The hyperbolic modular double and the Yang-Baxter equation
aa r X i v : . [ m a t h - ph ] D ec The hyperbolic modular double and theYang-Baxter equation
Dmitry Chicherin, Vyacheslav P. Spiridonov
Abstract.
We construct a hyperbolic modular double – an algebra lying in be-tween the Faddeev modular double for U q ( sl ) and the elliptic modulardouble. The intertwining operator for this algebra leads to an integraloperator solution of the Yang-Baxter equation associated with a gener-alized Faddeev-Volkov lattice model introduced by the second author.We describe also the L-operator and finite-dimensional R-matrices forthis model. §
1. Introduction
The representation theory is intimately related to special functions.The quantum groups and Yang-Baxter equation (YBE) provide a wideclass of novel functions that do not appear in the classical representa-tion theory of Lie groups. These functions possess a number of peculiarproperties and satisfy many intricate identities which do not have clas-sical counterparts. The noncompact (or modular) quantum dilogarithm[22, 23] is a remarkable special function significant for a large class ofquantum integrable systems. In particular, it plays a prominent role inthe space-time discretization of the Liouville model and in the construc-tion of the lattice Virasoro algebra [26, 49], as well as in the investigationsof the XXZ spin chain model in a particular regime [29, 9].The observation that there exist two mutually commuting Weylpairs led Faddeev [25] to the notion of a modular double of the quantumalgebra U q ( sℓ ). It is formed by two copies of U q ( sℓ ) with differentdeformation parameters whose generators mutually (anti)commute witheach other. This doubling enables unambiguous fixing of the represen-tation space of the algebra. Mathematics Subject Classification.
Key words and phrases.
Yang-Baxter equation, Faddeev-Volkov model,Sklyanin algebra, modular double, solvable lattice models. D. Chicherin, V. P. Spiridonov
The elliptic modular double introduced by the second author in[43] carry over the idea of doubling to the Sklyanin algebra [38]. Thisdoubling is extremely useful. The symmetry constraints with respect tothe extended algebra are much more powerful as compared to the initialalgebra. They enable again unambiguous description of the relevantobjects.The Faddeev-Volkov model [50] is a solvable two-dimensional latticemodel of statistical mechanics [5]. In contrast to the Ising model, its spinvariables take continuous values. The Boltzmann weights are expressedin terms of the modular quantum dilogarithm. In [6] the free energyper edge of this model was derived in the thermodynamic limit using aparticular form of the star-triangle relation.A generalization of the Faddeev-Volkov model has been proposedby the second author in [44]. In this extension the Boltzmann weightsare expressed in terms of the modular quantum dilogarithm as well, butthey have more involved form as compared to the original model. Thecorresponding star-triangle relation is a degeneration of the elliptic betaintegral evaluation formula [41]. The star-triangle relation associatedwith the latter integral appeared first in the operator form as mainidentity behind the integral Bailey lemma discovered in [42] (see also [20]for a detailed discussion) and later it was formulated in the functionalform in [7].In the present work we study an algebraic structure underlying thegeneralized Faddeev-Volkov model of [44] and related quantum inte-grable systems. First we consider a contraction of the Sklyanin alge-bra described in [27], which is more general than U q ( sl ). Then weshow that this symmetry algebra can be enhanced using the doublingconstruction. So, we will supplement the algebra with a dual set ofgenerators (anti)commuting with the initial generators. We baptize theresulting algebra as the hyperbolic modular double , following the ter-minology of [37] for the modular quantum dilogarithm considered as ageneralization of the Euler gamma function. It lies in between the el-liptic modular double and the modular double of U q ( sℓ ) in the sensethat the three algebras are arranged in a sequence of contractions. Wewill pass naturally from the language of lattice models to the standardYBE and find the most general solution of YBE having the symmetryof the hyperbolic modular double. It is an integral operator acting ona pair of infinite-dimensional spaces which are representation spaces ofthe latter algebra. An integral operator solution of the YBE (at theplain non-deformed level) was constructed for the first time in [16]. Thefactorization property of the corresponding R-operator was noticed later he hyperbolic modular double and the Yang-Baxter equation in [15], which resulted in a powerful almost purely algebraic techniquesof building general R-operators [10, 17, 19, 20].Previously in [13, 14] we described finite-dimensional solutions ofYBE for the elliptic modular double, modular double of U q ( sℓ ) andthe Lie group SL(2 , C ) in a concise form and elucidated their factoriza-tion property [11]. One of the principal aims of this paper is to find allfinite-dimensional solutions of YBE having the symmetry of the hyper-bolic modular double with generic deformation parameter. However, adetailed construction of the representation theory of the latter algebratogether with the analysis of special functions associated with that, fol-lowing the considerations of [34] or [27] and references therein, is leftaside. We do not discuss modular doubling of quantum affine algebrasas well, this subject was considered recently in [33].The plan of the paper is as follows. We start in Sect. 2 with adescription of the hyperbolic gamma function (modular quantum dilog-arithm) and state its basic properties. Then we present the solvablemodel of statistical mechanics generalizing the Faddeev-Volkov modeland the corresponding hyperbolic star-triangle relation. In Sect. 3 weconstruct an integral R-operator in terms of the Boltzmann weights ofthis solvable vertex model and show that it satisfies the Yang-Baxterequation. We also rewrite it in the factorized form. In Sect. 4 we de-scribe an algebra emerging as a degeneration of the Sklyanin algebra andconstruct the corresponding intertwining operator of equivalent repre-sentations. As a natural extension of this quantum algebra we introducethe hyperbolic modular double. Then we study finite-dimensional irre-ducible representations of the hyperbolic modular double. In Sect. 5we reduce the integral R-operator to a finite-dimensional invariant sub-space in one of its infinite-dimensional spaces. In this way we obtainan explicit formula for finite-dimensional solutions of the YBE with thesymmetry of the hyperbolic modular double. In Sect. 6 we apply thereduction formula in the simplest nontrivial setting. We choose the fun-damental representation in one of the spaces and recover the L-operatorfrom the integral R-operator. It automatically takes the factorized form.Finally, in Sect. 7 the reduction formula is elaborated further on by asimplification to finite-dimensional matrix solutions of YBE such thatall of them take a factorized form. §
