thenoncompact XXX spin chain which revealed its complete integrability (for further investigationsof this model, see [12, 14]).There are three increasing levels of complexity of finite-dimensional solutions of YBE de-scribed by matrices with the coefficients expressed in terms of the rational, trigonometric, andelliptic functions. In the infinite-dimensional setting the latter hierarchy is replaced by solutionsof YBE defined as integral operators with the integrands described by plain hypergeometric, q -hypergeometric and elliptic hypergeometric functions [36].The notion of the modular double was introduced by Faddeev in [18] and noncompact repre-sentations of this algebra arise naturally in the Liouville model studies [20, 35]. The quantumdilogarithm function [19] plays an important role in the description of these representations aswell as in the Faddeev–Volkov model [2, 43] and its generalization found in [38]. The ellipticmodular double extending Faddeev’s double was introduced in [37].The general solution of YBE at the elliptic level with the rank 1 symmetry algebra was foundin [16]. It is based on the properties of an integral operator with an elliptic hypergeometrickernel, the key identity for which (given by the Bailey lemma, see, e.g., [36]) coincides withthe star-triangle relation. In [16, 17] a particular finite-dimensional invariant space for therepresentations of the elliptic modular double has been described.The general R-operator is interesting on its own. In the case of group SL(2 , C ) and theFaddeev and elliptic modular doubles it is represented by an explicit integral operator acting onthe tensor product of two functional spaces [6, 14, 15, 16]. It can be thought of as a universalobject since it is expected that in some sense it conceals all solutions of YBE, particularly, thefinite-dimensional solutions. In this paper we show explicitly that, indeed, the latter solutionscan be derived as reductions of the infinite-dimensional R-operators in three particular cases:the SL(2 , C ) group R-operator [15], its real form analogue associated with the sl -algebra andthe R-operator for the Faddeev modular double, which was considered first in [5] as a formalfunction with an operator argument.Reductions to finite-dimensional invariant subspaces constitute a nontrivial problem. Indeed,general infinite-dimensional R-matrices are given by integral operators, but their reduction toa finite-dimensional invariant subspace in one of the tensor product spaces should be a matrixwith the entries described by differential or finite-difference operators.Our key results are given by the remarkably compact formulas for reduced R-operators (2.32),(2.38), and (3.36). The former and the latter cases are determined by a pair of integer parame-ters. In the SL(2 , C )-case (2.32) two integers emerge from the discretization of two spin variables, s and ¯ s . In the modular double case (3.36) the situation is qualitatively different, two integersrom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 3emerge from the intrinsically two-dimensional nature of the discrete lattice for one spin variable.In the context of univariate spectral problems such a quantization leads to the two-index or-thogonality relations which were found for the first time in the theory of elliptic hypergeometricfunctions, see [36] and references therein. In our problem, two integers appearing in the reduc-tion of R-matrix associated with the modular double are descendants from the similar integersexisting at the elliptic level [16, 17].It is well known that quantum integrable systems are related to 6 j -symbols of differentalgebras. In the context of 2 d conformal field theory these symbols are associated with thefusion matrices and, in this setting, the finite-dimensional 6 j -symbols of the modular doublewith q a root of unity have been constructed in [22]. Their continuous spin generalizations havebeen built in [35]. The most general discrete q -6 j -symbols of such type (with the doubling ofindices) are composed out of the product of two particular terminating ϕ basic hypergeometricseries related by a modular transformation [36]. Their noncompact analogues associated withthe lattice model of [38] and generalizing 6 j -symbols of [35] are easily derived as a limitingcase of the elliptic analogue of the Euler–Gauss hypergeometric function [36]. A similar set ofquestions was discussed recently for the quantum algebra U q ( osp (1 | , C ) group and the modular double (thecorresponding intertwining operators were constructed in [23] and [35]). We show explicitlythat both methods yield identical formulae embracing required finite-dimensional (in one orboth spaces) solutions of YBE. Additionally, we consider a finite-dimensional reduction of theR-operators for a tensor product of two Verma modules. This is the first paper in the seriesdedicated to finite-dimensional reductions of known integral R-operators. In the next work ofthis series [10] such a problem was solved for the elliptic modular double. In [7] new compact fac-torization formulae were derived for finite-dimensional R-matrices in several cases (for differentforms of factorizations, see [27] and references therein). Reduction of the integral R-operatorfor the generalized Faddeev–Volkov model of [38] is considered in [11].The paper consists of two parts. In the first part we consider SL(2 , C )-invariant solutions ofYBE. We begin in Section 2.1 with a concise review of the infinite-dimensional principal seriesrepresentation of the SL(2 , C ) group. In Section 2.2 we indicate the relevant Lax operatorsand the general R-operator emphasizing the role of the star-triangle relation. In Section 2.3we reduce the general SL(2 , C )-symmetric R-operator to a finite-dimensional representation inone of the spaces. In Section 2.4 we derive an analogous reduction for the general sl -algebraR-operator to the space of polynomials or the Verma module.Then we proceed to the fusion. In Section 2.5 we formulate the fusion for the sl algebra casein a rather nonstandard fashion. We construct projectors to the highest spin representation by D. Chicherin, S.E. Derkachov and V.P. Spiridonovmeans 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-operatoras well. After that, in Section 2.6 we get back to the SL(2 , C ) group and carry out the fusionin this case.In the second part of the paper we consider similar questions for the modular double. Therethe presentation closely follows the rational case in order to emphasize the striking similaritybetween these two cases. In Sections 3.1 and 3.2 we outline general structure of the modulardouble and present the general R-operator for it. The corresponding reduced R-matrix, whichis finite-dimensional in one of the quantum spaces (or both), is derived in Section 3.3. Finally,in Sections 3.4 and 3.5 we derive finite-dimensional R-matrices in the q -deformed cases usingthe fusion procedure. , C ) group We start with a short review of some basic well-known facts about representations of the groupSL(2 , C ). They are formulated in a form that will be natural for dealing with R-operators. Weoutline how finite-dimensional representations decouple from infinite-dimensional ones empha-sizing the role of the intertwining operator.The method of induced representations is a robust tool that enables one to construct a numberof interesting representations of a group (see for example [24]). Consider representations of thegroup SL(2 , C ) realized on the space of single-valued functions Φ( z, ¯ z ) on the complex plane. Theprincipal series representation [23] is parametrized by a pair of generic complex numbers ( s, ¯ s )subject to the constraint 2( s − ¯ s ) ∈ Z . We refer to them as spins in what follows. In order toavoid misunderstanding we emphasize that s and ¯ s are not complex conjugates in general. So,this representation T ( s, ¯ s ) is given explicitly as [23] (cid:2) T ( s, ¯ s ) ( g )Φ (cid:3) ( z, ¯ z ) = ( d − bz ) s (cid:0) ¯ d − ¯ b ¯ z (cid:1) s Φ (cid:18) − c + azd − bz , − ¯ c + ¯ a ¯ z ¯ d − ¯ b ¯ z (cid:19) , (2.1) g = (cid:18) a bc d (cid:19) ∈ SL(2 , C ) . Representations of the group SL(2 , C ) yield representations of the Lie algebra sl (2 , C ) in a stan-dard way. Assuming that g lies in a vicinity of the identity g = 1 + (cid:15) · E ik , where E ik are traceless2 × E ik ) jl = δ ij δ kl − δ ik δ jl , one extracts generators E ik and ¯E ik of the Lie algebra,T ( s, ¯ s ) (1 + (cid:15) · E ik )Φ( z, ¯ z ) = Φ( z, ¯ z ) + (cid:0) (cid:15) · E ik + ¯ (cid:15) · ¯E ik (cid:1) Φ( z, ¯ z ) + O (cid:0) (cid:15) (cid:1) . The generators E ik , ¯E ik are the first-order differential operators. We arrange them in 2 × ( s ) and ¯E (¯ s ) , which will be useful for the following considerations,E ( s ) = (cid:18) E E E E (cid:19) = (cid:18) z∂ − s − ∂z ∂ − sz − z∂ + s (cid:19) = (cid:18) z (cid:19) (cid:18) − s − − ∂ s (cid:19) (cid:18) − z (cid:19) . (2.2)The substitution z → ¯ z , ∂ → ¯ ∂ and s → ¯ s in this formula results in the matrix ¯E (¯ s ) for thegenerators ¯E ik .There exists an integral operator W which intertwines a pair of principal series representationsT ( s, ¯ s ) and T ( − − s, − − ¯ s ) for generic complex s and ¯ s ,W( s, ¯ s )T ( s, ¯ s ) ( g ) = T ( − − s, − − ¯ s ) ( g )W( s, ¯ s ) . (2.3)rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 5We will refer to this pair as the equivalent representations. The described intertwining relationcan be equally reformulated as a set of intertwining relations for the Lie algebra generatorsW( s, ¯ s )E ( s ) = E ( − − s ) W( s, ¯ s ) , W( s, ¯ s )¯E (¯ s ) = ¯E ( − − ¯ s ) W( s, ¯ s ) . (2.4)The operator W is defined up to an overall normalization and has the following explicit form [23][W( s, ¯ s )Φ] ( z, ¯ z ) = const (cid:90) C d x Φ( x, ¯ x )( z − x ) s +2 (¯ z − ¯ x ) s +2 . (2.5)Obviously this integral operator is well-defined for generic values of s and ¯ s and the problemsemerge for the discrete set of points 2 s = n , 2¯ s = ¯ n with n, ¯ n ∈ Z ≥ . These special values of thespins correspond to finite-dimensional representations which we are aiming at. That is why wewould like to have a meaningful intertwining operator for this discrete set. In order to obtain itwe note that the expression (2.5), considered as an analytical function of s , ¯ s , has simple polesexactly on this discrete set of (half)-integer points. Consequently, we need to choose properlythe normalization constant in (2.5) to suppress the poles at 2 s = n , 2¯ s = ¯ n . Further, pursuingthis strategy we find the normalization constant as an appropriate combination of the Eulergamma functions such that the intertwining operator (2.5) becomes well-defined in the case offinite-dimensional representations as well. In order to implement the outlined program we resortto the text-book formula for the following complex Fourier transformation [23] A ( α, ¯ α ) (cid:90) C d z e ipz + i ¯ p ¯ z z α ¯ z α = p α ¯ p ¯ α , A ( α, ¯ α ) = i −| α − ¯ α | π Γ (cid:0) α +¯ α + | α − ¯ α | +22 (cid:1) Γ (cid:0) − α − ¯ α + | α − ¯ α | (cid:1) , (2.6)where Γ( x ) is the Euler gamma function. One can substitute here z = x + iy , ¯ z = x − iy andpass to the integrations over x, y ∈ R . We replace p and ¯ p by the differential operators, p → i∂ x and ¯ p → i∂ ¯ x , use the shift operator e a∂ x f ( x ) = f ( x + a ), and come to the definition( i∂ z ) α ( i∂ ¯ z ) ¯ α Φ( z, ¯ z ) := A ( α, ¯ α ) (cid:90) C d x ( z − x ) − − α (¯ z − ¯ x ) − − ¯ α Φ( x, ¯ x ) . (2.7)In order to avoid cumbersome expressions we prefer to recast this formula to a concise form[ i∂ z ] α Φ( z, ¯ z ) = A ( α ) (cid:90) C d x [ z − x ] − − α Φ( x, ¯ x ) . (2.8)Here and in the following we profit from the shorthand notation[ z ] α = z α ¯ z ¯ α , α − ¯ α ∈ Z , (2.9)which unifies the holomorphic and antiholomorphic sectors. Let us remind once more that α and ¯ α are not assumed to be complex conjugates. The constraint on the exponents α , ¯ α in (2.9)ensures that the function [ z ] α is single-valued, whereas for generic values of α the holomorphicand anti-holomorphic factors of [ z ] α taken separately have branch cuts. Bearing in mind that theholomorphic sector is always accompanied by the antiholomorphic one we omit the ¯ α -dependencein the A -factor: A ( α, ¯ α ) → A ( α ).Thus, if the normalization in (2.5) is chosen properly, the intertwining