On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system

In this paper, we construct a quadratic r-matrix structure for the classical rational BC(n) Ruijsenaars-Schneider-van Diejen system with the maximal number of three independent coupling parameters. As a byproduct, we provide a Lax representation of the dynamics as well.


Introduction
The Ruijsenaars-Schneider-van Diejen (RSvD) models [1,2] are among the most intensively studied integrable many particle systems, having numerous relationships with different branches of theoretical physics and pure mathematics. They had found applications first in the theory of the soliton equations [1,3,4,5,6], but soon they appeared in the Yang-Mills and the Seiberg-Witten theories as well (see e.g. [7,8,9,10,11]). Besides these well-known links, the RSvD systems and their non-relativistic limits, the Calogero-Moser-Sutherland (CMS) systems [12,13,14], have appeared in the context of random matrix theory, too. Making use of the action-angle duality between the different variants of the CMS and the RSvD systems, new classes of random matrix ensembles emerged in the literature [15,16,17], exhibiting spectacular statistical properties. Under the name of classical/quantum duality, it has also been observed that the Lax matrices of the CMS and the RSvD models encode the spectra of certain quantum spin chains, thereby the purely classical models provide an alternative way to analyze the quantum systems, without any reference to the celebrated Bethe Ansatz techniques (for details see e.g. [18,19,20,21]). It is also worth mentioning that in the recent papers [22,23] the authors have constructed new integrable tops, closely related to the CMS and the RSvD particle systems. Besides the Lax representation of the dynamics, in their studies the associated rmatrix structures also turn out to be indispensable.
The characteristic feature the above exciting new developments all share in common is the prominent role played by the Lax matrices of the CMS and the RSvD models. However, all these investigations are based on the translational invariant models associated with the A n root system, exclusively. Apart from the technical difficulties, the probable explanation of this state of affair is the very limited knowledge about the Lax representation of the RSvD models in association with the non-A n -type root systems. Of course, one can easily construct Lax representations for both the C n -type and the BC n -type RSvD models by the Z 2 -folding of the A 2n−1 and the A 2n root systems, respectively [24]. However, this trivial approach is only of very limited use, since the resulting models contain only a single coupling parameter. Nevertheless, working in a symplectic reduction framework, in our papers [25,26] we succeeded in constructing Lax matrices for the rational C n and the rational BC n RSvD systems with the maximal number of independent coupling constants. Motivated by the plethora of potential applications outlined above, in this paper we work out the underlying classical r-matrix structures and also provide a Lax representation of the dynamics for the rational BC n RSvD model with three independent coupling parameters.
Let us recall that the configuration space of the rational BC n RSvD system is the open subset c = {λ = (λ 1 , . . . , λ n ) ∈ R n | λ 1 > . . . > λ n > 0} ⊆ R n , (1.1) that can be seen as an appropriate model for the standard open Weyl chamber of type BC n . The cotangent bundle T * c is trivial, whence the phase space of the RSvD system can be identified with the product manifold that we endow with the symplectic form We mention in passing that the unusual numerical factor in ω R is inserted purely for consistency with our earlier works [25,26]. As for the dynamics, it is governed by the Hamiltonian where µ, ν and κ are arbitrary real parameters satisfying µ < 0 < ν. Also, on these so-called coupling constants in this paper we impose the condition νκ ≥ 0. As can be seen in [27], this additional requirement ensures that the particle system possesses only scattering trajectories.
Having defined the models of our interest, now we wish to outline the content of the rest of the paper. To keep our present work essentially self-contained, in Section 2 we briefly skim through the necessary Lie theoretic machinery and the symplectic reduction background, that provide the building blocks of the latter developments. Also, this section allows us to fix the notations. Starting with Section 3 we present our new results. Section 3 is the longest and the most technical part of our paper, in which we study of the r-matrix structure of the rational C n RSvD model corresponding to the special choice κ = 0. Sticking to the Marsden-Weinstein reduction approach, in Subsection 3.1 we construct local extensions for the Lax matrix of the rational C n RSvD model. Making use of these local sections, in Subsection 3.2 a series of short Propositions and Lemmas allows us to construct a classical r-matrix structure for the C n -type model. In this respect our main result is Theorem 10, in which we formulate the r-matrix structure in a convenient quadratic form. The resulting quadratic r-matrices turn out to be fully dynamical, depending on all variables of the phase space P R . Subsequently, by switching to a purely algebraic approach, in Section 4 we generalize Theorem 10 to the rational BC n RSvD system with three independent coupling constants. The quadratic r-matrix structure of the BC n -type system is summarized in Theorem 11. To make this important result more transparent, in Theorem 12 we describe the r-matrix structure in a more convenient choice of gauge. In this gauge we also provide a Lax representation of the dynamics, as formulated in Theorem 13. Finally, in Section 5 we offer a short discussion on our results and also point out some open problems related to the RSvD systems.