2. A solvable lattice model
Gamma functions are the main building blocks in the constructionof special functions of hypergeometric type. The hierarchy of hyperbolicgamma functions is formed by particular combinations of two multiple
D. Chicherin, V. P. Spiridonov
Barnes gamma functions [4] (the standard Jackson’s q -gamma function[2] is a combination of two Barnes gamma functions of the second or-der as well, but we do not consider this function here). The hyper-bolic gamma function of the second order is a homogeneous function of u, ω , ω ∈ C . For Re( ω ) , Re( ω ) > < Re( u ) < Re( ω + ω ) ithas the form γ (2) ( u ; ω , ω ) := exp (cid:18) − π i2 B , ( u ; ω , ω ) −− Z R +i0 e ux (1 − e ω x )(1 − e ω x ) dxx (cid:19) , (1)where B , is a multiple Bernoulli polynomial of the second order B , ( u ; ω , ω ) = 1 ω ω (cid:18) u − ω + ω (cid:19) − ω + ω ! . Denoting q = e π i ω /ω and ˜ q = e − π i ω /ω and assuming that | q | < (cid:18) − Z R +i0 e ux (1 − e ω x )(1 − e ω x ) dxx (cid:19) = ( e π i u/ω ˜ q ; ˜ q ) ∞ ( e π i u/ω ; q ) ∞ , where ( t ; q ) ∞ = Q ∞ k =0 (1 − tq k ).The modular quantum dilogarithm is usually defined as a func-tion obtained from (1) by removing the exponential factor involving B , ( u ; ω , ω ), the shift u → u + ( ω + ω ) /
2, and some renormalizationof the variables u and ω , . In particular, in the context of 2 d conformalfield theory it is accepted to denote ω = b, ω = b − . We shall use thefollowing representation γ ( z ) := γ ( z ; b ) := exp − i π (cid:18) z − b + b − (cid:19) ++ i π
24 ( b + b − ) + + ∞ Z −∞ dt t e t (2 z − b − b − ) sin(i bt ) sin(i b − t ) , (2)where the integration contour goes above the singularity at t = 0. Onecan easily restore the original function γ (2) ( z ; ω , ω ) = γ ( z/ √ ω ω ; p ω /ω ) . he hyperbolic modular double and the Yang-Baxter equation This integral representation is valid for Re( b ) > < Re( z ) < Re( b + b − ). The analytic continuation enables one to extend the defi-nition (2) to a wider range of parameters.The inverse of this special function is called also the double sine-function and denoted either S ( u ; ω , ω ) [29] or S b ( z ) [9]. The notation γ (2) ( z ; ω , ω ) is taken from [44]. Here we use the terminology suggestedin [37]. Interrelations between main known modifications of function (2)are described in Appendix A of [44]. Various identities for the quantumdilogarithm can be found in [26, 48] and some other papers.The definition (2) implies that γ ( z ) is invariant under the swap b ⇆ b − . It satisfies two linear difference equations of the first order γ ( z + b ) = 2 sin( πbz ) γ ( z ) , γ ( z + b − ) = 2 sin( πb − z ) γ ( z ) . (3)Let us introduce the crossing parameter η := − b + b − . Then the reflection formula can be written as follows γ ( z ) γ ( − η − z ) = 1 . (4)The function γ ( z ) is meromorphic. It has a double series of zeros z = b ( n + 1) + b − ( m + 1) , n, m ∈ Z ≥ , (5)and a double series of poles z = − b n − b − m, n, m ∈ Z ≥ . (6)In the following we deal with multiple products of the hyperbolic gammafunction. In order to avoid lengthy formulae we adopt the convention γ ( ± x + g ) := γ ( x + g ) γ ( − x + g ) , γ ( ± x ± y ) := γ ( ± x + y ) γ ( ± x − y ) . In [44] a nontrivial generalization of the Faddeev-Volkov model [49,50] has been proposed. Such models of statistical mechanics are definedon the square lattice. The continuous spin variables sit in the latticevertices. The rapidity variables are associated with the medial graphbuilt with the help of pairs of directed lines crossing edges in the middlewith the inclination of 45 or 135 degrees (see, e.g., a picture on Fig. 2).Self-interaction of spins and interactions between the nearest neighbor-ing spins are allowed. The Boltzmann weight W ( α − β ; x, y ) is assignedto a horizontal edge connecting a pair of vertices with spins x , y that is D. Chicherin, V. P. Spiridonov crossed by a pair of medial graph lines carrying the rapidities α and β .Similarly the Boltzmann weight W ( α − β ; x, y ) is assigned to a verticaledge connecting a pair of vertices with the spins x , y that is crossedby a pair of medial graph lines carrying rapidities α and β . The self-interaction at a vertex z contributes the Boltzmann weight ρ ( z ) to thepartition function. The edge Boltzmann weights W and W of the model[44] are given by fourfold products of hyperbolic gamma functions andthe vertex Boltzmann weight ρ is a product of two hyperbolic gammafunctions W ( α ; x, y ) = γ ( α − η ± i x ± i y ) , (7) W ( α ; x, y ) = γ ( − α ± i x ± i y ) , ρ ( z ) = 12 γ ( ± z ) . The edge Boltzmann weights depend on the difference of rapidities andthey are symmetric in the spin variables W ( α ; x, y ) = W ( α ; y, x ), W ( α ; x, y ) = W ( α ; y, x ). Let us recall that the Boltzmann weights de-pend on b which is a temperature-like parameter.Physical interpretation of W and W as true Boltzmann weightsrequires their positivity. This is possible in two regimes of the key pa-rameter b : 1) b is real and 0 < b <
1; 2) | b | = 1 , Im( b ) >
0. In bothregimes η < γ ( z ) ∗ = γ ( z ∗ ). As a result one should demand thatthe spin variables are real. As to the rapidities, one can set η < α < − η for W ( α ) and 0 < − α < − η for W ( α ). These constraints should cor-respond to unitary representations of the hyperbolic modular double tobe described below.The Boltzmann weights possess a crossing symmetry, i.e. the hori-zontal and vertical edge weights are related as follows W ( α ; x, y ) = W ( η − α ; x, y ) . (8)We note that in view of the reflection formula (4) and quasiperiodicity(3) the vertex Boltzmann weight ρ ( z ) can be rewritten solely in terms ofthe trigonometric functions, i.e. its expression in terms of the hyperbolicgamma functions is overcomplicated. In contrast, the edge Boltzmannweights W and W are genuine products of hyperbolic gamma functionsand the number of such functions in the products cannot be reduced.The formulated model is solvable because of the star-triangle rela-tion depicted in Fig. 1. This relation equates the partition