operator can be repre-sented in two equivalent forms, either as a formal complex power of the differentiation operatorW( s, ¯ s ) = [ i∂ z ] s +1 or as a well defined integral operator[W( s, ¯ s )Φ] ( z, ¯ z ) = ( − | s − ¯ s | π Γ ( s + ¯ s + | s − ¯ s | + 2)Γ ( − s − ¯ s + | s − ¯ s | − (cid:90) C d x [ z − x ] − s − Φ( x, ¯ x ) . (2.10) D. Chicherin, S.E. Derkachov and V.P. SpiridonovAt special points 2 s = n , 2¯ s = ¯ n , n, ¯ n ∈ Z ≥ , the integral operator turns to the differential ope-rator of a finite order ( i∂ z ) n +1 ( i∂ ¯ z ) ¯ n +1 . Let us note that for generic s the holomorphic ∂ s +1 z and anti-holomorphic ∂ s +1¯ z parts (see (2.9)) of the intertwiner [ i∂ z ] s +1 taken separately areill-defined (working with the contour integrals with the kernel ( z − x ) α one cannot find a trans-lationally invariant measure). However, being taken together, they form a well-defined integraloperator.Formula (2.1) implies that for special values of spins 2 s = n , 2¯ s = ¯ n discussed above an( n + 1)(¯ n + 1)-dimensional representation decouples from the general infinite-dimensional ca-se [23]. Indeed, the space of polynomials spanned by ( n + 1)(¯ n + 1) basis vectors z k ¯ z ¯ k , where k =0 , , . . . , n and ¯ k = 0 , , . . . , ¯ 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 generatingfunction which contains all of them. The generating function for basis vectors of this finite-dimensional representation can be chosen in the following form[ z − x ] n = ( z − x ) n (¯ z − ¯ x ) ¯ n , (2.11)where x , ¯ x are some auxiliary parameters. Indeed, expanding (2.11) with respect to x and ¯ x werecover all ( n + 1)(¯ n + 1) vectors z k ¯ z ¯ k , where k = 0 , , . . . , n and ¯ k = 0 , , . . . , ¯ n .The decoupling of a finite-dimensional representation and the explicit expression for the gene-rating function (2.11) allow us to give a very natural interpretation to the situation from the pointof view of the intertwining operator. Indeed, an immediate consequence of the definition (2.3)is that the null-space of W( s, ¯ s ) – the space annihilated by the operator – is invariant under theaction of the operators T ( s, ¯ s ) ( g ). Therefore, if the intertwining operator has a nontrivial null-space then a sub-representation decouples and the corresponding invariant subspace appears. Inthe case at hand, when 2 s = n and 2¯ s = ¯ n , the intertwining operator turns into the differentialoperator ∂ n +1 ¯ ∂ ¯ n +1 .Of course this operator annihilates all ( n + 1)(¯ n + 1) basis vectors z k ¯ z ¯ k , where k = 0 , , . . . , n and ¯ k = 0 , , . . . , ¯ n , but the whole null-space of this operator is too big (it includes all harmonicfunctions) and we need some additional characterization for the considered finite-dimensionalsubspace. Relation (2.3) shows that the image of the intertwining operator W( − − s, − − ¯ s )is also invariant under the action of the operators T ( s, ¯ s ) ( g ). Moreover, formula (2.10) in theconsidered situation[W( − − s, − − ¯ s )Φ] ( z, ¯ z )= ( − | s − ¯ s | π Γ ( − s − ¯ s + | s − ¯ s | )Γ ( s + ¯ s + | s − ¯ s | + 1) (cid:90) C d x ( z − x ) s (¯ z − ¯ x ) s Φ( x, ¯ x ) , (2.12)clearly shows that for special values of the spins 2 s = n and 2¯ s = ¯ n discussed above the integralin the right-hand side is equal to a polynomial with respect to z and ¯ z , and the image of theoperator W( − − s, − − ¯ s ) (after dropping the numerical factor Γ ( − s − ¯ s + | s − ¯ s | ) whichdiverges at these points) is exactly the needed finite-dimensional subspace. After all we obtaina characterization of our finite-dimensional subspace: it is the intersection of the null-space ofthe intertwining operator W( s, ¯ s ) and of the image of the operator W( − − s, − − ¯ s ) both beingproperly normalized for special values of the spins 2 s = n and 2¯ s = ¯ n .The intertwining operator annihilates the generating function of the finite-dimensional rep-resentation (2.11), which can be seen solely from its basic properties. The following calculationsuggests this generating function itself. The formal differential operator form of the intertwiningoperatorsW( s ) = [ i∂ z ] s +1 , W( − − s ) = [ i∂ z ] − − s rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 7formally indicates that W( − − s ) and W( s ) are inverses to each other,W( s )W( − − s ) = 1l . (2.13)However, this inversion relation is broken for special values of the spins. Let us rewrite theidentity (2.13) taking into account the explicit expression for kernels of the integral operatorsW( − − s ) (2.12) and 1l, which is given by the Dirac delta-function. In this way we find therelation[ i∂ z ] s +1 [ z − x ] s = ( − −| s − ¯ s | π Γ ( s + ¯ s + | s − ¯ s | + 1)Γ ( − s − ¯ s + | s − ¯ s | ) δ ( z − x ) . At special points 2 s = n , 2¯ s = ¯ n the gamma-function Γ ( − s − ¯ s + | s − ¯ s | ) has poles, and there-fore the right-hand side of the latter formula vanishes. So, one obtains[ i∂ z ] n +1 [ z − x ] n = 0 , n = 0 , , , . . . , (2.14)i.e., the generating function of the finite-dimensional representation coincides with the kernel ofthe intertwining operator W( − − n/
2) after a proper normalization.Our calculation may seem superfluous since the relation (2.14) is evident per se. However, wepresented it here because all its basic steps remain valid after the trigonometric (see Section 3.1)and elliptic deformations (see [16, 17]) of the symmetry algebra. The deformations complicatesignificantly the intertwining operator and the generating function of finite-dimensional repre-sentations such that the deformed analogues of (2.14) are far from being obvious and in theelliptic case they are much more involved [10, 17]. , C )-invariant R-operator Emergence of the periodic integrable spin chain with SL(2 , C ) symmetry in the high energyasymptotics of quantum chromodynamics was discovered in [21, 30, 31]. The detailed conside-ration of the corresponding formalism was performed in [12, 14]. In these papers the quantum-mechanical model of interest has been solved, i.e., the relevant Baxter Q-operator has beenconstructed and the separation of variables has been implemented. The general R-operatorfor the SL(2 , C ) group has been extensively studied in the first part of [15] as a simplest non-trivial example of the general SL( N, C )-construction. Here we briefly outline main steps in theconstruction of this R-operator before proceeding to its finite-dimensional reductions.Firstly we tailor a pair of L-operators out of the Lie algebra generators E ( s ) , ¯E (¯ s ) (2.2) andthe spectral parameters u and ¯ u which are assumed to be restricted similar to the representationparameters, u − ¯ u ∈ Z [14, 15],L( u , u ) = u ·
1l + E ( s ) = (cid:18) z (cid:19) (cid:18) u − ∂ u (cid:19) (cid:18) − z (cid:19) , (2.15)¯L(¯ u , ¯ u ) = ¯ u ·
1l + ¯E (¯ s ) = (cid:18) z (cid:19) (cid:18) ¯ u − ¯ ∂ u (cid:19) (cid:18) − ¯ z (cid:19) . (2.16)Here we use the convenient shorthand notation u = u − s − , u = u + s, ¯ u = ¯ u − ¯ s − , ¯ u = ¯ u + ¯ s. (2.17)Each of the L-operators (2.15), (2.16) respects the RLL-relation with Yang’s 4 × ab,ef ( u − v )L ec ( u )L fd ( v ) = L bf ( v )L ae ( u )R ef,cd ( u − v ) , (2.18)R ab,ef (¯ u − ¯ v )¯L ec (¯ u )¯L fd (¯ v ) = ¯L bf (¯ v )¯L ae (¯ u )R ef,cd (¯ u − ¯ v ) , D. Chicherin, S.E. Derkachov and V.P. Spiridonovwhere 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. (2.49)). 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 generatorsof SL(2 , C ).The L-operators (2.15), (2.16) respect simultaneously another RLL-relation with some general R-operator which intertwines the co-product of L-operators in the pair of quantum spacesR ( u − v, ¯ u − ¯ v )L ( u , u )L ( v , v ) = L ( v , v )L ( u , u )R ( u − v, ¯ u − ¯ v ) , (2.19)R ( u − v, ¯ u − ¯ v )¯L (¯ u , ¯ u )¯L (¯ v , ¯ v ) = ¯L (¯ v , ¯ v )¯L (¯ u , ¯ u )R ( u − v, ¯ u − ¯ v ) , (2.20)where parameters u and u are defined in (2.17), and v , v are analogous linear combinationsof v and (cid:96) , v = v − (cid:96) − , v = v + (cid:96), ¯ v = ¯ v − ¯ (cid:96) − , ¯ v = ¯ v + ¯ (cid:96). The lower indices of R and L , L denote quantum spaces on which the operators act non-trivially. The L-operators are multiplied as conventional 2 × s , ¯ s and it is realized on the functions of variables z , ¯ z , the secondrepresentation is specified by the spins (cid:96) , ¯ (cid:96) and it is realized on the functions of variables z , ¯ z .In (2.19), (2.20) we drop dependencies of the R-operator on the representation parameters. Thefull-fledged notation would be R( u − v, ¯ u − ¯ v | s, ¯ s, (cid:96), ¯ (cid:96) ).Note that the R-operator serves for both L-operators, i.e., it is not just the holomorphicor anti-holomorphic object, as opposed to the L-operators (2.15), (2.16). In the following wefrequently omit the dependence of the R-operator (and other intertwining operators) on theanti-holomorphic parameters denoting it R( u ). The R-operator is invariant with respect to theSL(2 , C ) group, i.e., it commutes with the co-product of sl (2 , C ) generators (cid:2) R ( u, ¯ u ) , E ( s )1 + E ( (cid:96) )2 (cid:3) = 0 , (cid:2) R ( u, ¯ u ) , ¯E (¯ s )1 + ¯E (¯ (cid:96) )2 (cid:3) = 0 , which follows immediately from the RLL-relations (2.19) and (2.20).Apart from the RLL-relations (2.19), (2.20) the general R-operator satisfies the YBER ( u − v, ¯ u − ¯ v )R ( u, ¯ u )R ( v, ¯ v ) = R ( v, ¯ v )R ( u, ¯ u )R ( u − v, ¯ u − ¯ v ) , (2.21)where both sides are endomorphisms on the tensor product of three infinite-dimensional spacesrealizing arbitrary principal series representations of SL(2 , C ).In [14, 15] an integral operator solution of the intertwining relations (2.19) and (2.20) wasfound, which solves simultaneously YBE (2.21). The construction naturally gives to this generalR-operator several factorized forms related to an integral operator realization of the generatorsof symmetric group S [15]. Here we do not go into details of this formalism and just indicate thefactorization which is appropriate for our current purposes. The R-operator can be representedas a product of four elementary intertwining operators [15]R ( u − v, ¯ u − ¯ v ) = [ z ] u − v [ i∂ ] u − v [ i∂ ] u − v [ z ] u − v , (2.22)where we assume the shorthand notation z ij = z i − z j and (2.9). Taking into account (2.8)one can rewrite (2.22) explicitly as an integral operator. The notation (2.9) implies that the R-operator consists of the holomorphic and anti-holomorphic parts which, being taken separately,are ill-defined for generic spectral and representation parameters. The merge of holomorphicand antiholomorphic parts yields a well-defined integral operator.rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 9Formula (2.22) plays a crucial role in the subsequent discussion. It admits deformations [13]leading to R-operators for the modular double [6] and the elliptic modular double [16].The expression (2.22) may seem rather unusual. In [15] it was shown that the holomorphicpart of the R-operator (2.22), being restricted to the space of polynomials, coincides with thefamiliar R-operator constructed in [28, 39] in the form of the beta-function depending on the“square root” of the Casimir operator. However, the form (2.22) does not demand extra infor-mation about the structure of tensor products and corresponding Clebsch–Gordan coefficients.Furthermore, we will show that the integral R-operator (2.22) contains finite-dimensional solu-tions of the Yang–Baxter relation as well (2.21).The elementary intertwining operators appearing in (2.22) fulfill the following operator rela-tions [ i∂ k ] a [ z ] a + b [ i∂ k ] b = [ z ] b [ i∂ k ] a + b [ z ] a , k = 1 , . (2.23)These formulae have a remarkable interpretation in terms of the Coxeter relations of the sym-metric group S [13, 15]. Using (2.23) one can easily prove that the R-operator (2.22) respectsthe YBE (2.21). The operator factors in (2.22) are called intertwiners because they satisfy theequations[ i∂ ] u − u L ( u , u ) = L ( u , u ) [ i∂ ] u − u , [ i∂ ] v − v L ( v , v ) = L ( v , v ) [ i∂ ] v − v , (2.24)[ z ] u − v L ( u , u )L ( v , v ) = L ( v , u )L ( v , u )[ z ] u − v , (2.25)and similar ones with L substituted by ¯L. Here the operators [ i∂ k ] a and [ z ] a act on eachmatrix element of the matrices L k entrywise, i.e., they should be considered as 2 × i∂ ] u − u = [ i∂ ] s +1 and[ i∂ ] v − v = [ i∂ ] (cid:96) +1 are the intertwining operators of the equivalent representations (2.3) forthe first and second spaces, respectively. The equalities (2.24) are identical to the definingrelations (2.3) of the intertwining operator W. Applying several times (2.24) and (2.25) one caneasily check that the composite R-operator (2.22) obeys the RLL-relations (2.19) and (2.20).The identities (2.23) are equivalent to the famous star-triangle relation which can be repre-sented in the following three equivalent forms:1) as an integral identity [14, 41] (cid:90) C d w z − w ] α [ w − x ] β [ w − y ] γ = A ( − β ) A ( α − A ( γ −
