Relativistic Classical Integrable Tops and Quantum R-matrices

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical $R$-matrices. A particular case of ${\rm gl}_N$ system is gauge equivalent to the $N$-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum $R$-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin's quantum $R$-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large $N$ limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new ${\rm gl}_N$ quantum rational non-dynamical $R$-matrix via the relativistic top, which we get in a different way -- using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of ${\rm gl}_2$ the quantum rational $R$-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained $R$-matrix.


Introduction
We start with the quantum exchange relations [29] for the quantum gl N -valued L-operators: where the quantum non-dynamical R-matrix satisfies the quantum Yang-Baxter equation and unitarity condition R 12 (z) R 21 (−z) = f (z) 1 ⊗ 1 (1. 3) with some function f (z).
In this paper we consider a class of solutions of (1.1) and (1.2) having simple pole at z = 0 and satisfying relationL whereŜ is gl N -valued operator. Then (1.1) leads to (quadratic) Sklyanin algebra [44] forŜ which we denote as A Skl ,η . Notice here that we use two parameters and η in (1.1) (it is customary to consider η = ). In fact, one can even eliminate the η-dependence (see (1.20)) but we will see that it is useful to keep two free parameters from the very beginning.
In the classical limit → 0 the matrix components of the residueŜ become C-valued coordinates on the phase space of an integrable system described by the Lax matrix L η (z) (it coincides withL η (z), whereŜ is replaced with gl(N, C)-valued S) L η (z) = tr 2 (R η 12 (z)S 2 ) , S = Res z=0 L η (z) (1.5) and the standard quadratic r-matrix structure: with the classical r-matrix r 12 (z). We call this type of models relativistic integrable tops because η will be shown to play the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider [42]. The underlying Poincaré invariance is discussed in Section 5.2.
Thus, when η = we have simple relation (1.5) between the classical Lax operator and the quantum R-matrix, i.e. having quantum R-matrix we can define the classical integrable system. Write R-matrix in the standard gl N basis (E ij ) ab = δ ia δ jb as (1.7) Then it follows from (1.5) that L η (z) = L η (z, S) = N i,j,k,l=1 R η ij,kl (z) E ij S lk . (1.8) The latter leads to the converse statement, i.e. having the classical Lax matrix L η (z) we can find the quantum R-matrix as The purpose of the paper is twofold. The first one is to give description of the relativistic classical tops arising from the quantum R-matrices following (1.5). The second -is to derive new rational quantum R-matrix from the corresponding relativistic top via (1.9), which we obtain in a different way -by applying gauge transformation of Hecke type [31,32,33,46,4] to the rational Ruijsenaars-Schneider (RS) model [42].
1. Relativistic classical tops from quantum R-matrices. Using local expansion of L-operator and R-matrix near z = 0 we get equations of motion related to the Hamiltonian S 0 = tr(S) for the relativistic top in the form: where the inverse inertia tensor J η is the following linear functional: The latter M-operator appears to be equal (up to sign) to the non-relativistic limit of the Lax matrix L η (z): , l(z) = tr 2 (r 12 (z)S 2 ) = −M(z) . (1.14) Moreover, the next term in the expansion is the M-operator where M(0) is the non-relativistic limit of J η (S) (1.11).
The model (1.16) is bihamiltonian. It means that it can be described by a pair of compatible Poisson structures. The first one (the Poisson-Lie) is generated by the linear r-matrix structure {l 1 (z) , l 2 (w)} = [ l 1 (z) + l 2 (w), r 12 (z − w)] , (1.17) and the second -is by quadratic one where s 0 is additional generator (of the classical Sklyanin algebra). In the elliptic case (corresponding to the Belavin-Drinfeld classical r-matrix [12]) this type of bihamiltonian structure for (1.18)-(1. 19) was described in [25].
(1. 21) Relation (1.20) allows also to find the M-operator for (1.19) as The simplest example of the relativistic top is obtained in the elliptic case, where the quantum R-matrix is the Belavin's one [11]. In gl 2 case it coincides with the Baxter's one. Then for S a σ a , where σ a are the Pauli matrices (σ 0 = 1) In the elliptic case we also consider the large N limit to the elliptic hydrodynamics [25,38] by passage to the noncommutative torus description: is the Moyal bracket (⋆ is the Moyal product) and J η (S) is the pseudo-differential operator given in (5.60) (cf. (5.23)).