functions of he hyperbolic modular double and the Yang-Baxter equation two elementary cells, + ∞ Z −∞ ρ ( z ) W ( α − β ; x, z ) W ( α − γ ; y, z ) W ( β − γ ; w, z ) dz = χ ( α, β, γ ) W ( α − β ; y, w ) W ( α − γ ; x, w ) W ( β − γ ; x, y ) , (9)up to a normalization constant χ . Using this example one can see thatthe horizontal edge Boltzmann weights W ( α − β ; x, y ) depends on thedifference α − β , where α is the rapidity of the upward directed medianline of 45 degrees and β is the rapidity of the upward directed median lineof 135 degrees. For the vertical edge weights W ( α − β ; x, y ) the situationis similar – α is the rapidity of the line going to the right of the edge and β – of the line going to the left. We call identity (9) with the weights(7) the hyperbolic star-triangle relation. Corresponding normalizationconstant has the following form χ ( α, β, γ ) = γ (2 β − α ) γ (2 γ − β ) γ (2 α − γ − η ) . This local relation enables one to construct a family of commuting row- z xy w αβγ = χ × xy w αβγ Fig. 1. The star-triangle relation. to-row transfer matrices and then to calculate the partition function ofthe model using the machinery of QISM [5]. The free energy per edgeof the model in the thermodynamical limit has been calculated in [44]following the method from [6, 7].Let us substitute in (9)(10) W ( α ; x, y ) = m ( α ) W r ( α ; x, y ) , W ( α ; x, y ) = m ( η − α ) W r ( α ; x, y ) D. Chicherin, V. P. Spiridonov and choose the function m ( α ) in such a way that the normalizationconstant χ on the right-hand side of (9) disappears, m ( α − β ) m ( η − α + γ ) m ( β − γ ) m ( η − α + β ) m ( α − γ ) m ( η − β + γ ) χ ( α, β, γ ) = 1 . Ascribe now to the edges the renormalized weights W r ( α ; x, y ) and W r ( α ; x, y ). Then, denoting the total number of edges in the infinitelygrowing lattice as N , one finds that the free energy per edge βf edge = − lim N →∞ log Z ( r ) N = 0 , where Z ( r ) is the total partition function for the model with renormalizedBoltzmann weights.Equivalently, we can keep the original Boltzmann weights intact andcompute the contribution of the renormalizing factors in the asymptoticsof the partition function. Take the finite rectangular lattice with N spinsalong the horizontal line and M spins along the vertical line. Such latticehas ( N − M horizontal edges and N ( M −
1) vertical edges. Thereforethe indicated renormalization of the Boltzmann weights yields a scalingof the partition function Z N,M = Z ( r ) N,M m ( α ) ( N − M m ( η − α ) N ( M − , i.e. the free energy per edge of the original models isfree energy per edge = − lim N,M →∞ log Z N,M
N M = − log m ( α ) m ( η − α ) . It is easy to see that the needed normalization constant m ( α ) isfound from the equation(11) m ( α + η ) = γ (2 α ; b ) m ( − α ) . As shown in [44], the solution of this equation satisfying the unitarityrelation m ( α ) m ( − α ) = 1 is given by the ratio of two hyperbolic gammafunctions of the third order γ (3) ( u ; ω , ω , ω ) for a special choice of thequasiperiods ω i . In the current notation it has the following integralrepresentation m ( α ) = exp (cid:16) − π i (cid:18) α + 124 (1 − b + b − ) ) (cid:19) ++ 18 Z R +i0 e αt sin(i bt ) sin(i b − t ) cos(i( b + b − ) t ) dtt (cid:17) . (12) he hyperbolic modular double and the Yang-Baxter equation It happens that this result coincides with a similar normalization factor m ( α ) for the original Faddeev-Volkov model derived in [6].The hyperbolic star-triangle relation (9) can be written as the inte-gral identity + ∞ Z −∞ Y k =1 γ ( g k ± i z ) dz γ ( ± z ) = Y ≤ j
3. From the lattice model to the integral R-operator
The star-triangle relations imply integrability of the two-dimensionallattice models, similar to the case outlined in the previous section. An-other wide class of integrable systems is associated with the quantumspin chains. Formulation of the latter models in the framework of QISM[24, 32] requires definition of the R-matrix solving the YBE. In thissection we show how to construct such R-matrices associated with thehyperbolic star-triangle relation (9). Since the spin variables sitting invertices of the two-dimensional lattice take continuous values (in con-trast to the discrete spins of the Ising and chiral Potts models), on thespin chains side we deal with integral operators instead of the finite-dimensional R-matrices. In other words, the quantum spaces of therelevant spin chain are infinite-dimensional functional spaces. Thereforewe call solutions of the corresponding YBE the R-operators to empha-size this aspect. We will indicate in Sect. 5 that the integral R-operatorsrepresent in some sense the most general YBE solutions, since they em-brace all finite-dimensional R-matrices. Our presentation below followsto some extent the original construction of [18].Thus we are interested in the integral operator R ( u | g , g ) definedon the tensor product of two infinite-dimensional function spaces thatare representation spaces with labels (“spins”) g and g (arbitrary com-plex numbers) of some algebra. The symmetry algebra underlying thehyperbolic star-triangle relation will be introduced in Sect. 4. The R-operator depends on a complex number u – the spectral parameter andsatisfies the YBE R ( u − v | g , g ) R ( u | g , g ) R ( v | g , g )= R ( v | g , g ) R ( u | g , g ) R ( u − v | g , g ) . (16)Instead of the integral operator R ( u | g , g ), we can consider firsta more general notation operator R ( u , u | v , v ) depending on fourcomplex parameters and satisfying the equation R ( u , u | v , v ) R ( u , u | w , w ) R ( v , v | w , w )= R ( v , v | w , w ) R ( u , u | w , w ) R ( u , u | v , v ) . (17)This equation is rather similar to (16). An operator solution of Eq. (17)can be easily constructed in terms of the lattice model formulated inthe previous section. The parameters u , u , v , v , w , w are identifiedwith the rapidities. The kernel of the integral operator is a product offour edge Boltzmann weights (two horizontal and two vertical) and two he hyperbolic modular double and the Yang-Baxter equation vertex Boltzmann weights, (cid:2) R ( u , u | v , v )Φ (cid:3) ( z , z ) = + ∞ Z −∞ + ∞ Z −∞ ρ ( x ) ρ ( x ) W ( u − v ; z , z ) × (18) W ( u − v ; z , x ) W ( u − v ; z , x ) W ( u − v ; x , x )Φ( x , x ) dx dx . Thus the kernel of R ( u , u | v , v ) is the partition function of an ele-mentary square cell, see Fig. 2. (cid:2) R ( u , u | v , v ) Φ (cid:3) ( z , z ) == + ∞ Z −∞ + ∞ Z −∞ x z z x u u v v Φ( x , x ) dx dx Fig. 2. The kernel of the integral R-operator is the partitionfunction of an elementary square cell.