1) 1[ z − x ] − γ [ z − y ] − β [ y − x ] − α , (2.26)provided that the exponents respect the uniqueness conditions α + β + γ = ¯ α + ¯ β + ¯ γ = 2;2) as a particular point in the image of the operator [ i∂ z ] α − (with the same restriction onthe exponents as before)[ i∂ z ] α − (cid:18) z − x ] β [ z − y ] γ (cid:19) = A ( − β ) A ( γ −
1) 1[ z − x ] − γ [ z − y ] − β [ y − x ] − α ; (2.27)3) or as a pseudo-differential operators identity [25][ i∂ z ] α · [ z ] α + β · [ i∂ z ] β = [ z ] β · [ i∂ z ] α + β · [ z ] α . (2.28)0 D. Chicherin, S.E. Derkachov and V.P. Spiridonov Now we reduce the R-operator (2.22) to finite-dimensional representations in its first space.The principal possibility of this reduction is based on the following relation[ i∂ ] u − u R ( u , u | v , v ) = R ( u , u | v , v ) [ i∂ ] u − u , (2.29)where we use the R -operator R := P R with P – a permutation operator, P Ψ( z , z ) =Ψ( z , z )P . Relation (2.29) can be proved using the identity (2.23) and it shows that both,the null-space of the intertwining operator [ i∂ ] s +1 and the image of the intertwining opera-tor [ i∂ ] − s − , are mapped onto themselves by our R-matrix R . Therefore, if we find invariantfinite-dimensional subspaces of the latter spaces they will be invariant with respect to the actionof R-operator itself.We take the function [ z ] s Φ( z , ¯ z ), where 2 s = u − u − z , ¯ z ) is an arbitraryfunction, and act upon it by the R-operator. We break down the calculation to several stepsaccording to the factorized form (2.22) of the R-operator. At the end of calculation we choose2 s = n , 2¯ s = ¯ n with n, ¯ n ∈ Z ≥ such that [ z ] s turns into the generating function of the finite-dimensional representation in the first space (2.11) with an auxiliary parameter z . However,for a while we assume the spin s to be generic.Using formula (2.27) we implement the first step.We act by the first two factors [ i∂ ] u − v [ z ] u − v of the R-operator (2.22) and find[ i∂ ] u − v [ z ] u − v [ z ] s Φ( z , ¯ z )= A ( u − v ) A ( u − u ) · [ z ] u − u [ z ] v − u − [ z ] u − v Φ( z , ¯ z ) . (2.30)In order to apply the third factor [ i∂ ] u − v of the R-operator (2.22) we resort to the relation[ i∂ ] u − v [ z ] u − u [ z ] u − v Φ( z , ¯ z )= A ( u − v ) A ( u − u − · [ i∂ ] u − u − [ z ] v − u − [ z ] u − v Φ( z , ¯ z ) , which follows immediately from the integral representation (2.8) for [ i∂ z ] α . A merit of theprevious formula is that we traded the integral operator [ i∂ ] u − v for [ i∂ ] s , which becomesjust a differential operator for 2 s = n and 2¯ s = ¯ n . Incorporating into the latter formula theinert factors from (2.30) and the last factor [ z ] u − v of the R-operator (2.22), we findR ( u , u | v , v )[ z ] u − u − Φ( z , ¯ z ) = A ( u − v ) A ( u − u ) A ( u − v ) A ( u − u − × [ z ] u − v [ z ] v − u − [ i∂ ] u − u − [ z ] v − u − [ z ] u − v Φ( z , ¯ z ) . (2.31)In order to polish the latter formula we denote z = x like in (2.11) and rewrite (2.31) in termsof the representation parameters. Also we prefer to replace the R-operator by R = P R .Thus the general R-operator for the SL(2 , C ) group acting in the tensor product of twoinfinite-dimensional representation spaces with spins s , ¯ s and (cid:96) , ¯ (cid:96) can be reduced to a finite-dimensional subspace in the first space if 2 s = n , 2¯ s = ¯ n ( n, ¯ n ∈ Z ≥ ). We have the followingformula R (cid:0) u | n , ¯ n , (cid:96), ¯ (cid:96) (cid:1) [ z − x ] n Φ( z , ¯ z )= c · [ z − x ] − u + n + (cid:96) [ z ] u + n + (cid:96) +1 [ ∂ z ] n [ z ] − u + n − (cid:96) − [ z − x ] u + n − (cid:96) Φ( z , ¯ z ) , (2.32)rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 11where the normalization factor is c = ( − n +¯ n A ( u − n + (cid:96) ) A ( − u + n + (cid:96) ) . The latter formula gives a number of solutions of the YBE (2.21) which are endomorphisms onthe tensor product of an ( n + 1)(¯ n + 1)-dimensional and an infinite-dimensional spaces.We consider formula (2.32) as one of the main results of this paper. It gives a conciseexpression for known higher spin R-operators. They are “mixed” objects in a sense that theyare 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 fundamentalto arbitrary finite-dimensional representations. Moreover, the formula (2.32) produces all suchsolutions of the YBE related to the principal series representation. Its analogue for the modulardouble is derived in Section 3.3 and the elliptic modular double case is considered in [10].In order to get accustomed to the reduction formula (2.32) let us consider a simple example.One can easily recover the holomorphic L-operator (2.15) substituting ( n, ¯ n ) = (1 ,
0) in (2.32)and choosing the basis in the space C of the fundamental representation as e = − z , e = 1.Then R (cid:0) u − | , (cid:96) (cid:1) e = c · (cid:2) e ( z ∂ − (cid:96) + u ) + e (cid:0) z ∂ − (cid:96)z (cid:1)(cid:3) , (2.33) R (cid:0) u − | , (cid:96) (cid:1) e = c · (cid:2) e ( − ∂ ) + e ( u + (cid:96) − z ∂ ) (cid:3) . (2.34)Consequently the restriction of R ( u − | , (cid:96) ) to C in the first factor takes the matrix formL( u ) = (cid:18) u − (cid:96) + z∂ − ∂z ∂ − (cid:96)z u + (cid:96) − z∂ (cid:19) (2.35)and coincides with the holomorphic L-operator (2.15). Analogously taking ( n, ¯ n ) = (0 ,
1) werecover the anti-holomorphic ¯L-operator (2.16).Besides the L-operator, the formula (2.32) reproduces all its higher-spin generalizations. Si-multaneously, it produces R-matrices described by plain finite-dimensional matrices in bothspaces. Indeed, substituting in (2.32) the generating function (2.11) of the finite-dimensional( m + 1)( ¯ m + 1)-dimensional representation in the second space, we find a solution of theYBE (2.21) for the spins n , ¯ n and m , ¯ m in the first and second spaces, respectively, R (cid:0) u | n , ¯ n , m , ¯ m (cid:1) [ z − x ] n [ z − y ] m (2.36)= c · [ z − x ] − u + n + (cid:96) [ z ] u + n + m +1 [ ∂ z ] n [ z ] − u + n − m − [ z − x ] u + n − m [ z − y ] m . Expanding both sides of this relation in auxiliary parameters x , ¯ x , y , ¯ y one can rewrite it ina form of a square matrix with ( n + 1)(¯ n + 1)( m + 1)( ¯ m + 1) rows (or columns). The compactformula (2.36) produces all its entries. In particular, taking the fundamental representation inboth spaces n = m = 1, ¯ n = ¯ m = 0 we reproduce Yang’s R-matrix (cf. (2.49)). In this section we slightly digress from the discussion of the group SL(2 , C ) and outline how sl -symmetric 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 calculationsare essentially based on ideas explained above we will limit ourselves to the statement of theresults.Although the sl algebra is “a half” of the Lie algebra of the group SL(2 , C ), it requiresa special treatment. We deal with a functional representation of the sl -algebra in the space2 D. Chicherin, S.E. Derkachov and V.P. Spiridonovof polynomials of one complex variable C [ z ]. Fixing a generic complex number s ∈ C andrepresenting the algebra generators by the first order differential operators given in (2.2) weendow C [ z ] with a structure of the Verma module. For generic value of s the module is aninfinite-dimensional space with the basis { , z, z , . . . } and there are no invariant subspaces, i.e.,the representation is irreducible. Invariant subspaces arise for the discrete set of spin values2 s = n , n ∈ Z ≥ . The corresponding ( n + 1)-dimensional representation is irreducible and it isrealized on the submodule with the basis { , z, . . . , z n } .Since the sl generators are holomorphic, we have a single holomorphic L-operator givenin (2.15). Now only the holomorphic spectral parameter u is present. The general R-operatorR( u | s, (cid:96) ) is defined on the tensor product of two Verma modules with the spins s and (cid:96) . It hasto satisfy holomorphic analogues of the RLL-relation (2.19) and of the YBE (2.21).The general R-operator (2.22) for SL(2 , C ) group is well defined due to its non-analyticity, inother words, due to the presence of holomorphic and antiholomorphic parts. We cannot get thegeneral R-operator for sl (which has to be holomorphic) by crossing out the anti-holomorphicpart of (2.22). Anyway, the holomorphic RLL-relation (2.19) can be solved [15] in terms ofa well-defined operator on C [ z ] ⊗ C [ z ] which takes the following factorized form,R ( u | s, (cid:96) ) = Γ( z ∂ − s )Γ( z ∂ − u − s − (cid:96) ) Γ( z ∂ + u − s − (cid:96) )Γ( z ∂ − s ) , (2.37)where ratios of the operator-valued gamma functions are defined with the help of the integralrepresentation for Euler’s beta-functionΓ( z ∂ + a )Γ( z ∂ + b ) Φ( z , z ) := 1Γ( b − a ) (cid:90) dαα a − (1 − α ) b − a − Φ( αz + (1 − α ) z , z ) . This R-operator satisfies the holomorphic analogue of YBE (2.21) as well. As we remarked inSection 2.2 the operator (2.37) coincides with the one found in [28, 39] in the early days of thequantum inverse scattering method in spite of the fact that they look completely different.For 2 s = n , n ∈ Z ≥ , the general R-operator (2.37) can be restricted to an ( n +1)-dimensionalrepresentation in the first space. Taking into account permutation of the pair of tensor factors, R = P R , one can show that the restricted R-operator acquires a concise form R (cid:0) u | n , (cid:96) (cid:1) ( z − x ) n Φ( z )= c · ( z − x ) − u + n + (cid:96) z u + n + (cid:96) +112 ∂ nz z − u + n − (cid:96) − ( z − x ) u + n − (cid:96) Φ( z ) , (2.38)where the normalization factor is c = ( − n +1 Γ( − (cid:96) − n − u )Γ( − (cid:96) + n − u ) . Formula (2.38) is completely analogous to the SL(2 , C ) reduction formula (2.32). Expandingboth sides of (2.38) with respect to an auxiliary parameter x one recovers an ( n + 1) × ( n + 1)-matrix whose entries are the n -th order differential operators with polynomial coefficients inspectral parameter u of degree n (or lower).In [8] the Lax operator has been recovered from the general R-operator by means of a quitebulky calculation. Formula (2.38) provides considerable simplification of that result generalizingit to the higher-spin analogues of the rational Lax operator.In order to illustrate the power of the formula (2.38) we present below the R-operator for thespin 1 representation in the first space. In the basis e = 1, e = z , e = z of the 3-dimensionalspace, the R ( u | , (cid:96) )-operator takes the matrix form (we change notation z → z ) ( u + (cid:96) )( u + (cid:96) +1) − u + (cid:96) ) z∂ + z ∂ (cid:96) ( u + (cid:96) ) z − ( u +3 (cid:96) − z ∂ + z ∂ (cid:96) (2 (cid:96) − z +2(1 − (cid:96) ) z ∂ + z ∂ u + (cid:96) ) ∂ − z∂ ( u + (cid:96) )( u − (cid:96) +1)+2(2 (cid:96) − z∂ − z ∂ (cid:96) ( u − (cid:96) +1) z − u − (cid:96) +2) z ∂ − z ∂ ∂ ( u − (cid:96) +1) ∂ + z∂ ( u − (cid:96) )( u − (cid:96) +1)+2( u − (cid:96) +1) z∂ + z ∂ . rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 13Conventional methods demand laborious calculations to reproduce this complicated matrix. Inour case the result follows immediately from the formula (2.38). An explicit matrix factorizationformula for the operator R ( u | n , (cid:96) ) (2.38) generalizing factorization of the L-operator (2.15)was derived in the followup paper [7]. E.g., the R ( u | , (cid:96) )-operator given above factorizes toa product of five more elementary 3 × The standard procedure for constructing finite-dimensional higher-spin R-operators out of thefundamental one is the fusion procedure [28, 29]. Firstly, we remind how it works in the caseof the symmetry algebra sl using a formulation convenient for us. Then in the next sectionwe straightforwardly extend it to the case of the SL(2 , C ) group and show that the reductionformula (2.32) is in line with the fusion construction.For the rank one symmetry algebras underlying an integrable system the recipe of [28, 29]looks as follows. One forms an inhomogeneous monodromy matrix T j ...j n i ...i n out of L-operators L ji multiplying them as operators in quantum space and taking tensor products of the auxiliaryspace C , and then symmetrizes the monodromy matrix over the spinor indices. The parametersof inhomogeneity have to be adjusted in a proper way. The result T ( j ...j n )( i ...i n ) is an R-operator whichhas a higher-spin auxiliary space and solves the YBE. Thus constructing higher-spin R-operatorsone has to deal with Sym (cid:0) C (cid:1) ⊗ n which is a space of symmetric tensors with a number of spinorindices Ψ ( i ...i n ) . The usual matrix-like action of operators has the form[TΨ] ( i ...i n ) = T ( j ...j n )( i ...i n ) Ψ ( j ...j n ) , (2.39)where the summation over repeated indices is assumed. We prefer not to deal with a multitudeof spinor indices. Instead we introduce auxiliary spinors λ = ( λ , λ ), µ = ( µ , µ ) and contractthem with the tensors λ i · · · λ i n Ψ i ...i n = Ψ( λ ) , λ i · · · λ i n T j ...j n i ...i n µ j · · · µ j n = T( λ | µ ) . (2.40)Thus the symmetization over spinor indices is taken into account automatically. Henceforth, inplace of the tensors we work with the corresponding generating functions which are homogeneouspolynomials of degree n of two variablesΨ( λ ) = Ψ( λ , λ ) , Ψ( αλ , αλ ) = α n Ψ( λ , λ ) . (2.41)T( λ | µ ) is usually called the symbol of the operator. In this way formula (2.39) acquires a rathercompact form[TΨ] ( λ ) = n ! T( λ | ∂ µ )Ψ( µ ) | µ =0 . (2.42)Note that, in fact, we do not need to take µ = 0 in (2.42). The µ variable disappears automati-cally 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 exampleof the quantum-mechanical system of spin n , i.e., consider the symmetry group SU(2) and thegenerators (cid:126)J of the Lie algebra su in the representation of spin n . In the spin representationthe generators act on the space C and they are given by the Pauli matrices (cid:126)σ , so that (cid:2) (cid:126)J Ψ (cid:3) i = (cid:126)σ ji Ψ j , (cid:126)J ji = (cid:126)σ ji . n spin representations we obtain the generators on the space (cid:0) C (cid:1) ⊗ n , (cid:126)J j ...j n i ...i n = (cid:126)σ j i δ j i · · · δ j n i n + · · · + δ j i · · · δ j n − i n − (cid:126)σ j n i n . (2.43)In order to single out in the tensor product an irreducible maximal spin representation wesymmetrize over spinor indices yielding the representation of spin n , (cid:2) (cid:126)J Ψ (cid:3) ( i ...i n ) = (cid:126)σ ji Ψ ( ji ...i n ) + · · · + (cid:126)σ ji n Ψ ( i ...i n − j ) . (2.44)Further we introduce a pair of auxiliary spinors and find the symbol (cid:126)J ( λ, µ ) of the opera-tor (cid:126)J (2.43) converting formula (2.43) to (cid:126)J ( λ | µ ) = λ i · · · λ i n (cid:126)J j ...j n i ...i n µ j · · · µ j n = n (cid:104) λ | µ (cid:105) n − (cid:104) λ | (cid:126)σ | µ (cid:105) , (2.45) (cid:104) λ | = ( λ , λ ) , | µ (cid:105) = (cid:18) µ µ (cid:19) , where (cid:104) λ | µ (cid:105) = λ µ + λ µ and (cid:104) λ | (cid:126)σ | µ (cid:105) = λ i (cid:126)σ ji µ j are symbols of the identity operator andPauli matrices, respectively. In view of (2.42), (2.45), formula (2.44) acquires the indexless form (cid:2) (cid:126)J Ψ (cid:3) ( λ , λ ) = n ! n (cid:104) λ | ∂ µ (cid:105) n − (cid:104) λ | (cid:126)σ | ∂ µ (cid:105) Ψ( µ ) (cid:12)(cid:12) µ =0 . Consequently, instead of tensors and finite-dimensional operators we deal with their symbolsand generating functions. Note that due to the homogeneity of Ψ (2.41), (cid:104) λ | µ (cid:105) n is a symbol ofthe identity operator defined on the tensor product of n spaces1 n ! (cid:104) λ | ∂ µ (cid:105) n Ψ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) µ =0 = 1 n ! ∂ nα e α (cid:104) λ | ∂ µ (cid:105) Ψ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) µ =0 ,α =0 = 1 n ! ∂ nα Ψ( αλ ) (cid:12)(cid:12)(cid:12)(cid:12) α =0 = Ψ( λ ) . Then taking into account that n (cid:104) λ | µ (cid:105) n − (cid:104) λ | (cid:126)σ | µ (cid:105) = (cid:104) λ | (cid:126)σ | ∂ λ (cid:105)(cid:104) λ | µ (cid:105) n , we obtain an alternative expression for (cid:126)J , (cid:2) (cid:126)J Ψ (cid:3) ( λ , λ ) = (cid:104) λ | (cid:126)σ | ∂ λ (cid:105) n ! (cid:104) λ | ∂ µ (cid:105) n Ψ( µ ) (cid:12)(cid:12) µ =0 = (cid:104) λ | (cid:126)σ | ∂ λ (cid:105) Ψ( λ ) . (2.46)Thus we have realized the Lie algebra generators (cid:126)J as differential operators on the space ofhomogeneous polynomials of two variables (forming a projective space) J ± = (cid:104) λ | σ ± iσ | ∂ λ (cid:105) = (cid:104) λ | σ ± | ∂ λ (cid:105) , J = (cid:104) λ | σ | ∂ λ (cid:105) or, more explicitly, J + = λ ∂ λ , J − = λ ∂ λ , J = ( λ ∂ λ − λ ∂ λ ) . (2.47)This realization of the generators is known as the Jordan–Schwinger representation. We canchoose the homogeneous function ( λ + xλ ) n (see (2.41)) as a generating function of the ( n + 1)-dimensional representation with an auxiliary parameter x .One can easily proceed from the projective space to the space of polynomials of one complexvariable. Indeed, due to the homogeneityΨ( λ , λ ) = λ n Ψ (cid:0) λ λ , (cid:1) = λ n Ψ (cid:0) , λ λ (cid:1) rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 15all information about Ψ( λ , λ ) is encoded in a function of the ratio λ λ alone. In order to makecontact with the holomorphic set of the sl (2 , C ) generators (2.2) we choose λ = − z, λ = 1 andrewrite the generators (2.47) in terms of the variable zJ + = z ∂ − nz, J − = − ∂, J = z∂ − n . (2.48)Furthermore, the generating function of the Jordan–Schwinger representation turns into thegenerating function of one variable ( x − z ) n (cf. (2.11), recall (2.38)).Before proceeding to the fusion procedure for the SL(2 , C ) group, we remind construction ofthe L-operator. We build the L-operator with a finite-dimensional local quantum space startingfrom the Yang R-matrix. The latter acts on the tensor product of two spin- representationsR( u ) = u + (1l + (cid:126)σ ⊗ (cid:126)σ ) = (cid:18) u + + σ σ − σ + u + − σ (cid:19) . (2.49)Following the recipe from [28, 29] we form the product of the Yang R-matricesR ( j ...j n )( i ...i n ) ( u ) = Sym R j i ( u )R j i ( u − · · · R j n i n ( u − n + 1) , (2.50)where the indices refer to the first space in (2.49),R ji ( u ) = (cid:0) u + (cid:1) δ ji + (cid:126)σ ji (cid:126)σ, (2.51)and Sym implies symmetrization with respect to ( i . . . i n ) and ( j . . . j n ). In such a way oneobtains an operator acting on the space of symmetric rank n tensors, i.e., on the space of spin n representation, and on the two-dimensional auxiliary space where the (cid:126)σ -matrices are acting.According to [28, 29] it respects the Yang–Baxter relations. Now we calculate the symbolof (2.50) with respect to the quantum spaceR( u | λ, µ ) = λ i · · · λ i n R j ...j n i ...i n ( u ) µ j · · · µ j n = (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) · · · (cid:104) λ | R( u − n + 1) | µ (cid:105) , i.e., it is still an operator in the auxiliary space. Henceforth for the sake of brevity we refer toit as a symbol of the R-matrix. The derived symbol R( u | λ, µ ) factorizes to a product of Yang’sR-matrix symbols (cid:104) λ | R( u ) | µ (cid:105) = λ i R ji ( u ) µ j , (cid:104) λ | R( u ) | µ (cid:105) = (cid:104) λ | µ (cid:105) (cid:0) u + + (cid:126)n(cid:126)σ (cid:1) = (cid:18) ( u + 1) λ µ + uλ µ λ µ λ µ uλ µ +( u +1) λ µ (cid:19) , (2.52)where we introduced the unit vector (cid:126)n = (cid:104) λ | (cid:126)σ | µ (cid:105)(cid:104) λ | µ (cid:105) , (cid:126)n · (cid:126)n = 1. The product of such matrices iseasy to calculate and we obtainR( u | λ, µ ) = u ( u − · · · ( u − n + 1) (2.53) × (cid:18) ( u +1 − n ) (cid:104) λ | µ (cid:105) n + n (cid:104) λ | µ (cid:105) n − ( λ µ − λ µ ) n (cid:104) λ | µ (cid:105) n − λ µ n (cid:104) λ | µ (cid:105) n − λ µ ( u +1 − n ) (cid:104) λ | µ (cid:105) n − n (cid:104) λ | µ (cid:105) n − ( λ µ − λ µ ) (cid:19) . In compact notation this formula takes the formR( u | λ, µ ) = (cid:104) λ | µ (cid:105) n (cid:0) u + + (cid:126)n(cid:126)σ (cid:1)(cid:0) u − + (cid:126)n(cid:126)σ (cid:1) · · · (cid:0) u − n + + (cid:126)n(cid:126)σ (cid:1) = u ( u − · · · ( u − n + 1) (cid:104) λ | µ (cid:105) n (cid:0) u + 1 − n + n (cid:126)n(cid:126)σ (cid:1) and it can be easily proven by induction using the identity ( (cid:126)n(cid:126)σ ) = 1l.6 D. Chicherin, S.E. Derkachov and V.P. SpiridonovUp to the inessential normalization factor r n ( u ) = u ( u − · · · ( u − n + 1) and the shift of thespectral parameter u → u − n , we obtain the following symbol (see (2.45))L( u | λ, µ ) = r − n ( u )R( u − n | λ, µ )= u (cid:104) λ | µ (cid:105) n + n (cid:104) λ | µ (cid:105) n − (cid:104) λ | (cid:126)σ | µ (cid:105) (cid:126)σ = u (cid:104) λ | µ (cid:105) n + (cid:126)J ( λ, µ ) (cid:126)σ (2.54)for the higher-spin R-operator which acts on the tensor product of the spin n and spin repre-sentations. Such an R-operator is usually called the Lax operator with an ( n + 1)-dimensionallocal quantum space. Let us emphasize once more that (2.54) is a symbol of the Lax operatorsolely with respect to the local quantum space, but it is a matrix in the 2-dimensional auxiliaryspace. In order to avoid misunderstandings we showed in (2.53) its explicit matrix form. Theexpression (cid:104) λ | µ (cid:105) n is a symbol of the unit operator and (cid:126)J ( λ, µ ) is a symbol of the Lie algebragenerators. Hence the fusion procedure yields the familiar Lax operator,L( u ) = u
1l + (cid:126)J(cid:126)σ = (cid:18) u + J J − J + u − J (cid:19) . (2.55)The auxiliary spinors enabled us to reproduce this well-known result in a remarkably simple andexplicit way. They saved us from the need to construct projectors which single out irreduciblerepresentations and which are inevitable in the standard formulation.Now we are going to describe another way for deriving the L-operator (2.55) by means ofthe fusion procedure. The main reason to embark upon one more calculation is that it can begeneralized easily to the case of q -deformation (see Section 3.4) and, more importantly, to theelliptic deformation [10]. As before we deal with the symbols of finite-dimensional operators.The new ingredient is a factorization of the L-operator (cf. (2.15)). For calculating the symbolR( u | λ, µ ) of the “fused” R-matrices (2.50)R( u | λ, µ ) = (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) · · · (cid:104) λ | R( u − n + 1) | µ (cid:105) , (2.56)we choose the parametrization of the auxiliary spinor λ = − z , λ = 1 from the very beginning.Remind a realization of the spin generators as differential operators (cf. (2.48)) J + = z ∂ − z, J − = − ∂, J = z∂ − , which act in the two-dimensional space of linear functions ψ ( z ) = a z + a . In the basis e = − z , e = 1 of this space the matrices of the generators coincide with the Pauli-matrices J ± ( e , e ) = ( J ± e , J ± e ) = ( e , e ) σ ± ,J ( e , e ) = ( J e , J e ) = ( e , e ) σ . (2.57)Next we use the fusion procedure and derive the Lax operator (2.55) together with a represen-tation of the spin n generators (2.48) acting in the ( n + 1)-dimensional space of polynomials ψ ( z ) = a n z n + · · · + a .The symbol (cid:104) λ | R( u ) | µ (cid:105) of Yang’s R-matrix has been already found above (2.52), but nowwe are going to rewrite it in a different form. We represent it as a differential operator in thespinor variables acting on the identity operator symbol. Indeed, let us rewrite relations (2.57)in the equivalent form( − z, σ ± = J ± ( − z, , ( − z, σ = J ( − z, , (2.58)and use these formulae for calculating the symbol of Yang’s R-matrix (2.49) (cid:104) λ | R( u ) | µ (cid:105) = (cid:18) ( − z, (cid:0) u + + σ (cid:1) | µ (cid:105) ( − z, σ − | µ (cid:105) ( − z, σ + | µ (cid:105) ( − z, (cid:0) u + − σ (cid:1) | µ (cid:105) (cid:19) = (cid:18) u + z∂ − ∂z ∂ − z u + 1 − z∂ (cid:19) ( µ − µ z ) . rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 17We obtained the spin (cid:96) = L-operator (2.35) (with the shifted spectral parameter u → u + )acting on the symbol of the identity operator (cid:104) λ | µ (cid:105) = ( µ − µ z ). Then we observe that thissymbol can be cast in the factorized form (cid:104) λ | R( u ) | µ (cid:105) = (cid:18) z u + 1 (cid:19) (cid:18) − ∂ (cid:19) (cid:18) u − z (cid:19) ( µ − µ z ) | z = z , (2.59)which is easily checked by a direct calculation. Note that