2. Quantum rational R-matrix. We propose the factorized form for the rational Ruijsenaars-Schneider (RS) Lax matrix L RS (z) = g −1 (z) g(z + η) e P/c , (1.25) where c is the light speed, P is a diagonal matrix of the RS particles momenta, and g(z) is the matrix depending on the RS particles coordinates q. The latter was introduced in [4], where the non-relativistic rational top was constructed similarly starting from the rational Calogero-Moser (CM) model [15]. The transformation g(z) is known for the quantum elliptic and trigonometric RS models [24,5] (where the quantum IRF-Vertex correspondence was described). The rational one was mentioned in [4] 1 . By performing the gauge transformation and re-expressing L η (z) in terms of its residue we come to the relativistic rational top. It corresponds to some special values of the Casimir functions, while arbitrary values are related in the same way to the spin RS model [28] (see also [7]). The answer is given in Section 3.2.
Then using (1.5) we obtain rational unitary quantum R-matrix. In gl 2 case it is the 11-vertex R-matrix obtained previously in [16] 2 . In Section 4 we obtain gl N generalization of (1.27). Introduction of ǫ parameter as R ,ǫ (z) = ǫ R ǫ (ǫz) (1.28) allows to interpret it as deformation of the XXX R-matrix Notice that in this limit the relativistic top (1.10), (1.11) becomes free mechanical system in the sense that L η (z) = η −1 S 0 1 + z −1 S, and equations of motion are trivialṠ = 0. Therefore, the parameter ǫ can be also treated as an alternative definition of the coupling constant.
The Lax matrix (1.26) (which is gauge equivalent to the RS model) emerge from explicit change of variables: where ̺(i) = (i − 1)δ i≤N −1 + Nδ iN (see (3.2)), while σ j (q) are elementary symmetric functions (3.21)-(3.23). The case (1.30) corresponds to rank one matrix S and to special values of the Casimir function det L η (z) of Poisson brackets (1.6). In the (quantum) elliptic and trigonometric cases the (1.30)-type formulae forŜ =Ŝ(q, ∂ ∂q ) can be found in [44,24,5] (see also [28]). In general case L η (z) = tr 2 (R η 12 (z)S 2 ), where all S ij are independent variables. For nonrelativistic models it was shown in [31] that the top models on the special coadjoint orbit are gauge equivalent to Calogero-Moser (CM) systems [15] while generic orbits correspond to their spin generalizations. In the same way, the generic relativistic top can be treated as alternative form of the spin RS model [28]. The gauge transformations used in (1.26) are of the same form as in non-relativistic case, where they play the role of modifications of the underlying Higgs bundles. Hence, we deal with the relativistic version of the Symplectic Hecke Correspondence. It allows us to obtain the non-dynamical quantum R-matrix instead of direct usage of the quantum IRF-Vertex Correspondence [39]. In this respect, we realize the latter correspondence by means of the relativistic version of the classical (Symplectic Hecke) one. It is also interesting to mention that in view of (1.20), (1.21) we obtain the same form of equations (1.16) for the (spin) RS and (spin) CM models Spin chains and Gaudin models related to the 11-vertex rational R-matrix (1.27) and its classical limit are obtained straightforwardly. As an example we get the gl 2 Gaudin model Hamiltonians emerging from non-relativistic limit of the inhomogeneous chain: Notice that the first term corresponds to the standard rational (XXX) Gaudin Hamiltonians.
The 11-vertex model is defined by the R-matrix (1.27). The quantum local Hamiltonian of the homogeneous periodic spin (1/2) chain on n sites is of the form: It is a deformation of the XXX spin chain (see also [26], where this type of deformation was obtained using different R-matrix) described by the first term in (1.32): where the residueŜ is gl N -valued operator and L η,(k) are linear functionals ofŜ, i.e.
in some basis {T a } of gl N . The coefficients k R η a,b are functions of a free constant parameter η which role is explained below. Due to (1.1) the matrix elementsŜ b satisfy quadratic relations of A Skl ,η (see (2.7)) such as Sklyanin algebra [44] or its different extensions [37,41,17,40]. Notice that the representation space of operatorsŜ a is not fixed yet.
Similarly to (2.1) and (2.2) let the R-matrix be of the form: where the generators T −b are dual to T b : tr (T a T b ) = δ a+b , and R ,(−1) 12 It is important to mention that we deal with two constants and η (1.1). While plays the role of the Planck constant, the parameter η will be shown to describe relativistic deformation in the sense of Ruijsenaars.
Using notations of (2.1) and (2.3) it easy to write down the quadratic relations of A Skl ,η . Indeed, consider residue of (1.2) at w = 0: Expanding this equation near z = 0 we get identity P 12Ŝ1Ŝ2 =Ŝ 2Ŝ1 P 12 for z −2 terms while the coefficients behind z −1 give rise to the Sklyanin algebra 3 :