Taking into account explicit expressions for the Boltzmann weightswe rewrite Eq. (18) as follows[ R ( u , u | v , v )Φ] ( z , z ) = + ∞ Z −∞ + ∞ Z −∞ dx dx Φ( x , x )4 γ ( ± x ) γ ( ± x ) × γ ( ± i z ± i z + u − v − η ) γ ( ± i x ± i z + v − u ) × γ ( ± i x ± i z + v − u ) γ ( ± i x ± i x + u − v − η ) . (19)This R-operator corresponds to the generalized Faddeev-Volkov modelof [44]. It can be derived from the elliptic hypergeometric R-operatorconstructed in [20] after taking a particular limit in the parameters, butthe rigorous proof of this fact would require quite intricate techniques.Remind that the latter R-operator intertwines representations of theSklyanin algebra. Let us recall that the vertex Boltzmann weight ρ is in fact a trigono-metric function. The integrand function in the expression (19) is a gen-uine product of 16 hyperbolic gamma functions (modulo a trigonometricmultiplier) and their number cannot be reduced. In [10] the R-operatorassociated with the Faddeev modular double of U q ( sℓ ) has been con-structed in a similar form. It corresponds to the Faddeev-Volkov latticemodel itself. However, the integrand function of the corresponding inte-gral operator solution of YBE is a product of only 8 hyperbolic gammafunctions (up to a trigonometric multiplier). One can obtain this R-operator rigorously from expression (19) by taking appropriate limits inthe parameters such that the γ -functions depending on the sum x + x disappear. E.g., such a limiting procedure is described in [44] for thereduction of the hyperbolic star-triangle relation down to the originalFaddeev-Volkov model case.Similar to the very first integral R-operator considerations in [18],it is easy to check that the operator R ( u , u | v , v ), Eq. (18), solvesEq. (17). In order to demonstrate how it works we resort to graphicalrepresentation of the integral R-operators. In Fig. 3 at the right topwe depicted convolution of the kernels of the integral operators from theleft-hand side expression in Eq. (17) and at the right bottom we depictedconvolution of the kernels from the RHS of Eq. (17). External pointsare marked by numbers 1 , ,
3. They denote three quantum spaces. In-tegration over internal points is assumed (convolution of the kernels).The dotted lines are the rapidity lines. The left-hand and right-handside expressions in Eq. (17) are connected to each other by a sequenceof moves. Each move is the application of the star-triangle transforma-tion, Eq. (9). Thus the YBE (17) boils down to a combination of thestar-triangle relations (9). Keeping track of the normalization factors χ arising at each step one can check that they eventually drop out.However, we still need to understand the algebraic meaning of solu-tion (18), which is the primary goal of the present paper. In other words,we have to give an algebraic interpretation of the rapidities u , u , v , v .As we will see further they can be chosen as linear combinations of thespectral parameters u, v and the representation labels g , g of a certainalgebraic structure which is studied in the next section, u = u + g , u = u − g , v = v + g , v = v − g . (20)This relation yields a solution of the YBE in the form (16), R ( u − v | g , g ) = R ( u , u | v , v ) . he hyperbolic modular double and the Yang-Baxter equation u v u v w Fig. 3. The sequence of star-triangle moves transforming theleft-hand side expression in (17) to its right-hand form.
Our considerations below indicate that this is in fact the most generalsolution of YBE compatible with certain quantum algebra. The moststrong argument follows from the fact that Eq. (18) embraces all finite-dimensional solutions of the YBE (16) associated with this algebra.Factorization of the kernel of the integral operator (18) implies fac-torization of the operator itself. Indeed, it can be written as a productof five elementary operators R ( u , u | v , v ) = P S( u − v ) M ( u − v ) M ( u − v ) S( u − v ) . (21)Here P is a permutation operator of two tensor factors, i.e. P Φ( z , z )= Φ( z , z ). S( u ) is an operator of multiplication by a particular func-tion, S( u ) = W ( u ; z , z ) = γ ( ± i z ± i z + u − η ) . (22)M and M are two copies of the integral operator (cid:2) M( g ) Φ (cid:3) ( z ) = 1 γ ( − g ) + ∞ Z −∞ ρ ( x ) W ( g ; z, x ) Φ( x ) dx (23)acting in the first and second quantum spaces, respectively. This op-erator is a degeneration of an elliptic hypergeometric integral operator introduced in [42]. Owing to the chosen definition (23), normalizationsof the R-operators (21) and (19) differ by the multiplicative numericalfactor γ (2 v − u ) γ (2 v − u ). This renormalization of the R-operatorremoves certain divergences appearing during the reduction we describebelow and makes this procedure smooth.At this point we should specify an appropriate function space for theoperator M, Eq. (23). Firstly, the kernel of M is invariant with respectto the reflections x → − x and z → − z . Consequently, M projects outodd functions and maps onto the space of even functions. Moreover,calculating the asymptotics of the kernel at x → ±∞ we obtain a re-striction on the asymptotic behavior of Φ( x ). Thus we assume that Φ isan even function, i.e. Φ( − x ) = Φ( x ), and e π i xg Φ( x ) is rapidly decayingat x → + ∞ .In view of the reflection equation (15) and unitarity, Eq. (14), of theBoltzmann weights the operators S and M satisfy (in the space of evenfunctions) the inversion relationsS( u ) S( − u ) = 1 , M( g ) M( − g ) = 1 . (24)The second relation is a degeneration of the inversion formula provedin [46] for the elliptic hypergeometric integral operator of [42]. The in-version relation (24) for S( u ) is valid for generic values of g ∈ C , butfor M( g ) it is violated for particular discrete lattice points on C . In thelatter case a nontrivial null-space of M( g ) appears which is describedin the next section. Let us note that for generic parameters the oper-ators S, M , M provide a twisted representation of generators of thepermutation group S satisfying the Coxeter relations. More precisely,the star-triangle relation (9) can be reformulated as the cubic Coxeterrelations for S, M , M , whereas the inversion relations (24) representquadratic Coxeter relations. For more details on this interpretation, seethe end of Sect. 6 and [15, 17, 19, 20] where the allied constructions areelaborated in detail. Finally we note that the identities (24) lead to theunitarity-like relation for the R-operator (21), R ( u | g , g ) R ( − u | g , g ) = 1 . (25) §