the factorization in (2.59) is slightlydifferent from (2.15) (at (cid:96) = ), since it involves a particular ordering of z and ∂ and such order-ing is compatible with the factorization of the L-operator up to the shift of spectral parameter.Then we consider the product of two consecutive symbols in (2.56) and profit a lot fromthe factorization (2.59) which provides cancellation of two adjacent matrix factors (which areunderlined in the following formula) (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) = (cid:18) z u + 1 (cid:19) (cid:18) − ∂ (cid:19) (cid:18) u − z (cid:19) (cid:18) z u (cid:19) × (cid:18) − ∂ (cid:19) (cid:18) u − − z (cid:19) ( µ − µ z ) ( µ − µ z ) | z = z = z = u (cid:18) z u + 1 (cid:19) (cid:18) − ∂ − ∂ (cid:19) (cid:18) u − − z (cid:19) ( µ − µ z ) ( µ − µ z ) | z = z = z . By now the generalization of the previous result to the product of n − (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) · · · (cid:104) λ | R( u − n + 1) | µ (cid:105) = r n ( u ) (cid:18) z u + 1 (cid:19) (cid:18) − ∂ − ∂ − · · · − ∂ n (cid:19) (cid:18) u − n + 1 0 − z (cid:19) × ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z . Further we multiply all matrices on the right-hand side of the previous formula and obtain r n ( u ) (cid:18) u − n + 1 + z ( ∂ + · · · + ∂ n ) − ∂ − · · · − ∂ n z ( ∂ + · · · + ∂ n ) − nz u + 1 − z ( ∂ + · · · + ∂ n ) (cid:19) × ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z = r n ( u ) (cid:18) u − n + 1 + z∂ − ∂z ∂ − nz u + 1 − z∂ (cid:19) ( µ − µ z ) n , where on the last step we use an obvious formula( ∂ + · · · + ∂ n ) ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z = ∂ ( µ − µ z ) n . The final result for the symbol (2.56) of the “fused” Yang R-matrices isR( u | λ, µ ) = r n ( u ) (cid:18) u + 1 − n + J J − J + u + 1 − n − J (cid:19) ( µ − µ z ) n , where the generators J ± , J for the representation of spin n are given by (2.48).Factorization of the L-operator plays an important role in the construction of the generalR-operator for deformed [13, 16] and non-deformed [13] rank 1 symmetry algebra, as well as inthe higher rank case [15]. Here we see that it finds a natural place in the fusion construction aswell.8 D. Chicherin, S.E. Derkachov and V.P. Spiridonov , C ) The fusion procedure enables one to produce even more intricate sl -symmetric solutions of theYBE. Starting from the L-operator acting in the tensor product of spin and spin (cid:96) repre-sentations (we assume (cid:96) 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 and spin (cid:96) representations. Since we are mainly interested in the R-operators which are invariant withrespect to the SL(2 , C ) group, we will thoroughly study how the fusion procedure applies in thiscase.As before we profit a lot from the auxiliary spinors notation. However, from now on theholomorphic and antiholomorphic sectors are present and we need to introduce a pair of auxiliaryspinors λ i , ¯ λ ¯ i . They are independent variables not related by the complex conjugation. Thuswe introduce a pair of scalar objects (without spinor indices)Λ( u, λ, µ ) = λ i L ji ( u ) µ j , ¯Λ(¯ u, ¯ λ, ¯ µ ) = ¯ λ ¯ i ¯L ¯ j ¯ i (¯ u )¯ µ ¯ j , (2.60)which are linear combinations of the L-operators’ entries (2.15), (2.16). An easy calculationshows that Λ( u, λ, µ ) = − ( λ + λ z )( µ − µ z ) ∂ + u λ ( µ − µ z ) − ( u + 1) µ ( λ + λ z )= − ( λ + λ z ) u +1 ( µ − µ z ) − u · ∂ · ( λ + λ z ) − u ( µ − µ z ) u +1 . (2.61)The factorized expression (2.61) looks much like formula (2.15). Indeed in both expressions thedifferential operators are sandwiched between some multiplication by a function operators. Theanalogous relation takes place for ¯L (2.16). Then we multiply a number of Λ-operators withshifted spectral parameters to form a Λ-stringΛ( u )Λ( u − · · · Λ( u − n + 1) = ( − n ( λ + λ z ) u +1 ( µ − µ z ) − u × (cid:0) ∂ ( µ − µ z ) (cid:1) n · ( λ + λ z ) − u + n − ( µ − µ z ) u − n == ( − n ( λ + λ z ) u +1 ( µ − µ z ) − u + n − · ∂ n · ( λ + λ z ) − u + n − ( µ − µ z ) u +1 . Here we apply the formula which can be easily proven by induction, (cid:0) ∂ ( µ − µ z ) (cid:1) n = ( µ − µ z ) n − ∂ n ( µ − µ z ) n +1 . Then we take into account the anti-holomorphic sector and form the product of Λ- and¯Λ-strings resulting in the symbol for a higher-spin R-operatorR fus ( u, ¯ u | λ, ¯ λ, µ, ¯ µ ) = Λ( u )Λ( u − · · · Λ( u − n + 1) ¯Λ(¯ u ) ¯Λ(¯ u − · · · ¯Λ(¯ u − ¯ n + 1) (2.62)= ( − n +¯ n [ λ + λ z ] u +1 [ µ − µ z ] − u + n − · [ ∂ z ] n · [ λ + λ z ] − u + n − [ µ − µ z ] u +1 . This R-operator acts in the tensor product of the infinite-dimensional representation specifiedby the spins (cid:96) , ¯ (cid:96) and the finite-dimensional representation with the spins n , ¯ n . Let us remindthat R fus is a symbol with respect to the first finite-dimensional space only, but it is a differentialoperator in the second infinite-dimensional space. Evidently, the right-hand side of (2.62) ispolynomial in λ and µ , as it should be. In order to reconstruct the operator itself from itssymbol we resort to the rule (2.42). More precisely, we apply the corresponding relation toa function Φ( λ, ¯ λ | z, ¯ z ), which is homogeneous in λ and ¯ λ of the homogeneity degree n and ¯ n ,respectively,[R fus ( u, ¯ u )Φ] ( λ, ¯ λ | z, ¯ z ) = R fus ( u, ¯ u | λ, ¯ λ, ∂ µ , ∂ ¯ µ )Φ( µ, ¯ µ | z, ¯ z ) (cid:12)(cid:12) µ =¯ µ =0 . (2.63) We should note that the idea to reformulate the fusion procedure with the help of such Λ-operators belongsto D. Karakhanyan. rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 19We stress that the fusion formulae (2.62), (2.63) are somewhat different from the standard ones.We find them better adapted for applications.The higher-spin R-operator (2.63) obtained by means of the fusion is identical with thereduction of the general R-operator calculated in (2.32), which will be demonstrated shortly.First of all the operator form of the star-triangle relation (2.28) enables one to rewrite thesymbol (2.62) as followsR fus ( u, ¯ u | λ, ¯ λ, µ, ¯ µ ) = ( − n +¯ n [ λ + λ z ] u +1 × [ ∂ z ] u +1 · [ µ − µ z ] n · [ ∂ z ] − u + n − [ λ + λ z ] − u + n − . (2.64)We have seen above that ( λ + λ x ) n is a generating function of the ( n + 1)-dimensional Jordan–Schwinger representation of sl . Its generalization to the group SL(2 , C ) is straightforward:[ λ + λ x ] n (see (2.9)) is a generating function of the ( n + 1)(¯ n + 1)-dimensional representationrealized in the space of homogeneous functions Ψ( λ, ¯ λ ),Ψ( λ, ¯ λ ) = Ψ( λ , λ , ¯ λ , ¯ λ ) , Ψ( αλ , αλ , ¯ α ¯ λ , ¯ α ¯ λ ) = α n ¯ α ¯ n Ψ( λ, ¯ λ ) . Then we act by R fus on the generating function according to (2.63) and choose the symbol inthe form (2.64). At the same time we do not act by the R fus -operator on any function in itssecond space. At this point we take into account that[ ∂ µ − ∂ µ z ] n [ µ + µ x ] n = n !¯ n ![ x − z ] n (2.65)and obtainR fus (cid:0) u + n , ¯ u + ¯ n (cid:1) [ λ + λ x ] n = n !¯ n ![ λ + λ z ] u + n + (cid:96) +1 [ ∂ z ] u + n − (cid:96) [ z − x ] n [ ∂ z ] − u + n + (cid:96) [ λ + λ z ] − u + n − (cid:96) − = n !¯ n ![ λ + λ z ] u + n + (cid:96) +1 [ z − x ] − u + n + (cid:96) [ ∂ z ] n [ z − x ] u + n − (cid:96) [ λ + λ z ] − u + n − (cid:96) − . (2.66)Here we profited from the star-triangle relation (2.28) at the last step. In order to comparethe reduction of the general R-operator (2.32) with the expression (2.66) following from thefusion formula (2.62) we just need to pass from the Jordan–Schwinger representation to thestandard representation of SL(2 , C ) (in the space of functions of one complex variable) describedin Section 2.1. Consequently we choose λ = − z , ¯ λ = − ¯ z , λ = ¯ λ = 1 and denote z = z ,¯ z = ¯ z . 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 thespaces) R-operators give identical results, and the general R-operator (2.22) contains all solutionsof the Yang–Baxter equation associated with the principal series representations of the SL(2 , C )group. Using the patterns of the previous sections, in the following we show that all described construc-tions for the group SL(2 , C ) can be straightforwardly adapted to the modular double. Addition-ally, we construct corresponding finite-dimensional solutions of the YBE using the fusion. The modular double of U q ( sl ) was introduced by Faddeev in [18]. This algebra is formed bytwo sets of generators E , F , K and (cid:101) E , (cid:101) F , (cid:101) K . The usual commutation relations for E , F , K which generate U q ( sl ) with q = e iπτ ( τ ∈ C and it is not a rational number)[ E , F ] = K − K − q − q − , KE = q EK , KF = q − FK , (3.1)0 D. Chicherin, S.E. Derkachov and V.P. Spiridonovare supplemented by similar relations for (cid:101) E , (cid:101) F , (cid:101) K with the deformation parameter (cid:101) q = e iπ/τ .The generators E and F commute with (cid:101) E and (cid:101) F . The generator K anti-commutes with (cid:101) E and (cid:101) F , (cid:101) K anti-commutes with E and F while K commutes with (cid:101) K .For particular representations of the modular double see [4, 18, 20, 35, 40] and referencestherein. We use the parametrization τ = ω (cid:48) ω , where ω and ω (cid:48) are complex numbers with thepositive imaginary parts, Im ω >
0, Im ω (cid:48) >
0, satisfying the normalization condition ωω (cid:48) = − .Then q = exp (cid:0) iπω (cid:48) /ω (cid:1) , (cid:101) q = exp (cid:0) iπω/ω (cid:48) (cid:1) , and the change q (cid:29) (cid:101) q is equivalent to ω (cid:29) ω (cid:48) . We denote also ω (cid:48)(cid:48) = ω + ω (cid:48) , β = π (cid:18) ωω (cid:48) + ω (cid:48) ω (cid:19) . (3.2)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 thespace of entire functions rapidly decaying at infinity along contours parallel to the real line. Thisrepresentation is parameterized by one complex parameter s called the spin , and the generatorshave the following explicit form [4, 5, 6] K s = e − iπ ω ˆ p , (cid:0) q − q − (cid:1) E s = e iπxω (cid:104) e − iπ ω (ˆ p − s − ω (cid:48)(cid:48) ) − e iπ ω (ˆ p − s − ω (cid:48)(cid:48) ) (cid:105) , (3.3) (cid:0) q − q − (cid:1) F s = e − iπxω (cid:104) e iπ ω (ˆ p + s + ω (cid:48)(cid:48) ) − e − iπ ω (ˆ p + s + ω (cid:48)(cid:48) ) (cid:105) , where ˆ p denotes a momentum operator in the coordinate representation ˆ p = πi ∂ x . The formulaefor generators (cid:101) K s , (cid:101) E s , (cid:101) F s are obtained by a simple interchange ω (cid:29) ω (cid:48) in (3.3).The modular double is associated with two basic special functions. The first one is thenon-compact quantum dilogarithm which has the following integral representation γ ( z ) = exp − + ∞ (cid:90) −∞ dtt e itz sin( ωt ) sin( ω (cid:48) t ) , (3.4)where the contour goes above the singularity at t = 0. In the context of quantum integrablesystems it has been found first in [19]. Some basic formulae for γ ( z ) can be found in [20, 42].This function respects a pair of finite-difference equations of the first order and the reflectionrelation γ ( z + ω (cid:48) ) γ ( z − ω (cid:48) ) = 1 + e − iπω z , γ ( z + ω ) γ ( z − ω ) = 1 + e − iπω (cid:48) z , γ ( z ) γ ( − z ) = e iβ e iπz . (3.5)One can interpret 2 ω and 2 ω (cid:48) as some quasiperiods of the quantum dilogarithm.The second function we need is D a ( z ) = e − πiaz γ ( z + a ) γ ( z − a ) . (3.6)In fact it coincides with the Faddeev–Volkov R-matrix [2, 43]. Some relations for this functionsare presented in [5]. It naturally arises when one looks for the intertwining operator of equivalentrepresentations of the modular double [35], and it serves as the main building block in theconstruction of a general R-matrix as an integral operator [5, 6]. This general R-operator isrom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 21a product of four Faddeev–Volkov’s R-matrices. The function D a ( z ) obeys simple reflectionrelations D a ( z ) = D a ( − z ) , D a ( z ) D − a ( z ) = 1 , (3.7)and a pair of finite-difference equations of the first order D a ( z − ω (cid:48) ) D a ( z + ω (cid:48) ) = cos π ω ( z − a )cos π ω ( z + a ) , D a ( z − ω ) D a ( z + ω ) = cos π ω (cid:48) ( z − a )cos π ω (cid:48) ( z + a ) . (3.8)Note that the functions γ ( z ) and D a ( z ) are symmetric with respect to ω and ω (cid:48) .A generalization of the Faddeev–Volkov model still associated with the γ ( z )-function wasfound in [38]. It leads to a more general R-operator