Classical limit
Quantum R-matrix. In the classical limit → 0 the operatorsŜ a become C-valued coordinates on the phase space of an integrable system described by the Lax matrix L η (z, S). Notice that relation (2.5) remains intact at classical level, i.e.
Therefore, having the classical Lax matrix L η (z) of the described type we can compute the quantum R-matrix in the following way: We will use this formula in Section 3 for derivation of the rational R-matrix.
Classical r-matrix. Let the quantum R-matrix has the following expansion in the Planck constant : where (2.14) The term r 12 (z) is the classical r-matrix. It is skew-symmetric and satisfies the classical Yang-Baxter equation: The latter arises from (1.2) and (2.12). Similarly, by substituting (2.12) into (1.1) we come to quadratic Poisson structure where L η (z) is the classical L-operator (the Lax matrix).

Relativistic top
Let us define the relativistic top as an integrable model described by the Lax matrix (2.10) and the r-matrix structure (2.18): (2.19) We will see that equations of motion have the form of the integrable multidimensional Euler (or Euler-Arnold) top. On the other hand, it will be shown below that the parameter η plays the same role as the relativistic deformation parameter in the Ruijsenaars-Schneider generalization of Calogero-Moser models. This is why we call these type of models relativistic tops 5 .
Classical Sklyanin algebra. The phase space is parameterized by N 2 coordinates {S a }. It is equipped with the following quadratic Poisson structure: where r 12 is defined in (2.13) and L η,(0) (S) in (2.1) and (2.19). The brackets (2.20) can be obtained both -from the quantum algebra (2.7) (by taking the classical limit (2.12)) or from (2.19) by computing residue at w = 0 and evaluating the coefficient in front of z −1 . The Poisson brackets are degenerated. In order to restrict it on a symplectic leaf we need to fix Casimir functions C k (S). They appear as coefficients in the expansion of det L η (z) which is known to be central element for the Poisson brackets (2.18): The number of independent Casimir functions (in general) equals N. They can be accumulated from coefficients in front of nonpositive powers of z in (2.22) (others are dependent). The Hamiltonians (including the Casimir functions) can be computed from the expansion near z = 0 The number of Casimir functions N should be subtracted from the number of independent Hamiltonians (2.23) N(N + 1)/2. This gives N(N − 1)/2 for the Hamiltonians only. It equals to the half of dimension of a general symplectic leaf. The Poisson commutativity of the Hamiltonians H k,l is guaranteed by (2.19). Therefore, the model is integrable in the Liouville-Arnold sense.
Equations of motion and Lax pair. The simplest Hamiltonian is given by To get equations of motion let us compute the trace over the second component (in tensor product gl N ⊗2 ) of (2.20). It leads to the top-like equations where the inverse inertia tensor J η is the following linear functional of S: In a similar way, by applying tr 2 to the both parts of (2.21) we get equations of motion (2.26) in the Lax form: Notice that it is independent of η. As we will see below the M-operator in this description coincides with non-relativistic Lax matrix.