4. The intertwining operator of a degenerate Sklyanin alge-bra
In this section we study in detail the operator M, Eq. (23), in orderto infer its algebraic meaning. We rewrite Eq. (23) explicitly in terms he hyperbolic modular double and the Yang-Baxter equation of the hyperbolic gamma functions (cid:2) M( g ) Φ (cid:3) ( z ) = 12 + ∞ Z −∞ γ ( − g ± i z ± i x ) γ ( ± x ) γ ( − g ) Φ( x ) dx . (26)For certain discrete values of g the operator M( g ) simplifies con-siderably. First of all at the origin g = 0 it is the identity operator,M(0) = 1, which can be seen by taking the limit g → g ) respects apair of the contiguous relations involving the shifts of g by b and b , − isin(2 π i bz ) sin( b ∂ z ) M( g ) = M( g + b ) , (27) − isin(2 π i b − z ) sin( b ∂ z ) M( g ) = M( g + b ) . (28)Elliptic analogues of these relations can be found in [12, 21]. Applying re-currences (27), (28) for constructing M( g ) in the discrete quarter-infinitelattice of points g = nb + m b , n, m ∈ Z ≥ , we find that the integral op-erator (26) is converted to a product of n + m finite-difference operatorsof the first orderM( nb + m b ) = (cid:20) − isin(2 π i bz ) sin( b ∂ z ) (cid:21) n (cid:20) − isin(2 π i b − z ) sin( b ∂ z ) (cid:21) m . (29)We have already mentioned above that the R-operator (18) is re-lated to a certain quantum algebra. Let us now specify it explicitly. Itis a contraction of the Sklyanin algebra [38] that has been introduced in[27] and then investigated in [40, 3, 8]. This degenerate Sklyanin alge-bra is formed by four generators A, B, C, D which respect the followingcommutation relations
C A = e i πb A C , D C = e i πb C D , [ A , D ] = −
2i sin πb C , [ B , C ] = A − D
2i sin πb ,A B − e i πb B A = e i πb D B − B D = i2 sin 2 πb ( C A − D C ) . (30)It has a pair of Casimir operators K = e i πb A D − sin πb C ,K = e − i πb A + e i πb D πb − B C −
12 cos πb C (31) commuting with all four generators. This algebra is different from theconventional quantum deformation of the rank 1 Lie algebra, U q ( sℓ )[30], in particular, it does not obey the Hopf algebra structure. Asshown in [27], the spectral problem for a special quadratic combinationof the generators of this algebra reproduces the eigenvalue problem forthe Askey-Wilson polynomials. Therefore this algebra comprises theZhedanov algebra as well [51], which was constructed precisely with theaim of interpreting Askey-Wilson polynomials as a representation spaceof some quadratic algebra.The Sklyanin algebra possesses a representation by finite-differenceoperators with elliptic coefficients which depend on an arbitrary complexparameter g labeling representations [39]. Particular linear combinationsof this algebra generators in a degeneration limit, such that the ellipticnome goes to zero and the elliptic functions reduce to trigonometric ones,take the form (the details of this procedure can be found in [27], see also[43]) A ( g ) = i e i π b e − π i bg sin 2 π i bz h e πbz e i b ∂ z − e − πbz e − i b ∂ z i , (32) B ( g ) = −
12 cos πb C ( g ) −
14 sin πb π i bz × h cos πb (2 g + 4i z − b ) e i b ∂ z − cos πb (2 g − z − b ) e − i b ∂ z i , (33) C ( g ) = 12 sin πb π i bz h e i b ∂ z − e − i b ∂ z i , (34) D ( g ) = − i e − i π b e π i bg sin 2 π i bz h e − πbz e i b ∂ z − e πbz e − i b ∂ z i . (35)These operators satisfy defining relations (30). In this representationthe Casimir operators (31) take the values K ( g ) = e i πb , K ( g ) = cos 2 πbg πb . We can construct Verma module representations of the algebra (30)following an analogy with the sℓ algebra. We choose | i = 1 to be alowest weight vector in the representation annihilated by the loweringoperator C , C | i = 0. In order to obtain the basis of the Verma modulewe act by the raising operator B on the lowest weight vector a number he hyperbolic modular double and the Yang-Baxter equation of times: | k i := B k | i , k ∈ Z ≥ . Using relations (30) one can check that A | k i = [ k/ X l =0 a k,l ( g ) | k − l i , D | k i = [ k/ X l =0 d k,l ( g ) | k − l i ,C | k i = [ k − ] X l =0 c k,l ( g ) | k − − l i , (36)where a k,l , d k,l , c k,l are some functions of g . Contrary to the familiarsituation of sℓ the lowering operator C acting on the vector | k i producesnot | k − i but a linear combination of vectors with descending weights k − , k − , k − , . . . . Similarly, the operators A and D are not diagonalcontrary to their counterpart in sℓ . Acting on the vector | k i they mixit with the vectors having descending weights k − , k − , k − , . . . .For the chosen trigonometric polynomial realization, the vector | k i is alinear combination of cos(2 π i jbz ), where j = k, k − , k − , . . . , g the representation is infinite-dimensional.However, if g = ( n + 1) b , n ∈ Z ≥ , the situation drastically changes.Then | n + 1 i is a linear combination of {| k i} nk =0 . Acting by powersof the raising operator B on the vector | n i we do not get out of the n -dimensional space. In order to avoid misunderstanding we note that C | n + 1 i 6 = 0, unlike the sℓ case.Since representations with the labels g and − g have the same valuesof the Casimir operators, they are equivalent. Indeed, they are inter-twined by the operator M( g ), Eq. (26), as follows from the relationsM( g ) A ( g ) = A ( − g ) M( g ) , M( g ) B ( g ) = B ( − g ) M( g ) , M( g ) C ( g ) = C ( − g ) M( g ) , M( g ) D ( g ) = D ( − g ) M( g ) , (37)which can be checked by an explicit calculation.The operator M( g ) is invariant under the swap b ⇆ b − . Therefore itis natural to introduce the second set of generators e A, e B, e C, e D respectingthe commutation relations (30) with the replacement b → b − , i.e. e C e A = e i πb e A e C , e D e C = e i πb e C e D , [ e A , e D ] = −