than we consider here [11], which can beobtained as a limit from the most complicated known R-operator derived in [16].Now we proceed to finite-dimensional representations of the modular double. In order to fixthe spin s such that a finite-dimensional representation decouples from the infinite-dimensionalrepresentation π s we resort to the intertwining operator of equivalent representations of themodular double. It is known that the representations π s and π − s are equivalent. The cor-responding intertwining operator [35] is expressed in terms of the special function (3.6), suchthat D − s (ˆ p ) K s = K − s D − s (ˆ p ) , D − s (ˆ p ) E s = E − s D − s (ˆ p ) , D − s (ˆ p ) F s = F − s D − s (ˆ p ) , (3.9)where ˆ p is the momentum operator. There are analogous relations for (cid:101) E , (cid:101) F , (cid:101) K , since the D -function is invariant with respect to the permutation of ω and ω (cid:48) . The latter relations canbe easily checked using equations (3.8). Applying the Fourier transformation of the D -func-tion (3.6) [5, 20, 42] A ( a ) + ∞ (cid:90) −∞ dt e πitz D a ( t ) = D − ω (cid:48)(cid:48) − a ( z ) , (3.10) A ( a ) ≡ e iπ (2 a + ω (cid:48)(cid:48) ) + iβ γ (2 a + ω (cid:48)(cid:48) ) , A ( a ) A ( − a − ω (cid:48)(cid:48) ) = 1 , we immediately represent the intertwiner D − s (ˆ p ) as an integral operator (in analogy with (2.5)) D − s (ˆ p )Φ( x ) = A ( s − ω (cid:48)(cid:48) ) + ∞ (cid:90) −∞ dx (cid:48) D s − ω (cid:48)(cid:48) ( x − x (cid:48) )Φ( x (cid:48) ) . (3.11)Thus the intertwiner admits two forms for generic values of s : as a formal function of themomentum operator and a well-defined integral operator.A finite ( n + 1)( m + 1)-dimensional representation decouples from the infinite-dimensionalone for special values of the spin s = s := − ω (cid:48)(cid:48) − nω − mω (cid:48) , n, m ∈ Z ≥ , where the integers n and m enumerate the points of a quarter-infinite lattice on the complexplane (or a line, for real ω/ω (cid:48) ). Such finite-dimensional representations emerged first in thetwo-dimensional rational conformal field theory through 6 j -symbols for the modular doublewith q = e πim/ ( m +1) and ˜ q = e πi ( m +1) /m , m ∈ Z [22]. The most general two-index 6 j -symbolswere discovered in the theory of elliptic hypergeometric functions [36]. They are described bymeromorphic functions of one variable satisfying the biorthogonality relation with an absolutely2 D. Chicherin, S.E. Derkachov and V.P. Spiridonovcontinuous measure, which has the form usually ascribed to the functions of two independentvariables. The latter property brought to the theory of spectral problems the new notion oftwo-index (bi-)orthogonality.For the elliptic modular double the two-index finite-dimensional representations were dis-covered in [16, 17]. In principle, finite-dimensional representations of the Faddeev modulardouble can be derived as certain limits from this elliptic construction, but we give here an in-dependent consideration and, moreover, describe the finite-dimensional R-matrices analogousto (2.38). Note that in the SL(2 , C ) group case the finite-dimensional representations were alsoparametrized by a pair of non-negative integer numbers n and ¯ n , but the integer ¯ n has a differentnature emerging from a discretization of the separate spin variable ¯ s , which is absent in our case.In order to find finite-dimensional representations of interest we investigate the null-space ofthe intertwiner. We take the formal operator identity D − s (ˆ p ) D s (ˆ p ) = 1l , (3.12)which is a consequence of the reflection formula (3.7), and rewrite it in an equivalent formsubstituting D s (ˆ p ) and 1l for their kernels (see (3.11)) D − s (ˆ p ) D − s − ω (cid:48)(cid:48) ( x − y ) = A − ( − s − ω (cid:48)(cid:48) ) δ ( x − y ) . (3.13)Then we note that zeros of the quantum dilogarithm γ ( z ) = 0 are located at z = ω (cid:48)(cid:48) +2 nω +2 mω (cid:48) , n, m ∈ Z ≥ , which indicates that the relation (3.12) is broken down at the corresponding points.Consequently for the spin values specified above, s = s , the right-hand side of (3.13) vanishesand a nontrivial null-space of D − s (ˆ p ) arises D − s (ˆ p ) D nω + mω (cid:48) ( x − y ) = 0 , n, m ∈ Z ≥ . The latter formula is a deformed analogue of (2.14). From the intertwining relations (3.9) thenull-space is seen to be invariant under the action of the modular double generators. Corre-sponding representation is finite-dimensional as we will see shortly. The generators are fixed byexpressions (3.3) and their modular duals with the spin parameter s = s . One can also showthat the corresponding representation is irreducible.In this way we have found that D nω + mω (cid:48) ( x − y ) is the generating function of a finite-dimensional representation containing all its basis vectors. Here y is an auxiliary parameter,which is convenient to write in the exponential form Y = Y ( y ) = e iπ ω y , (cid:101) Y = (cid:101) Y ( y ) = e iπ ω (cid:48) y . (3.14)Since we assume that the quasiperiods are incommensurate (i.e., that τ = ω (cid:48) ω is not a rationalnumber), the auxiliary variables Y and (cid:101) Y are multiplicatively incommensurate for generic y (i.e., if (cid:101) Y k Y l = 1 for some integer k and l , then k = l = 0). Using the finite-difference equations(3.8) our generating function can be rewritten as a finite product D nω + mω (cid:48) ( x − y ) = n − (cid:89) k =0 (cid:0) (cid:101) Y − (cid:101) X ˜ q n − − k + (cid:101) Y (cid:101) X − ( − m ˜ q − n − + k (cid:1) × m − (cid:89) l =0 (cid:0) Y − Xq m − − l + Y X − ( − n q − m − + l (cid:1) , (3.15)where we use the shorthand notation X = X ( x ) = e iπ ω x , (cid:101) X = (cid:101) X ( x ) = e iπ ω (cid:48) x . (3.16)rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 23Expanding the Laurent polynomial (3.15) in integer powers of Y ( y ) and (cid:101) Y ( y ) we extract( n + 1)( m + 1) basis elements of the finite-dimensional representation given by the monomials (cid:101) X n − k X m − l with k = 0 , , . . . , n, l = 0 , , . . . , m. (3.17)Let us note that for s = s the integral in (3.11) diverges. The divergence is compensated bythe normalization factor, which turns to zero, A ( s − ω (cid:48)(cid:48) ) = 0. The ambiguity can be resolved andfor finite-dimensional representations the intertwiner D − s (ˆ p ) becomes a sum of finite-differenceoperators which follows from (3.8). One can directly check as well that the basis vectors (3.17)are annihilated by D − s (ˆ p ). Now we proceed to integrable structures for the modular double. The L-operator is constructedout of the modular double generators taken in the representation π s (3.3) [5],L( u | s ) = (cid:32) e iπω u K s − e − iπω u K − s ( q − q − ) F s ( q − q − ) E s e iπω u K − s − e − iπω u K s (cid:33) . (3.18)This L-operator respects the standard RLL-intertwining relation (cf. (2.18))R ab,ef ( u − v )L ec ( u )L fd ( v ) = L bf ( v )L ae ( u )R ef,cd ( u − v )with 4 × u ) = sin iπω ( u + ω (cid:48) )2 sin iπω (cid:48) ω ⊗
1l + σ ⊗ σ + σ ⊗ σ + cos iπω ( u + ω (cid:48) )2 cos iπω (cid:48) ω σ ⊗ σ , (3.19)which is equivalent to the set of commutation relations (3.1). The second L-operator is obtainedfrom L( u ) by the interchange ω (cid:29) ω (cid:48) : (cid:101) L( u ) = L( u ) | ω (cid:29) ω (cid:48) . The same is true for the corresponding (cid:101) R-matrix. In the following we indicate formulae only for the L-operator (3.18), and all relationsfor the (cid:101)
L-operator have the same form with ω (cid:29) ω (cid:48) .The L-operator (3.18) can be represented in the factorized formL( u | s ) = (cid:32) U − U − − U − e iπω x U e iπω x (cid:33) (cid:32) e − iπ ω ( p − ω (cid:48)(cid:48) ) e iπ ω ( p − ω (cid:48)(cid:48) ) (cid:33) (cid:32) − U U − e − iπω x − U − U e − iπω x (cid:33) , (3.20) U = e iπ ω u , U = e iπ ω u , where we introduced the “light-cone” parameters u and u instead of u and su = u + s + ω − ω (cid:48) , u = u − s + ω − ω (cid:48) . (3.21)In the notation L( u ) we omit for simplicity dependence on the spin parameter s . Factoriza-tion of the L-operator of the XXZ spin chain has been introduced in [3] in relation to thechiral Potts models. The factorization formula (3.20) is completely analogous to formula (2.15)for the SL(2 , C ) group. The same can be said about spectral parameters u , u in (2.17)and (3.21). However, although the operators L( u ) , (cid:101) L( u ) for the modular double look as analoguesof L( u ), ¯L(¯ u ) 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 generalR-operator acts in the tensor product of two infinite-dimensional representations π s ⊗ π s (3.3).It has been found first in [5], but the corresponding form of the R-operator is not suitable for4 D. Chicherin, S.E. Derkachov and V.P. Spiridonovour purposes. Here we profit from another construction implemented in [6], where it has beenobtained solving a pair of RLL-relations (cf. (2.19), (2.20)),R ( u − v )L ( u | s )L ( v | s ) = L ( v | s )L ( u | s )R ( u − v ) , (3.22)R ( u − v ) (cid:101) L ( u | s ) (cid:101) L ( v | s ) = (cid:101) L ( v | s ) (cid:101) L ( u | s )R ( u − v ) . (3.23)The spin parameters s , s and the spectral parameters u , v appearing in the RLL-rela-tions (3.22), (3.23) are combined to four “light-cone” parameters u , u , v , v in accordancewith (3.21), i.e., u = u + s + ω − ω (cid:48) , u = u − s + ω − ω (cid:48) ,v = v + s + ω − ω (cid:48) , v = v − s + ω − ω (cid:48) . The notation R ( u − v ) is a shortened version of R ( u − v | s , s ) or R ( u , u | v , v ) takinginto account the spin parameters.The R-operator is invariant with respect to the modular double, i.e., it commutes with theco-product of the generators. More precisely,[R( u ) , ∆( K )] = [R( u ) , ∆( E )] = [R( u ) , ∆( F )] = 0 , [R( u ) , ∆( (cid:101) K )] = [R( u ) , ∆( (cid:101) E )] = [R( u ) , ∆( (cid:101) F )] = 0 , (3.24)where we abbreviate the co-product taken in the tensor of representations with spins s and s ,( π s ⊗ π s ) ◦ ∆ , to ∆ bearing in mind the specified representations. The co-product is given bythe formulae∆( K ) = K K , ∆( E ) = E K + K − E , ∆( F ) = F K + K − F . Analogous relations take place for (cid:101) E , (cid:101) F , (cid:101) K . The invariance (3.24) follows straightforwardly fromthe RLL-relations (3.22), (3.23) subject to the shift of spectral parameters u → u + w , v → v + w with arbitrary w .Construction of the general R-operator for the modular double from [6] follows the samepattern as for the SL(2 , C ) group. It is based on the elementary intertwining operators thatyield an integral operator representation of the symmetric group S . Now we repeat witha slight modification what has been said in Section 2.2 about the SL(2 , C )-invariant R-operator.The general R-operator is a product of four factors which are elementary intertwining operatorsR ( u − v ) = D u − v ( x ) D u − v (ˆ p ) D u − v (ˆ p ) D u − v ( x ) . (3.25)Here we denote x ij = x i − x j . The latter formula has to be compared with (2.22) whichhas the same structure, only the building blocks are different. The change ω (cid:29) ω (cid:48) does notalter the R-operator (3.25) which satisfies both RLL-relations (3.22) and (3.23). Similar tothe expression (2.22), the representation (3.25) for our infinite-dimensional R-operator playsa major role in what follows. In the next section we find its reductions to finite-dimensionalinvariant subspaces.According to (3.25) the general R-operator is a product of four Faddeev–Volkov’s R-matri-ces [43]. Applying (3.11) one can rewrite it explicitly as an integral operator. Let us note thatit is not the only possible form of the R-operator. Initially constructed in [5] the R-operatorfor the modular double was obtained in the form which is not convenient enough to address thecurrent problem. In [5] the R-operator appeared in disguise of the D -function (3.6) and arcoshof the Casimir operator. Thus, dealing with such an operator, one has to decompose tensorproducts to a sum of irreducible representations and use the Clebsch–Gordan coefficients [35].The R-operator in the form (3.25) has the virtue of not demanding any auxiliary information.rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 25In order to justify the chosen terminology for the R-operator factors in (3.19), we indicatehere the relations D u − u (ˆ p )L ( u , u ) = L ( u , u ) D u − u (ˆ p ) ,D v − v (ˆ p )L ( v , v ) = L ( v , v ) D v − v (ˆ p ) , (3.26) D u − v ( x )L ( u , u )L ( v , v ) = L ( v , u )L ( v , u ) D u − v ( x ) , (3.27)which have a clear meaning in terms of the permutation group S and which enable us to checkthe RLL-relations (3.22) and (3.23). Similar to (2.24) and (2.25), here we assume that the D -operators are acting as 2 × D a (ˆ p k ) D a + b ( x ) D b (ˆ p k ) = D b ( x ) D a + b (ˆ p k ) D a ( x ) , k = 1 , . (3.28)Using these relations one can check [6] that the R-operator (3.25) satisfies the YBER ( u − v )R ( u )R ( v ) = R ( v )R ( u )R ( u − v ) . (3.29)Both sides in the latter relation are endomorphisms on the space π s ⊗ π s ⊗ π s . For brevitywe do not indicate dependence on the spin parameters.The Coxeter relations (3.28) are equivalent to the star-triangle relation [43] which has threemanifestations:1) an integral identity [5, 40, 42] A ( a ) A ( b ) A ( c ) + ∞ (cid:90) −∞ dzD a ( z − z ) D b ( z − z ) D c ( z − z )= D − ω (cid:48)(cid:48) − a ( z − z ) D − ω (cid:48)(cid:48) − b ( z − z ) D − ω (cid:48)(cid:48) − c ( z − z ) with a + b + c = − ω (cid:48)(cid:48) ;2) a particular point in the image of the operator D − a − ω (cid:48)(cid:48) (ˆ p ) (with the same restriction onthe parameters as before) D − a − ω (cid:48)(cid:48) (ˆ p ) (cid:0) D b ( z ) D c ( z ) (cid:1) = D − ω (cid:48)(cid:48) − a ( z ) D − ω (cid:48)(cid:48) − b ( z ) D − ω (cid:48)(cid:48) − c ( z ) A ( b ) A ( c ) ; (3.30)3) an operator identity D a (ˆ p ) D a + b ( x ) D b (ˆ p ) = D b ( x ) D a + b (ˆ p ) D a ( x ) . (3.31) Now we have all ingredients at hand to perform a reduction of the described R-operator formodular double to a finite-dimensional representation in one of its tensor factors. The calculationfollows precisely the same pattern as in the SL(2 , C ) case (see Section 2.3). Again the principalpossibility of this reduction is based on the following relation for the R-operator (3.25) D u − u (ˆ p ) R ( u , u | v , v ) = R ( u , u | v , v ) D u − u (ˆ p ) , (3.32)which can be proved using the identity (3.28). Here, again, R = P R , where P is a per-mutation operator. This relation shows that both, the null-space of the intertwining opera-tor D u − u (ˆ p ) and the image of the intertwining operator D u − u (ˆ p ), are mapped onto them-selves by our R-matrix R . Therefore the invariant finite-dimensional subspaces of the null-space are invariant with respect to the action of the R-operator as well.6 D. Chicherin, S.E. Derkachov and V.P. SpiridonovWe consider the R-operator R ( u | s , s ) acting on the tensor product π s ⊗ π s and introducethe “light-cone” parameters (see (3.21)) u = u + s + ω − ω (cid:48) , u = u − s + ω − ω (cid:48) ,v = s + ω − ω (cid:48) , v = − s + ω − ω (cid:48) . (3.33)We apply the R-operator to the function D − ω (cid:48)(cid:48) + u − u ( x ) in the first space. For s = − ω (cid:48)(cid:48) − nω − mω (cid:48) , n, m = 0 , , , . . . , the latter function becomes a generating function of the finite-dimensional representation in the first space. However, for a moment s is assumed to begeneric. According to the structure of R-operator (3.25) we consider sequential action of itsseparate factors. As the first step, we apply D u − v (ˆ p ) D u − v ( x ) to D − ω (cid:48)(cid:48) + u − u ( x )Φ( x )and, using formula (3.30), obtain D u − v (ˆ p ) D u − v ( x ) D − ω (cid:48)(cid:48) + u − u ( x )Φ( x )= A ( u − u ) A ( u − v ) · D u − u ( x ) D − ω (cid:48)(cid:48) + v − u ( x ) D u − v ( x )Φ( x ) . (3.34)Further we apply the third factor D u − v (ˆ p ) of the R-operator to both sides of this relation.On the right-hand sides we use the relation D u − v (ˆ p ) D u − u ( x ) D u − v ( x )Φ( x )= A − ( u − v ) A − ( u − u ) · D u − u − ω (cid:48)(cid:48) (ˆ p ) D v − u − ω (cid:48)(cid:48) ( x ) D u − v ( x )Φ( x ) , which can be easily checked taking into account the integral form of the intertwiner (3.11). Ina full analogy with the SL(2 , C ) calculation we traded a complicated integral operator D u − v (ˆ p )for D u − u − ω (cid:48)(cid:48) (ˆ p ) which turns to D nω + mω (cid:48) (ˆ p ) in the finite-dimensional setting. The latteroperator is just a sum of the finite-difference operators which follows from equations (3.8). Thesubstitution x − y → ˆ p in (3.15) yields an explicit expression for D nω + mω (cid:48) (ˆ p ), D nω + mω (cid:48) (ˆ p ) = n − (cid:89) k =0 (cid:16) e iπ ω (cid:48) ˆ p ˜ q n − − k + e − iπ ω (cid:48) ˆ p ( − m ˜ q − n − + k (cid:17) × m − (cid:89) l =0 (cid:16) e iπ ω ˆ p q m − − l + e − iπ ω ˆ p ( − n q − m − + l (cid:17) . (3.35)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 (3.25) acting on the tensor product oftwo infinite-dimensional representations π s ⊗ π s can be reduced to a finite-dimensional repre-sentation in the first space for s = − ω (cid:48)(cid:48) − nω − mω (cid:48) , n, m ∈ Z ≥ . It acts on the generatingfunction of finite-dimensional representation (3.15) according to the following explicit formula R ( u | s , s ) · D nω + mω (cid:48) ( x )Φ( x ) = c · D u − s − s ( x ) (3.36) × D − u − s − s − ω (cid:48)(cid:48) ( x ) · D nω + mω (cid:48) (ˆ p ) · D − u − s + s − ω (cid:48)(cid:48) ( x ) D u − s + s ( x )Φ( x ) , where the normalization factor is c − = A (cid:0) u + s + s (cid:1) A (cid:0) u + s − s (cid:1) and x is an auxiliary parameter.Both sides of the equality (3.36) can be expanded in integer powers of the variables X ( x ), (cid:101) X ( x ) (see (3.16)). This yields simultaneously an expansion in integer powers of the variablesrom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 27 X ( x ), (cid:101) X ( x ) (see (3.17)), which form a basis of the finite-dimensional representation of in-terest. The resulting formula (3.36) is very helpful in applications. We use it as follows. Firstly,finite-difference operators in the sum D nω + mω (cid:48) (ˆ p ) act from the left on the D -functions andshift their arguments. After these shifts we trade all D -functions (3.6) in (3.36) for quantumdilogarithms (3.4) and apply the finite-difference equations (3.5). In this way we completelyget rid off the quantum dilogarithms. The final result contains only trigonometric functions,i.e., a linear combination of the products of X ( x ), (cid:101) X ( x ), X ( x ), (cid:101) X ( x ), and Φ( x ) withthe shifted argument. Thus the restriction of the general R-operator can be represented asan ( n + 1)( m + 1)-dimensional matrix whose entries are finite-difference operators with thetrigonometric coefficients.Formula (3.36) constitutes one of the main results of this paper. It gives a new rich class ofsolutions of the YBE which are endomorphisms on a tensor product of finite-dimensional andinfinite-dimensional representations of the modular double specified in (3.3).In [32, 33] an explicit hypergeometric formula for the R-matrix of U q ( sl ) acting on a tensorproduct of two highest-weight representations has been presented. It would be interesting torelate our formula (3.36) to R-matrices from these papers. In [27] group theoretical origins ofsimilar factorization formulae were elucidated from the representation theory of U q ( ˆ sl ) algebra.In order to demonstrate how formula (3.36) works in practice we recover the L-operator (3.18)out of the R-operator (3.25). With this task in mind, we choose the spin s = − ω (cid:48) − ω (cid:48)(cid:48) , i.e., fix n = 0, m = 1. The generating function (3.15) of the 2-dimensional representation in the firstspace is D ω (cid:48) ( x ) = e iπ ω x + e − iπ ω x . Consequently e = e iπ ω x , e = e − iπ ω x form a basis of C . The finite-difference operatorin (3.36) is D ω (cid:48) (ˆ p ) = e iπ ω ˆ p + e − iπ ω ˆ p . Up to a normalization factor the right-hand side of (3.36)takes the formcosh iπ ω (cid:0) x − u + s − ω (cid:1) · cosh iπ ω (cid:0) x − u − s − ω − ω (cid:48) (cid:1) · Φ( x − ω (cid:48) )+ cosh iπ ω (cid:0) x + u − s + ω (cid:1) · cosh iπ ω (cid:0) x + u + s + ω + ω (cid:48) (cid:1) · Φ( x + ω (cid:48) )Expanding this function in terms of X ± ( x ) X ± ( x ) = e iπ ω ( ± x ± x ) we obtain R (cid:0) u − ω − ω (cid:48) (cid:1) e = c · (cid:2) e (cid:0) e iπω u K s − e − iπω u K − s (cid:1) + e (cid:0) q − q − (cid:1) E s (cid:3) , R (cid:0) u − ω − ω (cid:48) (cid:1) e = c · (cid:2) e (cid:0) q − q − (cid:1) F s + e (cid:0) e iπω u K − s − e − iπω u K s (cid:1)(cid:3) . Thus we have reproduced the desired result (3.18). In a similar way one reproduces the (cid:101)
L-operator at s = − ω − ω (cid:48)(cid:48) . Implementing these reduced R-operators in the YBE (3.29) onerecovers the RLL-relations (3.23). An explicit matrix factorization formula for R ( u | s , s )generalizing the L-operator factorization (3.20) was derived in the followup paper [7].A reduction of the R-operator to finite-dimensional representations in both spaces can beconstructed as well. One just should choose an appropriate discrete value of the spin s in (3.36)and substitute Φ( x ) for the corresponding generating function. In this way one generatesa number of finite-dimensional solutions of the YBE including the trigonometric R-matrix (3.19)among them. More precisely, let us fix the spin parameters as s = − ω (cid:48)(cid:48) − n ω − m ω (cid:48) , n , m ∈ Z ≥ , and s = − ω (cid:48)(cid:48) − n ω − m ω (cid:48) , n , m ∈ Z ≥ , in the first and second spaces, respectively.Then R ( u | s , s ) · D n ω + m ω (cid:48) ( x ) D n ω + m ω (cid:48) ( x ) = c · D u − s − s ( x ) D − u − s − s − ω (cid:48)(cid:48) ( x ) × D n ω + m ω (cid:48) (ˆ p ) · D − u − s + s − ω (cid:48)(cid:48) ( x ) D u − s + s ( x ) D n ω + m ω (cid:48) ( x ) (3.37)8 D. Chicherin, S.E. Derkachov and V.P. Spiridonovis a concise expression for the finite-dimensional (in both spaces) R-matrix. After expansionwith respect to auxiliary parameters X ( x ), (cid:101) X ( x ), X ( x ), and (cid:101) X ( x ) it can be rewrittenexplicitly is the form of an ( n + 1)( m + 1)( n + 1)( m + 1)-dimensional matrix.The integral R-operator exists as well for the U q ( sl )-algebra [13], which is a “one-half” of themodular double. A reduction of this R-operator leads to the trigonometric L-operator as wasshown in [9]. Derivation of the corresponding higher-spin finite-dimensional solutions of YBEusing the described reduction procedure will be presented elsewhere. U q ( sl ) algebra Now we would like to show that the reduction result of the previous section can be derived withthe help of the fusion procedure. In the present section we develop the fusion for the quantumalgebra U q ( sl ) and in the next one we consider the modular double. Our approach is not thatwell known since we extensively use the symbols of operators.Similar to the discussion in Section 2.5 we construct the Lax operator with a finite-dimensio-nal local quantum space out of the q -deformed Yang’s R-matrix (remind that the deformationparameter and quasiperiods are related as q = e iπω/ω (cid:48) ). The latter acts on the tensor productof two fundamental representations and is given by the matrixR( u ) = 12 q u + − q − u − q − q −