Non-relativistic limit
The non-relativistic limit η → 0 is similar to the classical one due (2.10). It follows from (2.12) and (2.10) that and When η → 0 the leading term in (2.35) is the linear Poisson-Lie structure on gl * N Lie coalgebra. The generator S 0 = trS is the Casimir function of the latter brackets. Let us fix it as S 0 = N and set In the same way the linear r-matrix structure is obtained at the level of Lax matrices: Non-relativistic top. We will refer to an integrable model described by the Lax matrix (2.33) and the Poisson structure (2.37), (2.38) as the non-relativistic top. The phase space is the coadjoint orbit of GL N Lie group. It is equipped with the linear Poisson-Lie structure on gl * N . For example, using the standard basis of matrices (E ij ) ab = δ ia δ jb (2.37) acquires the from It is easy to see that h k,k = c k . The Poisson commutativity of the Hamiltonians h k,l is guaranteed by (2.38). The M-operators corresponding to the Hamiltonians h k,l are evaluated by expansion of −tr 2 (r 12 (z − w)l k−1 2 (w)) (see [9]). An alternative way is given in (2.54)-(2.56). Notice that the Poisson brackets (2.37) follows from (2.38) and local expansion from (2.34). To get (2.37) one should substitute expansion (2.34) into (2.38) and compute the residue at w = 0 and then at z = 0.
Similarly to (2.10), we have a simple link between the Lax matrix and classical r-matrix given by (2.33). Substitution of (2.33) into (2.38) gives rise to the classical Yang-Baxter equation (2.17) (it follows from the Jacobi identity for the Poisson brackets (2.38) as well). By analogy with (2.11) we have This relation was used in [4] for computation of the rational classical r-matrix.

η-independent quadratic Poisson brackets
Let us consider another limit of brackets (2.32) and (2.35). Set With this rescaling S 0 → 0 when η → 0. Then the residue S becomes traceless, i.e. (2.43) Applying this limit to (2.32) we get The Poisson brackets (2.35) acquire the following form in the limit: The missing brackets {S, s 0 } can be found by taking the limit in (2.26 where the inverse inertia tensor J is defined as with r ′ 12 (0) from (2.31). Equations (2.47), (2.48) also play the role of equations of motion generated by the Hamiltonian s 0 : The corresponding Lax equation can be obtained in two ways. The first one [9] -is by taking tr 2 in (2.44). This yields The latter matrix should be w-independent up to some element from the kernel of ad L(z) . Alternatively, one can consider the limit of (2.28): Notice again that M(z) given by (2.29) is η-independent. Moreover, it coincides with L(z) up to sign and some scalar -element from Ker(ad L(z) ), i.e.
Then, substituting rescaling (2.41) and using expansion (2.30) we get with r ′ 12 (z) defined in (2.30). Thus, the roles of L and M-operators are interchanged while taking the limit. In addition, r ′ 12 (z) has no singularities at z = 0. Then Finally, we see that the expansion (2.30) of L η (z) in η provides M-operators for both -ηdependent and η-independent descriptions: Notice also that the M-operator (2.54) is also valid for the linear r-matrix structure (2.38) since the Lax pairs for the linear and quadratic (η-independent) r-matrix structures are the same (up to scalar terms). The formulae obtained in this section can be considered as an extension of [9] for the class of integrable systems under consideration.
Relation between A Skl η =0 and A Skl η=0 . We have two different description of the classical Sklyanin algebras (and related integrable models). In the first one the quadratic Poisson structure (2.20) The same happens at quantum level. One can quantize the Lax matrix (2.45) aŝ Then the exchange relations (1.1) gives η-independent quadratic algebra A Skl ,η=0 . Algebras A Skl η =0 and A Skl η=0 are related. The relation is easy to demonstrate explicitly in the elliptic case (see Section 5). The idea is the following. There exists a linear functional ϕ η on gl N (depending on the boundary conditions) such that (2.5) and (2.57) are related as follows: where g η (z) is some function with a simple zero at z = −η 0 and simple pole at z = 0. Finally, the relation can be written as It holds true in the rational case as well. We may use this relation to get explicit change of variables from the η-independent description (2.45) to the η-dependent (2.19):