2i sin πb e C , [ e B , e C ] = e A − e D
2i sin πb , e A e B − e i πb e B e A = e i πb e D e B − e B e D = i2 sin 2 πb ( e C e A − e D e C ) . (38) We also need to specify the commutation relations for generators fromdifferent sets. The generators
A, D anticommute with e B, e C ; the gener-ators B, C anticommute with e A, e D ; the generators A, D commute with e A, e D ; and the generators B, C commute with e B, e C .An explicit realization of the generators e A, e B, e C, e D by finite-diffe-rence operators is given by the formulae (32)–(35), where b should bereplaced by b − . New Casimir operators have the form e K = e i π/b e A e D − sin π/b e C = e i π/b , e K = e − i π/b e A + e i π/b e D π/b − e B e C −
12 cos π/b e C = cos 2 πg/b π/b . (39)Taken together, two sets of generators A, B, C, D and e A, e B, e C, e D form the algebra which we call the hyperbolic modular double . It liesin between the Faddeev’s modular double of U q ( sℓ ) [25] and the ellip-tic modular double [43] in the sense that these algebraic structures arerelated by a sequence of contractionsElliptic modular double →→ Hyperbolic modular double →→ Modular double of U q ( sℓ ) . Using particular combinations of the generators of this algebra similarto the one considered in [27], it is possible to construct a modular doubleof the Zhedanov algebra [51] as well.Evidently, M( g ) works as the intertwining operator for the secondset of generators as well,M( g ) e A ( g ) = e A ( − g ) M( g ) , M( g ) e B ( g ) = e B ( − g ) M( g ) , M( g ) e C ( g ) = e C ( − g ) M( g ) , M( g ) e D ( g ) = e D ( − g ) M( g ) . (40)Irreducible representations of the hyperbolic modular double are fixedby one complex number g , with the g and − g label representations beingequivalent. Realization of the generators of this algebra in the space ofanalytical functions is unique (up to a multiplication by a numericalfactor), because solutions of a system of finite-difference equations withthe shifts by b and b − (which can be taken real and incommensurate)are determined up to multiplication by a number. Relations (37) and(40) are natural extensions of the intertwining relations for the U q ( sl )algebra and its modular double derived by Ponsot and Teschner [35]. Inthe limit described in [44], when the hyperbolic R-matrix is degenerated he hyperbolic modular double and the Yang-Baxter equation to that of Faddeev and Volkov, the integral operator M( g ) passes to theintertwining operator of [35].Intertwining operators are quite useful, since they enable one to getinsight to the structure of representations of the corresponding algebra.Indeed, the null-space of M( g ), Ker M( g ), and the image of M( − g ),Im M( − g ), are invariant spaces of the representation with the label g that follows from (37) and (40).The inversion formula (24) implies thatM z ( g ) γ ( g ± i x ± i z ) = 0 , where g = nb m b , n, m ∈ Z ≥ , ( n, m ) = (0 , , since the normalization factor 1 /γ (2 g ) of M( − g ) is divergent at the spec-ified values of g (see Eq. (5)). Here the subindex in the operator M z indicates that z is used as the integration variable. Hence, expanding γ ( g ± i x ± i z ) in x we recover the null-space of M z ( g ). Moreover, thefunction γ ( g ± i x ± i z ) is proportional to the integrand of the operatorM( − g ). Consequently, its expansion in x lies in the image of M( − g ).Let us remind that for these values of g the integral operator M( g ) turnsto the finite-difference operator (29).In the following we will be interested in irreducible finite-dimensionalrepresentations of the hyperbolic modular double at g n,m = b n + 1) + 12 b ( m + 1) , n, m ∈ Z ≥ , (41)which have the dimension ( n + 1)( m + 1). They are realized in theinvariant space Ker M( g n,m ) ∩ Im M ren ( − g n,m ) , where M ren ( − g ) = γ (2 g )M( − g ).All basis vectors of the finite-dimensional irreducible representationare embraced by the generating function γ ( ± i x ± i z + g n,m ) , where x is an auxiliary parameter. Indeed, owing to Eqs. (3) and (4), itturns into the finite product of trigonometric functions γ ( ± i x ± i z + g n,m ) =(42) n − Y r =0 πb ( ± i x + i z + b ( n − − r ) + b ( m + 1)) × m − Y s =0 πb ( ± i x + i z + b ( m − − s ) − b ( n − . From the latter formula we extract the natural basis of the finite-dimen-sional representationcos(2 jπ i bz ) cos(2 lπ i z/b ) , j = 0 , , . . . , n, l = 0 , , . . . , m. Note that the generating function coincides with the edge Boltzmannweight (7). §
5. Reductions of the integral R-operator
In this section we show that the integral operator solution, Eq. (18),of the YBE (16) enables one to recover all finite-dimensional solutionsof YBE as well. In order to do it we apply the operator R ( u | g , g )to the function γ ( ± i z ± i z + u − u )Φ( z ), where z is an auxiliaryparameter and Φ( z ) is an arbitrary function from the second space.For g = g n,m the first factor turns to the generating function, Eq. (42).Temporarily we assume g to be generic. Computation of the result ofthis action is pictorially presented in Fig. 4, where we slightly changedthe graphical rules. Now all edges represent the Boltzmann weights W ,Eq. (7). The black blob corresponds to the vertex Boltzmann weight ρ . We omit rapidity lines and indicate corresponding differences of therapidities explicitly.At the first step we apply the star-triangle relation, Eq. (9), imple-menting integration at the vertex x . Thus the only integration