Preliminaries
In this section we overview those Lie theoretic notions and results that underlie the geometric construction of the classical r-matrix structure for the rational C n RSvD system. Our approach is based on the symplectic reduction derivation of the RSvD models, that we also briefly outline. In Subsection 2.1 we closely follow the conventions of the standard reference [28], whereas in Subsection 2.2 we employ the notations introduced in our earlier work [25] on the RSvD systems.

Lie theoretic background
Take a positive integer n ∈ N and keep it fixed. Let N = 2n and introduce the sets N n = {1, . . . , n} and N N = {1, . . . , N}. (2.1) With the aid of the N × N matrix we define the non-compact real reductive matrix Lie group that we equip with the Cartan involution is a maximal compact subgroup of G, having the identification K ∼ = U(n) × U(n).
On the Lie algebra naturally induces the Cartan decomposition with the Lie subalgebra and the complementary subspace k = ker(ϑ − Id g ) and p = ker(ϑ + Id g ), (2.9) respectively. That is, each element Y ∈ g can be decomposed as with unique components Y + ∈ k and Y − ∈ p. Notice that the Z 2 -gradation (2.8) of g is actually orthogonal with respect to the non-degenerate Ad-invariant symmetric bilinear form To make our presentation simpler, for all k, l ∈ N N we introduce the standard elementary matrix e k,l ∈ gl(N, C) with entries Also, with each λ = (λ 1 , . . . , λ n ) ∈ R n we associate the N × N diagonal matrix The set of diagonal matrices a = {Λ(λ) | λ ∈ R n } (2.14) forms a maximal Abelian subspace in p. Note that in a the family of matrices forms an orthonormal basis, i.e. D − c , D − d = δ c,d for all c, d ∈ N n . The centralizer of the Lie algebra a inside K is the Abelian Lie group with Lie algebra In this Abelian Lie algebra the set of matrices forms a basis obeying the orthogonality relations D + c , D + d = −δ c,d (c, d ∈ N n ). Let m ⊥ and a ⊥ denote the sets of the off-diagonal elements of k and p, respectively. With these subspaces can write the refined orthogonal decomposition (2. 19) In other words, each element Y ∈ g can be uniquely decomposed as where each component belongs to the subspace indicated by the subscript. In order to provide convenient bases in the subspaces m ⊥ and a ⊥ , for each c ∈ N n we introduce the linear functional Let us observe that the set of functionals can be seen as a realization of a set of positive roots of type C n . Now, associated with the positive root 2ε c (c ∈ N n ), we define the matrices In association with the other positive roots, for all 1 ≤ a < b ≤ n we define the following matrices with purely real entries: a ± e n+a,n+b − e n+b,n+a ), together with the following ones with purely imaginary entries: a ± e n+a,n+b + e n+b,n+a ), (e a,n+b + e b,n+a ± e n+a,b ± e n+b,a ). (2.25) The point is that the set of vectors {X +,ǫ α } forms a basis in the subspace m ⊥ , whereas the family {X −,ǫ α } provides a basis in a ⊥ . Moreover, they obey the orthogonality relations Note that the family of vectors forms a basis in the real Lie algebra u(n, n). We mention in passing that it is a basis in the complexification gl(N, C) ∼ = u(n, n) C , too. Next we turn to the linear operator defined for each λ ∈ R n . The real convenience of the basis (2.27) stems from the commutation relations where c ∈ N n , α ∈ R + and ǫ ∈ {r, i}. Notice that the subspace m ⊥ ⊕ a ⊥ is invariant under the linear operator ad Λ(λ) , whence the restriction is well-defined for all λ ∈ R n , with spectrum The regular part of a is defined by the subset a reg = {Λ(λ) | λ ∈ R n and ad Λ(λ) is invertible}, (2.32) in which the standard Weyl chamber {Λ(λ) | λ ∈ c} is an appropriate connected component. Note that this Weyl chamber can be naturally identified with the configuration space c (1.1) of the rational BC n RSvD system. Having set up the algebraic stage, now we turn to some geometric results that are specific to the symplectic reduction derivation of the rational RSvD models. First, recall that the regular part of p (2.9) defined by is a dense and open subset of p. It is an important fact that with the smooth free right M-action is a smooth principal M-bundle, providing the identification In the geometric construction of the dynamical r-matrix for the rational C n RSvD model we shall utilize certain local sections of π with the characteristic properties below.
To proceed further, we introduce the set of complex column vectors that can be naturally identified with a sphere of real dimension 2n − 1. At each point V ∈ S the tangent space to S can be identified with the real subspace of the complex column vectors that we endow with the inner product δV, δv T V S = Re((δV ) * δv) (δV, δv ∈ T V S). (2.50) Next, we introduce the distinguished column vector E ∈ S with components E a = 1 and E n+a = −1 (a ∈ N n ).
(2.51) Also, with each vector V ∈ S we associate the N × N matrix Since the K-action on S defined by the smooth map is transitive, and since kξ(V )k −1 = ξ(kV ) for all k ∈ K and V ∈ S, it is clear that the adjoint orbit of K passing through the element ξ(E) ∈ k has the form As is known, the orbit O can be seen as an embedded submanifold of k, and for its tangent spaces we have the identifications In our earlier papers [29,30,25,26] we have seen many times that this non-trivial minimal adjoint orbit plays a distinguished role in the symplectic reduction derivation of both the CMS and the RSvD systems. In this paper, throughout the construction of a dynamical r-matrix for the rational C n RSvD system, we will also exploit that with the free U(1)-action it is clear that the derivative of ξ takes the form Let us also note that for all X ∈ k and V ∈ S we have XV ∈ T V S and The last two equations entail that for each δV ∈ T V S one can find a Lie algebra element X ∈ k and a real number t ∈ R such that Having determined the derivative of ξ, now we shall work out the derivatives of certain local sections, that find applications it the latter developments.
Proposition 2. Let V (0) ∈ S be an arbitrary point and define Then for the derivative of W at the point ρ (0) we have Proof. It is evident that there is a smooth local section W of the principal U(1)-bundle ξ that satisfies the conditions displayed in (2.63). Take an arbitrary tangent vector [X, ρ (0) ] ∈ T ρ (0) O generated by some X ∈ k, and introduce the shorthand notation By taking the derivative of the relationship ξ • W = IdǑ at the point ρ (0) , we find that ). However, due to (2.59) we can write that with a unique real number x. Its value can determined by the fact that the tangent vector δW belongs to subspace (ker(ξ * V (0) )) ⊥ , leading to the formula (2.64).