Relativistic rational top
In this section we obtain explicit answer for the Lax pair of the relativistic top. As it was already mentioned this model is a top-like form of the spin Ruijsenaars-Schneider (RS) model.
To get the answer we represent the Lax matrix of RS model in the factorized form (3.6) which is convenient for the gauge transformation. The dynamical variables of the top are the components of the residue (3.18), (3.26) of the gauge transformed RS Lax matrix (3.16). We express the gauge transformed L-operator in terms of its residue. This gives the correct answer for generic top since it is independent of the Casimir functions values.

Factorized L-operators for classical Ruijsenaars-Schneider model
In this paragraph we propose factorized forms of L-operators for the rational RS model [42].
Following [4] for the set of variables {q j }, j = 1...N such that N j=1 q j = 0 let us introduce the where It has the property det Ξ(q, z) = Nz i.e. the matrix is degenerated at z = 0. It can be also treated as the rational analogue of the modification of bundles over elliptic curves used in [31] for the description of the elliptic top.
Rational sl N RS model with spectral parameter is defined by the following Lax matrix: The classical r-matrix structure was found in [8].
Let us also write the similar answer for 7 Rational sl N RS model without spectral parameter. The Lax matrix is represented in the form: where and (3.10) It easy to verify that and ∂ z V = C 0 V . The limit to Calogero-Moser model is obtained as follows: Notice that we can also define the Lax matrix as It differs from (3.7) by the canonical map with a = 1 and ξ = η. Then Let us mention that the transformation (3.9) was used in the classical [20] and the quantum [19] IRF-Vertex transformations. In this way the (Jordanian) R-matrices of the Cremmer-Gervais type were obtained.

Lax pair
Apply the gauge transformation g(z) = Ξ(q, z) D −1 0 (q) to the RS Lax matrix (3.4), (3.6) with η := −η. Then it follows from (3.6) that Our purpose now is to express this matrix in terms of its residue at z = 0. Set where the adjugate is transpose of the cofactor matrix. To find matrix components we need the inverse of Ξ: where x j = q j + z. Expansion in powers of z gives (3.20) In (3.19) and (3.20) the elementary symmetric functions are used: These functions satisfy the following set of identities: Here and below we imply that the values of indices corresponding to the undefined argument value (N − 1) of function ̺ −1 (3.2) are skipped in the summations.
It is important to mention that the answer does not depend on the values of the Casimir functions (2.22). As we know it is defined by only quantum R-matrix which is the subject of the next Section. Therefore, we can consider the obtained expression (3.27)-(3.30) as independent definition of the Lax matrix for the generic rational relativistic top. The case (3.16) which is gauge equivalent to the rational RS model appears for the particular values of the Casimir functions C k (2.22). From (3.17) we conclude that the RS case corresponds to: In the non-relativistic limit this case corresponds to the rational top on the coadjoint orbit of minimal dimension (2N − 2). In a general case the obtained model yields alternative description of the spin RS model [28].
The quantum Lax matrix is obtained from (3.27)-(3.30) by substitution S →Ŝ. In (3.26) it corresponds to p j := ∂ q j with the choice of normal ordering.
M-operator from non-relativistic limit. As it was shown in (2.33) the M-operator (2.29) coincides (up to minus) with the non-relativistic limit of the Lax matrix. Hence, we can use the answer obtained in [4]: 8 : In N = 2 case r