left is atthe vertex x . It corresponds to the integral operator M ( u − v ) withthe kernel ρ ( x ) W ( u − v ; z , x ) which acts on the product W ( v − u + η ; z , z ) W ( u − u + η ; z , z ) Φ( z ). At the second step we just rear-range the factors such that we gain the integral operator M ( u − u + η )with the kernel ρ ( x ) W ( u − u + η ; z , x ) which acts on the product W ( u − v ; z , z ) W ( v − u + η ; z , z ) Φ( z ). Then we note that the x z z x x z z z z x z z z u -v u -v v -u + η v -u + η u -u u -v v -u + η u - v v -u + η u -u + η u -v v -u + η u -u + η v -u + η u - v Fig. 4. The sequence of transformations converting the in-tegral operator R ( u | g , g ) at g = g n,m to a finite-dimensional matrix in the first space. remaining integral operator M ( u − u + η ) = M ( g + η ) for g = g n,m turns to the finite-difference operator M ( nb + m b ), Eq. (29). Thus we he hyperbolic modular double and the Yang-Baxter equation have obtained the reduction formula which encompasses all solutions ofthe YBE (16) that have the symmetry of the hyperbolic modular doubleand are realized on the tensor product of the finite-dimensional repre-sentation with the label g n,m , Eq. (41), in the first space, and arbitraryinfinite-dimensional representation with the label g in the second one, R ( u | g n,m , g ) γ ( ± i z ± i z + b ( n + 1) + b ( m + 1)) Φ( z )(43) = c · γ ( ± i z ± i z + − u + g n,m + g ) γ ( ± i z ± i z + − u − g n,m − g − η ) × M ( nb + m b ) γ ( ± i z ± i z + − u + g n,m − g ) γ ( ± i z ± i z + − u − g n,m + g − η ) Φ( z ) , where c = 1 γ ( u + g n,m ± g ) . The same result can be obtained using the fusion following the proceduredescribed in [13, 14].Expanding both sides of this formula in the auxiliary parameter z we recover the reduced R-operator that is a matrix whose entries aresome finite-difference operators acting in the second space, i.e. we havethe L-operator. One can straightforwardly reduce further the L-operator(43) to R-matrices which are finite-dimensional in both spaces. In orderto achieve it we just need to force the representation label g of the secondspace to lie on the second copy of the lattice (41).Remarkably, the factorized form of the integrand function of the in-tegral operator (18) is inherited by the reduced R-operator. We will seein Sect. 6 and 7 that the reduced solution of YBE (43) can be furtherarranged to the factorized product of matrices form. In [13, 14] an anal-ogous reduction formula has been derived for the integral R-operators inthe following three cases: 1) the R-operator with the symmetry groupSL(2 , C ); 2) solutions of YBE with the symmetry of the modular doubleof U q ( sℓ ); 3) the most general known R-operator obeying the symmetryof the elliptic modular double (and of the Sklyanin algebra, of course). §
6. The fundamental representation L -operator and its fac-torization Let us show how formula (43) works in practice. We consider thesimplest nontrivial representation in the first space g = g , = b + b (see Eq. (41)), i.e. the fundamental representation of the ( A, B, C, D )-generated part of the hyperbolic double and trivial representation for the ( e A, e B, e C, e D )-part. Corresponding solution of the YBE is known asthe (spin ) L-operator. The generating function in this case has thefollowing form (see Eq. (42)) γ ( ± i z ± i z + b + b ) = 2 cos 2 π i bz + 2 cos 2 π i bz = e π i bz + e , where the basis of the 2-dimensional representation in the first space C is formed by e = 1 and e = 2 cos 2 π i bz . The intertwining oper-ator from (43) simplifies to M ( b ) = c · π i bz (cid:16) e i b ∂ − e − i b ∂ (cid:17) (seeEq. (29)). To simplify the formulae we shift the spectral parameter u → u + b . Now we wish to rewrite formula (43) in a matrix form. Inthe formula (43) we pull the hyperbolic gamma functions depending on ± i z ± i z to the left and the hyperbolic gamma functions depending on ± i z ± i z to the right. Then we simplify them by means of Eqs. (3) and(4). Thus the right-hand side expression in (43) takes the form h π i bz − πb (2i z + u + g ) i × e i b ∂ h π i bz − πb (2i z − u + g ) i − h π i bz − πb (2i z − u − g ) i × e − i b ∂ h π i bz − πb (2i z + u − g ) i . (44)Now it is straightforward to rewrite the reduced R-operator R ( u + b | g , , g ) =: L( u | g ) in a matrix form in the basis { e , e } of C , usingthe definition of matrix elements L( u | g ) e k := P i e i [L( u | g )] i,k ,L( u | g ) = 1sin 2 π i bz (cid:18) − πb (i z + u ) − πb (i z − u )1 1 (cid:19) ×× e i b ∂ − e − i b ∂ ! (cid:18) − πb (i z − u )1 − πb (i z + u ) (cid:19) . (45)Here we substituted z → z . The rapidities u , u are defined as u = u + g and u = u − g (recall Eq. (20)). We stress that the L-operator isautomatically obtained in the factorized form, Eq. (45). This might beexpected since the initial formula (43) obeys similar factorization.Choosing in the L-operator the second space representation label as g = g , = b + b , i.e. restricting it to the fundamental representationas well, we recover a 4 × he hyperbolic modular double and the Yang-Baxter equation This factorized representation is analogous to the one found in [3,28] (compare Eq. (45) with the normal-ordered factorized L-operatorEq. (2.20) in [3]). There the lateral matrices are identified with thetrigonometric intertwining vectors that provide the vertex-face corre-spondence between the 7-vertex model and a trigonometric SOS model.The L-operator, Eq. (45), can be written in terms of the degenerateSklyanin algebra generators as well,L( u | g ) = 2 (cid:18) − e − i πbu A ( g ) − e i πbu D ( g )sin πb C ( g )(46) − πb B ( g ) − πb (cos 2 πbu + cos πb ) C ( g ) e i πbu A ( g ) + e − i πbu D ( g ) (cid:19) . In [3] this L-operator (46) has been identified with the quantum L-operator for the 2-particle trigonometric Ruijsenaars model.Taking g = g , = b + b one recovers in the same way the secondL-operator e