1l + 12 σ ⊗ σ + 12 σ ⊗ σ + 12 q u + + q − u − q + q − σ ⊗ σ = (cid:32)(cid:2) u + + σ (cid:3) q σ − σ + (cid:2) u + − σ (cid:3) q (cid:33) , (3.38)where [ x ] q is the q -number [ x ] q = q x − q − x q − q − . Here σ ± = ( σ ± iσ ) / q -number of thematrix σ is defined in an evident way, since it is diagonal.In a full analogy with the non-deformed case, the recipe of [28, 29] suggests to form an inho-mogeneous monodromy matrix out of the q -deformed Yang’s R-matrices and to symmetrize it,R ( j ...j n )( i ...i n ) ( u ) := Sym R j i ( u )R j i ( u − · · · R j n i n ( u − n + 1) , (3.39)where Sym implies symmetrization with respect to indices ( i . . . i n ) and ( j . . . j n ) and all theseindices refer to the first space of the R-matrix (3.38)R ji ( u ) = 12 q u + − q − u − q − q − δ ji + 12 ( σ ) ji ⊗ σ + 12 ( σ ) ji ⊗ σ + 12 q u + + q − u − q + q − ( σ ) ji ⊗ σ . The standard way of treating (3.39) implies construction of the symmetrizer, i.e., a projector tothe highest spin representation in the decomposition of the product of n fundamental represen-tations. We implement the projection by means of the auxiliary spinors λ , µ that is equivalentto dealing with the symbols of R-matrices. The symbol of (2.51) (with respect to the localquantum space, not the auxiliary one) factorizes (see (2.51))R( u | λ, µ ) = λ i · · · λ i n R j ...j n i ...i n ( u ) µ j · · · µ j n = (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) · · · (cid:104) λ | R( u − n + 1) | µ (cid:105) (3.40)to a product of the symbols for q -Yang’s R-matrices (cid:104) λ | R( u ) | µ (cid:105) = λ i R ji ( u ) µ j , (cid:104) λ | R( u ) | µ (cid:105) = (cid:18) [ u + 1] q λ µ + [ u ] q λ µ λ µ λ µ [ u ] q λ µ + [ u + 1] q λ µ (cid:19) . (3.41)rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 29The product of n such matrices is given byR( u | λ, µ ) = [ u ] q [ u − q · · · [ u − n + 1] q (cid:18) A ( u | λ µ , λ µ ) B ( λ µ , λ µ ) λ µ B ( λ µ , λ µ ) λ µ A ( u | λ µ , λ µ ) (cid:19) , (3.42)where the functions A and B have the following formA ( u | λ µ , λ µ ) = n (cid:88) k =0 n ! k !( n − k )! [ u + 1 + k − n ] q ( λ µ ) n − k ( λ µ ) k = (cid:2) u + 1 − n + ( λ ∂ λ − λ ∂ λ ) (cid:3) q ( λ µ + λ µ ) n , (3.43)B ( λ µ , λ µ ) = n (cid:88) k =0 n ! k !( n − k )! [ k ] q ( λ µ ) k − ( λ µ ) n − k == 1 λ µ [ λ ∂ λ ] q ( λ µ + λ µ ) n . (3.44)The summation formulae in (3.43), (3.44) facilitate reconstruction of operators from the sym-bolic entries of the matrix (3.42). In analogy with the non-deformed case we again removethe inessential normalization factor r n ( u ) = [ u ] q [ u − q · · · [ u − n + 1] q and shift the spectralparameter u → u − n to obtain a symbol of the L-operatorL( u | λ, µ ) = r − n ( u )R (cid:0) u − n | λ, µ (cid:1) = (cid:32)(cid:2) u + ( λ ∂ λ − λ ∂ λ ) (cid:3) q λ λ [ λ ∂ λ ] qλ λ [ λ ∂ λ ] q (cid:2) u + ( λ ∂ λ − λ ∂ λ ) (cid:3) q (cid:33) (cid:104) λ | µ (cid:105) n . Since (cid:104) λ | µ (cid:105) n is a symbol of the identity operator, applying (2.42), we immediately recover thefamiliar Lax operator,L( u ) = (cid:18) [ u + J ] q J − J + [ u − J ] q (cid:19) , (3.45)where the generators of U q ( sl ) are realized by the finite-difference operators J − , J + , q ± J intwo variables, J − = λ λ [ λ ∂ λ ] q , J + = λ λ [ λ ∂ λ ] q , J = ( λ ∂ λ − λ ∂ λ ) . (3.46)One can check that they do respect commutation relations of U q ( sl ) J + J − − J + J − = [2 J ] q , J J ± − J ± J = ± J ± . Let us remind that the representation is defined on the space of homogeneous polynomialsof two variables λ , λ of degree n (see (2.41)). In order to proceed to the space of polynomialsof one variable we choose λ = − z , λ = 1, so that the generators (3.46) take the conventionalform J − = − z [ z∂ ] q , J + = z [ z∂ − n ] q , J = z∂ − n . (3.47)We close this section by an alternative calculation of the symbol (3.40) which follows thepattern at the end of Section 2.5. The main merit of the following calculation is that it can begeneralized to the elliptic case [10]. First of all we fix the auxiliary spinor λ = − z , λ = 1 anduse the realization of spin generators (cf. (3.47)) J − = − z [ z∂ ] q , J + = z [ z∂ − q , J = z∂ − , ψ ( z ) = a z + a . In the basis e = − z , e = 1, the matrices of the generators coincide with the Pauli-matrices J ± ( e , e ) = ( e , e ) σ ± , J ( e , e ) = ( e , e ) σ . (3.48)The fusion procedure enables us to derive the L-operator (3.45) together with representationof the spin n generators (3.47) acting in the ( n + 1)-dimensional space of polynomials ψ ( z ) = a n z n + · · · + a . Relations (3.48) are equivalent to( − z, σ ± = J ± ( − z, , ( − z, σ = J ( − z, q -Yang’s R-matrix (3.38) as a matrix differenceoperator acting on the symbol of the identity operator (cf. (3.41)), (cid:104) λ | R( u ) | µ (cid:105) = (cid:32) ( − z, (cid:2) u + + σ (cid:3) q | µ (cid:105) ( − z, σ − | µ (cid:105) ( − z, σ + | µ (cid:105) ( − z, (cid:2) u + − σ (cid:3) q | µ (cid:105) (cid:33) = (cid:18) [ u + z∂ ] q − z [ z∂ ] q z [ z∂ − q [ u + 1 − z∂ ] q (cid:19) ( µ − µ z ) . This operator is just the trigonometric L-operator (3.45) for the spin representation. Thecrucial observation is that it can be factorized respecting a special ordering of z and ∂ , (cid:104) λ | R( u ) | µ (cid:105) = 1 q − q − (cid:18) zq − u − zq u +1 (cid:19) (cid:18) q z ∂ q − z ∂ (cid:19) (cid:18) q u − z − − q − u z − (cid:19) × ( µ − µ z ) | z = z . (3.49)Formula (3.49) represents a trigonometric deformation of the factorization formula (2.59).The derived formula enables us to simplify the product of two consecutive symbols from (3.40)since a pair of adjacent matrix factors is cancelled (cid:104) λ | R( u ) | µ (cid:105)(cid:104) λ | R( u − | µ (cid:105) = 1( q − q − ) (cid:18) zq − u − zq u +1 (cid:19) (cid:18) q z ∂ q − z ∂ (cid:19) × (cid:18) q u − z − − q − u z − (cid:19) (cid:18) zq − u zq u (cid:19) (cid:18) q z ∂ q − z ∂ (cid:19) (cid:18) q u − − z − − q − u +1 z − (cid:19) × ( µ − µ z ) ( µ − µ z ) | z = z = z = [ u ] q q − q − (cid:18) zq − u − zq u +1 (cid:19) × (cid:18) q z ∂ + z ∂ q − z ∂ − z ∂ (cid:19) (cid:18) q u − − z − − q − u +1 z − (cid:19) ( µ − µ z ) ( µ − µ z ) | z = z = z . The generalization of this formula is obvious. Thus the product of symbols (3.40) is equal toR( u | λ, µ ) = r n ( u ) 1 q − q − (cid:18) zq − u − zq u +1 (cid:19) (cid:18) q z ∂ + ··· + z n ∂ n q − z ∂ −···− z n ∂ n (cid:19) × (cid:18) q u − n +1 − z − − q − u + n − z − (cid:19) ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z . (3.50)Then we need to get rid off the taken special ordering in (3.50). To that end we apply an evidentformula q ± ( z ∂ + ··· + z n ∂ n ) ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z = q ± z∂ ( µ − µ z ) n , rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 31which enables us to cast the matrix product on the right-hand side of (3.50) in the form r n ( u ) (cid:18) [ u − n + 1 + z ∂ + · · · + z n ∂ n ] q − z [ z ∂ + · · · + z n ∂ n ] q z [ z ∂ + · · · + z n ∂ n − n ] q [ u + 1 − z ∂ − · · · − z n ∂ n ] q (cid:19) × ( µ − µ z ) · · · ( µ − µ z n ) | z = ··· = z n = z = r n ( u ) (cid:18) [ u − n + 1 + z∂ ] q − z [ z∂ ] q z [ z∂ − n ] q [ u + 1 − z∂ ] q (cid:19) ( µ − µ z ) n . Thus the final result for the symbol of the “fused” q -Yang R-matrices (3.39) isR( u | λ, µ ) = r n ( u ) (cid:18) [ u + 1 − n + J ] q J − J + [ u + 1 − n − J ] q (cid:19) ( µ − µ z ) n , where the generators J ± , J in spin n representation are given by the expression (3.47). The fusion procedure for the modular double closely follows the construction from Section 2.6.One forms inhomogeneous monodromy matrix out of the L-operators and then symmetrizesit over the spinor indices resulting in a finite-dimensional (in one of the spaces) higher-spinL-operator.Again, instead of working with the higher-rank tensors we introduce auxiliary spinors λ i , (cid:101) λ i , µ j , and (cid:101) µ j and contract them with the monodromy matrix according to (2.40). The homoge-neity (2.41) implies that there are redundant variables. We get rid off them by choosing thegauge λ λ = − µ µ = − a and b as follows λ = λ ( a ) = e iπ ω a , λ = λ ( a ) = − e − iπ ω a ,µ = µ ( b ) = e iπ ω b , µ = µ ( b ) = − e − iπ ω b . (3.51)Analogous relations hold for the spinors (cid:101) λ and (cid:101) µ obtained from (3.51) after the interchange ω (cid:29) ω (cid:48) with the same a and b . Since we assume that the ratio of quasiperiods τ is not rational, λ and (cid:101) λ are multiplicatively incommensurate for generic a and the same is true for µ and (cid:101) µ forgeneric b .Further, we form symbols of the L-operators (3.18) (i.e., some scalar operators) contractingthem in the matrix space with the auxiliary spinors λ i L ji ( u ) µ j = Λ( u, λ, µ ) , (cid:101) λ i (cid:101) L ji ( u ) (cid:101) µ j = (cid:101) Λ( u, (cid:101) λ, (cid:101) µ ) . (3.52)Taking into account that D ω (cid:48) (ˆ p ) = e − iπ ω ˆ p + e iπ ω ˆ p (see (3.5), (3.6)), one can easily check theequality D u + ω (cid:48) ( x − a ) D − u ( x + b ) D ω (cid:48) (ˆ p ) D − u ( x − a ) D u + ω (cid:48) ( x + b ) = i Λ( u ) . It will be helpful to rewrite this formula in a slightly different form by means of the operatorstar-triangle relation (3.31) D u + ω (cid:48) ( x − a ) D u + ω (cid:48) (ˆ p ) D ω (cid:48) ( x + b ) D − u (ˆ p ) D − u ( x − a ) = i Λ( u ) . (3.53) We are grateful to D. Karakhanyan and R. Kirschner for a discussion on this point. (cid:101) Λ( u ), which is obtained after the permutation ω (cid:29) ω (cid:48) . Thederived formula is reminiscent to the L-operator factorization (3.20). Now we form a string outof the symbols Λ and (cid:101) Λ (3.53) with the shifted spectral parameters,R fus ( u | λ, (cid:101) λ, µ, (cid:101) µ ) = Λ( u )Λ( u − ω (cid:48) ) · · · Λ( u − ( m − ω (cid:48) ) × (cid:101) Λ( u − ( m − ω (cid:48) − ω ) · · · (cid:101) Λ( u − ( m − ω (cid:48) − nω ) . In view of the reflection formula (3.7) and relation (3.53) this product can be recast to the formR fus ( u | λ, (cid:101) λ, µ, (cid:101) µ ) = D u + ω (cid:48) ( x − a ) D u + ω (cid:48) (ˆ p ) [ iD ω (cid:48) ( x + b )] m [ iD ω ( x + b )] n × D − u +( m − ω (cid:48) + nω (ˆ p ) D − u +( m − ω (cid:48) + nω ( x − a ) . (3.54)Finally, we reconstruct the operator of interest from its symbol using formula (2.42), whichresults in the representation[R fus ( u )Φ] ( λ, (cid:101) λ | x ) = R fus ( u | λ, (cid:101) λ, ∂ µ , ∂ (cid:101) µ )Φ( µ, (cid:101) µ | x ) (cid:12)(cid:12)(cid:12) µ = (cid:101) µ =0 , (3.55)where the symbol R fus is fixed in (3.54). Let us stress once more that the fusion formula (3.55)is completely analogous to the SL(2 , C ) group case (2.63). The higher-spin R-operator acts ona function Φ( λ, (cid:101) λ | x ) having the homogeneity degrees m in λ and n in (cid:101) λ , respectively. In (3.55)one has differentiations over spinors µ , (cid:101) µ , but the operator R fus (3.54) formally depends on b and not on the exponential of b . In order to see that there is no contradiction, we note thataccording to the definitions (3.52) Λ and (cid:101) Λ are linear in spinors. Consequently R fus (3.54),being a product of them, has to be polynomial in spinors. This can be checked directly as well.Recalling the definition of µ , (cid:101) µ (3.51) and D ω (cid:48) ( x + b ) = µ e iπ ω x − µ e − iπ ω x , D ω ( x + b ) = (cid:101) µ e iπ ω (cid:48) x − (cid:101) µ e − iπ ω (cid:48) x , (3.56)we conclude that R fus in (3.54) depends polynomially on µ and (cid:101) µ . Thus the fusion formulae (3.54)and (3.55) match to each other.The right-hand side of (3.54) explicitly depends on a , so its polynomiality in spinors λ , (cid:101) λ isnot obvious at all. It is necessary to demonstrate it explicitly. Furthermore, we need to compa-re (3.55) with the reduction formula (3.36), 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 thegenerating function D nω + mω (cid:48) ( a − y ) of a finite-dimensional representation and act upon it bythe “fused” R-operator in the first space according to the prescription (3.55). The generatingfunction with the auxiliary parameter y explicitly depends on λ and (cid:101) λ (see (3.15)), D nω + mω (cid:48) ( a − y ) = n − (cid:89) k =0 (cid:16)(cid:101) λ e − iπ ω (cid:48) y ˜ q n − − k − ( − m (cid:101) λ e iπ ω (cid:48) y ˜ q − n − + k (cid:17) ·× m − (cid:89) p =0 (cid:16) λ e − iπ ω y q m − − p − ( − n λ e iπ ω y q − m − + p (cid:17) , (3.57)and has the homogeneity degrees m in λ and n in (cid:101) λ , respectively. Now, using the relations(see (3.56), (3.57))[ D ω (cid:48) ( x + b )] m [ D ω ( x + b )] n | µ → ∂ µ , (cid:101) µ → ∂ (cid:101) µ D nω + mω (cid:48) ( b − y ) = n ! m ! D nω + mω (cid:48) ( x − y ) , (3.58)we can perform differentiations over spinors in (3.55) that immediately yield the desired resultR fus (cid:0) u + nω + mω (cid:48) (cid:1) D nω + mω (cid:48) ( a − y ) = i n + m n ! m ! D u − s − s ( x − a ) D u − s + s (ˆ p ) × D nω + mω (cid:48) ( x − y ) D − u − s − s − ω (cid:48)(cid:48) (ˆ p ) D − u − s + s − ω (cid:48)(cid:48) ( x − a ) = i n + m n ! m ! D u − s − s ( x − a ) × D − u − s − s − ω (cid:48)(cid:48) ( x − y ) D nω + mω (cid:48) (ˆ p ) D u − s + s ( x − y ) D − u − s + s − ω (cid:48)(cid:48) ( x − a ) , (3.59)rom Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation 33where at the last step we profited from the operator star-triangle relation (3.31). Identifyingthe variables a = x , x = x , y = x , we find a nice agreement of the fusion formula (3.59) withthe reduction formula (3.36). Thus both approaches are equivalent and yield identical results. Acknowledgement
We thank the referees for useful remarks to the paper. This work is supported by the RussianScience Foundation (project no. 14-11-00598).
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