Spin chains and Gaudin models
The  For example, in gl 2 case the classical r-matrix (2.13) (classical limit of (1.27)) (3.41) In the limit (1.29) this formula reproduces the well-known rational Gaudin Hamiltonians (the first term in (3.41)).
Let us also compute the quantum local Hamiltonian of the homogeneous (z a = 0) periodic spin chain on n sites. The quantum R-matrix Therefore, we can calculate the local Hamiltonian using standard approach of [10,29]. The answer is given by It is a deformation of the XXX spin chain. The latter is described by the only first term in (1.32): The generators E ij , (E ij ) ab = δ ia δ jb are dual toŜ ji = tr(E ijŜ ).
The Hamiltonian of type (3.44) was obtained in [26] using different R-matrix (it depends on two spectral parameters) 9 . We describe this type of models and related soliton equations in our next publication [34].

Quantum rational R-matrix
The quantum non-dynamical R-matrix can be found by the standard procedure of the IRF-Vertex Correspondence starting from the rational RS model. Here we use another approach based on (1.9). Applying it to (3.27)-(3.30) we get Another deformation of the Heisenberg chain was found in [30].
By redefinition R 12 (z, ǫ) = ǫ R ǫ 12 (zǫ) (4.6) we can treat the answer as deformation of the standard XXX R-matrix. Indeed, one can verify that lim In the classical limit we get the rational skew-symmetric non-dynamical r-matrix from [4] 10 : 10 Expression (4.8) differs from the one given in [4] by common factor N and scalar term 1 Example: 11-vertex R-matrix. In N = 2 case we obtain the 11-vertex R-matrix [16]: Example: rational gl 3 R-matrix. In gl 3 case (4.1)-(4.4) gives the following 9 × 9 quantum R-matrix: Plugging it into (1.30) one gets the gl 3 relativistic top. In (3.31) case it gauge equivalent to 3-body rational RS model via (3.26).
Remark: Presumably, the R-matrix (4.1)-(4.4) can be obtained by special limiting procedure from the Belavin's elliptic R-matrix. Such an algorithm was described in [45]. Computer calculations gave the same answer for N = 2, 3 (as in our examples (4.9), (4.10)). There is another approach to the rational R-matrices based on the deformed Yangians [30]. The Rmatrices in this description depend on two spectral parameters. It seems likely that some relation to our R-matrix may exist. For example, the twisted gl 2 case considered in [26] leads to the same local spin chain Hamiltonian. In the same time the deformation discussed in [30] differs from ours. Let us also notice that an algorithm for computing gl N quantum R-matrices (related to [30] description) was proposed and studied in [14]. 5 Belavin's R-matrix and elliptic models