L( u | g ) whose entries are generators of the second half of thehyperbolic modular double.Thus we see that our construction is self-consistent. A particularreduction of the integral R-operator results in the generators of the hy-perbolic modular double. Consequently the latter quantum algebra isindeed the symmetry algebra of the integral R-operator (19). The alge-braic interpretation of the rapidities stated above (20) is correct. Theinfinite-dimensional spaces in the construction of the integral R-operatorare equipped with the structure of representations of the hyperbolicmodular double.Implementing the reduction condition g = g , in the YBE (16) weobtain an RLL-relation, R ( u − v | g , g ) L ( u | g ) L ( v | g ) == L ( v | g ) L ( u | g ) R ( u − v | g , g ) . (47)Here the integral R-operator (19) acts in a pair of infinite-dimensionalspaces with the representation labels g and g and intertwines the ma-trix product of two L-operators. The lower index (1 or 2) of the L-operator enumerates the infinite-dimensional spaces where it acts non-trivially, i.e. the entries of L i (see Eq. (45)) are some difference operatorsin the variable z i . The RLL-relation (47) can be rewritten in terms ofthe rapidities as well in a full analogy with Eq. (17), R ( u , u | v , v ) L ( u , u ) L ( v , v ) == L ( v , v ) L ( u , u ) R ( u , u | v , v ) , (48) where L( u , u ) := L( u | g ), L( v , v ) := L( v | g ) (recall Eq. (20)).Now we can give a natural interpretation of the operator S( u ),Eq. (22), which is one of the factors of the R-operator (21). It im-plements the permutation of rapidities ( u , u , v , v ) ( u , v , u , v )in the matrix product of two L-operators, i.e.S( u − v ) L ( u , u ) L ( v , v ) = L ( u , v ) L ( u , v ) S( u − v ) . This statement can be checked by a straightforward calculation. Onthe other hand, the R-operator itself implements the permutation of therapidities ( u , u , v , v ) ( v , v , u , u ) in Eq. (48). For more detailsof such permutation of parameters in various models, see [17, 19, 20]. §
7. Factorized finite-dimensional solutions of the YBE
In the previous section we have shown that the reduction formula(43) produces the L-operator in the factorized form from the fundamen-tal representation in the first space for the integral R-operator. Now weare going to demonstrate that the same pattern persists for all finite-dimensional representations, i.e. we show that the higher-spin solutionsof the YBE can be factorized as well. Finite-dimensional representa-tions of the hyperbolic modular double naturally factorize to productsof finite-dimensional representations of its two halves. Therefore with-out loss of generality we can consider nontrivial representations for onlyone of its halves. Thus, we choose g = g n, = b ( n + 1) + b , n ∈ Z ≥ ,(recall Eq. (41)) in the reduction formula (43).The generating function of the ( n + 1)-dimensional representationof interest takes the form (recall Eq. (42)) γ ( ± i z ± i x + g n, ) = n − Y r =0 h π i bz + 2 cos πb (2i x + b ( n − − r )) i = n +1 X j =1 ψ ( n ) n +2 − j ( x ) ϕ ( n ) j ( z ) = n +1 X j =1 ϕ ( n ) n +2 − j ( x ) ψ ( n ) j ( z ) , (49)where ϕ ( n ) j ( z ) := (2 cos 2 π i bz ) j − , j = 1 , , . . . , n + 1 . The second equality in (49) is used to define the dual basis ψ ( n ) ( x ),whereas the third equality follows from the invariance of the generatingfunction under the permutation of x and z . Thus the generating functionproduces two natural bases { e j } n +1 j =1 and { f j } n +1 j =1 of C n +1 , e j = ϕ ( n ) j ( z ) , f j = ψ ( n ) j ( z ) , j = 1 , , . . . , n + 1 . he hyperbolic modular double and the Yang-Baxter equation Expanding both sides of Eq. (43) as linear combinations of ϕ ( n ) ( z ), weobtain a matrix form of the reduced R-operator written in the indicatedpair of bases, i.e. R ( u | g n , g ) ψ ( n ) j ( z ) = ϕ ( n ) l ( z ) (cid:2) R ( u | g n , g ) (cid:3) lj . In the previous section we did not have such a subtlety, since for n = 1(the fundamental representation) both bases coincide.Similar to the pattern given in the previous section, the direct cal-culation yields the following factorization formula for the reduced R-operator(50) R ( u | g n, , g ) = V ( u + g, z ) D ( z, ∂ ) C V T ( u − g, z ) C , consisting of the product of five matrices. Here we substituted z → z for brevity. C is a numerical matrix with the unities on the antidiagonal,i.e. ( C ) lj = δ n +2 − l,j . Entries of the diagonal matrix[ D ( z, ∂ )] lj := δ lj β ( n ) l ( z ) e ( n +2 − l ) i b ∂ z . (51)are the shift operators determined by the expansion of M( nb ) of theform(52) M( nb ) = n +1 X l =1 β ( n ) l ( z ) e ( n +2 − l ) η∂ z . Entries of the matrix V , [ V ( u, z )] jl = V ( n ) jl ( u, z ), are some trigonometricfunctions. They are defined by the relations n +1 X j =1 ϕ ( n ) j ( x ) V ( n ) jl ( u, z ) :=(53) l − Y r =0 h π i bx − πb (2i z − u − η − g n, + 2 br ) i × n − l Y r =0 h π i bx − πb ( − z − u − η − g n, + 2 br ) i . It is easy to see that V ( n ) jl ( u, − z ) = V ( n ) j,n +2 − l ( u, z ), i.e. V ( u, − z ) = V ( u, z ) C .Let us recall that for the factorized L-operator the lateral matricesare composed out of the trigonometric intertwining vectors providing the vertex-face correspondence [3]. Then, in the case of the ( n + 1)-dimensional representation, the lateral matrices V are constructed outof the fused trigonometric intertwining vectors (see Eq. (53)) providingthe vertex-face correspondence for the higher-spin models. In [11] ananalogous factorization formula has been derived for finite-dimensionalR-operators with the symmetry algebras sℓ , U q ( sℓ ), and the Sklyaninalgebra. § Acknowledgements
We are indebted to S. E. Derkachov for useful discussions of theresults of this paper. This work is supported by the Russian ScienceFoundation (project no. 14-11-00598).
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