Sklyanin algebra and relativistic elliptic tops
Belavin's elliptic gl N R-matrix [11] is the central object of the section as well as related quantum L-operator: In (5.1) and (5.2) the basis of gl N and corresponding functions are chosen as 11 , ω a = a 1 + a 2 τ N , (5.4) where ϑ(z) is the odd theta function and a 1 , a 2 ∈ Z N . The Sklyanin algebra (2.7) is defined by the local behavior of ϕ η a (z) near z = 0: The last terms (2πı∂ τ ω a ) are canceled out in the final answers.
The quadratic relations A Skl τ, ,η in the T a ⊗ T b component of (1.1) read as follows: Notice that in the case η = and b = 0 one should consider the limit η → in (5.8) which gives (5.10) The case η = and N = 2 in (5.9) gives rise to the Sklyanin algebra in its original form [44].
Consider the upper half-plane H + ⊂ C. The moduli space M of elliptic curves is the result of the action of SL 2 (Z) on H + by the Möbius transform The modular transformations acts on the theta-function as where ζ 8 = 1. Then we can find that f(a, b, c|τ, , η) is modular invariant, and therefore, it is a well defined function on M. In this way the universal bundle plays the role of the moduli space of the algebra A Skl τ, ,η . Then the moduli space is Relation between A Skl η =0 and A Skl η=0 . Following [17] 12 let us clarify the relation between ηdependent and η-independent L-operators discussed above (2.58). Two descriptions are distinct from each other by quasiperiodic boundary conditions on the lattice C/Z + Zτ : L(z + 1) = Q −1 L(z)Q , L(z + τ ) = Λ −1 L(z)Λ , (5.14) where Q and Λ are from (5.3). Notice that L(z) (5.14) is a section of EndV -bundle, where V is the holomorphic vector bundle with the transition functions Q andΛ(z) = exp(2πı( z N + τ 2N ))Λ. In the same time L η (z) should be considered as a map between V and V ′ , where V ′ is defined by the transition functions Q and Λ ′ (z) = exp(2πıη)Λ(z).
The first (scalar) term T 0 S 0 E 1 (η) in the upper line of (5.23) vanishes from the commutator in (5.22) as well as the first term SE 1 (η) in the lower line. To verify (5.22), (5.23) one needs the following identity: Non-relativistic limit η → 0 coincides with η-independent description at the level of equations of motion because the Lax matrices (2.33) and (2.45) are the same up to the scalar term S 0 1. This is due to existence of bihamiltonian structure, i.e. any linear combination of the linear and quadratic Poisson brackets are again some Poisson bracket (see [25] for details).
The equations of motion (5.22) keep the same form in the limit with The underlying elliptic function identity is very well known [27]: This model was introduced in [31]. Its phase space is the coadjoint orbit of Lie group GL N . When dimension of the orbit is minimal (2N − 2) the Lax matrix is gauge equivalent to the one of elliptic Calogero-Moser model. More detailed description can be found in [46]. Higher rank Sklyanin algebras in the context of integrable systems were also discussed in [13,25,17].

Poincaré invariance
The Poincaré Lie algebra for the relativistic integrable systems [42] is defined as The RS models can be obtained by symplectic (or the Poisson) reduction procedures from the cotangent bundles to a certain loop groups [23] 14 . For the top-like (elliptic) Lax operators (1.19) a similar procedure was suggested in [13] and [17]. In all the description the Lax matrix appears through reduction from the group element. It satisfies some moment map constraint generated by the symmetries of the (co)adjoint action.
Let us show that the mentioned above reductions provide naturally the Poincaré Lie algebra (5.28). As a preliminary, consider the finite-dimensional case, i.e. the cotangent bundle T * G to G = GL N Lie group. Let g ∈ G and A ∈ gl * N . The symplectic structure on T * G is equal to 14 See also example of the relativistic Toda systems [43] in [21] The corresponding Poisson brackets are of the form: Taking tr 1 of (5.31) we get {trA, g} = g . A(z + 1,z + 1) = QĀ(z,z)Q −1 ,Ā(z + τ,z +τ ) = ΛĀ(z,z)Λ −1 .

(5.36)
A section ξ(z,z) is holomorphic if dĀξ(z,z) = 0. Two holomorphic structures dĀ and dĀ′ are equivalent if they are related by the gauge transformation of the gauge group Ḡ The quotient of the space of generic connections A = {dĀ} 15 by the gauge group action is the moduli space Bun(Σ τ , GL(N, C)) = A/G of holomorphic bundles. The initial phase space P Στ is the Poisson algebra of holomorphic functionals on R. The Poisson brackets are similar to (5.30), (5.31) (see [17]). The gauge transformations (5.37) along with leads to the finite-dimensional reduced phase space P red . The reduced Poisson algebra coincides with (2.46) for the elliptic r-matrix (5.19), and g(z,z)| P red = L(z) = T 0 S 0 + tr 2 (r 12 (z) and reproduce the Poincaré algebra (5.28) in the form of (5.33) with g = L(z). The variable a extends the phase space of the top. The variable dual to a is the one which acts on the top variables by dilatation S → λS. This action does not preserves the values of the Casimir functions generated by det L(z, S). Therefore, the Poincaré symmetry emerges on the top's phase space extended by the two-dimensional space (a, λ) -cotangent bundle to the one-dimensional center of the group.

(5.64)
The obtained equations (5.62) can be treated as hydrodynamical limit of the elliptic spin Ruijsenaars-Schneider model. It is an interesting problem to find its relation to another type of hydrodynamical limit [1]. The latter approach leads to the quantum gl N Benjamin-Ono and KdV systems while our approach gives rise to the gl N Sklyanin type algebra (5.56) and classical equations (5.62), (5.63).
In the non-relativistic limit η → 0 we reproduce the answer from [25,38]. Likewise the RS model goes into CM one, the relativistic top goes to non-relativistic (5.25). In the same way instead of J η (5.60) we get In a similar way one can describe the dispersionless limit θ → 0, ǫ 1,2 → ∞, lim θ→0 (θǫ 1,2 ) = ǫ ′ 1,2 < 1. In the limit the Lie algebra sin θ of the group SIN θ becomes the Lie algebra Ham(T 2 ) of Hamiltonian vector fields on the two-dimensional torus.

Conclusion
Let us briefly summarize the obtained results.  (6.6) 5) Alternatively, the relativistic top can be described in η-independent form as bihamiltonian system with the quadratic Poisson structure described in Section 2.5 and the linear Poisson structure given in Section 2.4. The relation between η-dependent and η-independent descriptions are given by (2.59). ∂L (z) ∂S kl ⊗ E lk . (6.8) In the limit (1.28)-(1.29) it gives the XXX R-matrix. In gl 2 case it is the (11-vertex) Cherednik's R-matrix [16].
The latter statement can be obtained by the direct IRF-Vertex transformation starting from the quantum R-matrix for the rational RS model. We will give this proof elsewhere.
The obtained rational R-matrix allows to define new type of spin chains and Gaudin models. In the elliptic case, in addition to the relativistic top we describe the large N limit as the elliptic hydrodynamics.

Remarks
• Relations between the Ruijsenaars-Schneider (RS) systems and quantum integrable chains appeared recently in the context of the Quantum-Classical duality using the Bethe ansatz approach [22] or the τ -function approach [2]. This duality, in particular, implies the substitution η = into the Lax matrix of the RS model and provides an alternative (to the algebraic Bethe ansatz) method for computation of spectrum of the quantum spin chains transfer-matrices.
The phenomenon of the Quantum-Classical duality type was also observed at the level of gauge theories in the series of papers [36]. In this approach the Planck constant (in quantum integrable system) was identified with the twisted mass parameter in the N = 2 * SUSY Yang-Mills theory. The latter mass parameter is related to the action of the global U(1) group on the adjoint chiral multiplet field. It resembles the appearance of the η parameter in the twisted boundary conditions for the Lax operator (5.13), i.e. the twisted mass and the η play similar roles and can be closely related.
A relation between the classical and quantum systems arises also in studies of the spectral duality [35]. A general statement is that the spectral duality works in the same way both at classical and quantum levels, or the properly defined classical and quantum spectral curves for spin chains coincide. It also resembles the similarity of the quantum R-matrices and the classical Lax operators.
• Expression (4.1)-(4.4) for the rational R-matrix is complicated. In the same time, the computations of particular examples emerge a lot of cancellations. We hope that the answer can be simplified. It is an interesting problem to find some elegant form for the rational R-matrix of group-theoretical type.
• The classification of integrable systems of Hitchin type on elliptic curves can be naturally made in terms of characteristic classes of underlying Higgs bundles [32,46]. In particular, it allows one to obtain intermediate solutions of the quantum Yang-Baxter equation (between pure dynamical for RS model and pure non-dynamical one) [33]. In the rational case we deal with degenerated (and punctured) elliptic curve y 2 = z 3 . It is interesting to know whether elliptic classification survives in the rational limit.
• In this paper we do not consider the trigonometric case. However, it would appear reasonable that the obtained results are valid in this case as well. The corresponding nondynamical quantum R-matrix was obtained in [5] from the trigonometric RS model via the IRF-Vertex transformation. It can be used for construction of trigonometric top-like models.