The rational C n RSvD model from symplectic reduction
Based on our earlier results, in this subsection we review the symplectic reduction derivation of the rational C n RSvD system. The surrounding ideas and the proofs can be found in [25]. An important ingredient of the symplectic reduction derivation of the RSvD system of our interest is the cotangent bundle T * G of the Lie group G (2.3). For convenience, we trivialize T * G by the left translations. Moreover, by identifying the dual space g * with the Lie algebra g (2.6) via the bilinear form (2.11), it is clear that the product manifold P = G × g provides an appropriate model for T * G. For the tangent spaces of the manifold P we have the natural identifications and for the canonical symplectic form ω ∈ Ω 2 (P) we can write where (y, Y ) ∈ P is an arbitrary point and ∆y ⊕ ∆Y, δy ⊕ δY ∈ T y G ⊕ g are arbitrary tangent vectors. An equally important building block in the geometric picture underlying reduction derivation of the RSvD model is the adjoint orbit O (2.54). Of course, it carries the Kirillov-Kostant-Souriau symplectic form ω O ∈ Ω 2 (O), that can be written as Making use of the bundle ξ (2.57) and the equations (2.60) and (2.61), one can easily see that Now, by taking the symplectic product of the symplectic manifolds (P, ω) and (O, ω O ), we introduce the extended phase space (2.72) To describe the Poisson bracket on this space, for each smooth function F ∈ C ∞ (P ext ), at each point u = (y, Y, ρ) ∈ P ext , we define the gradients by the natural requirement where δy ∈ T y G, δY ∈ g and X ∈ k are arbitrary elements. Now, one can easily verify that the Poisson bracket on P ext induced by the symplectic form ω ext can be cast into the form To proceed further, let us note that the smooth map is a symplectic left action of the product Lie group K × K on the extended phase space P ext , and it admits a K × K-equivariant momentum map As we proved in [25], the rational C n RSvD model can be derived by reducing the symplectic manifold P ext at the zero value of the momentum map J ext . Let us recall that the standard Marsden-Weinstein reduction consists of two major steps. At the outset, we need control over the level set that turns out to be an embedded submanifold of P ext (2.72). However, to get a finer picture, we still need some more background material. First, for each a ∈ N n we define the rational function Also, we need the vector-valued function F : that allows us to introduce the function A : P R → exp(p) with the matrix entries where a, b ∈ N n . As we have seen in [25], function A provides a Lax matrix for the rational C n RSvD model with the two independent parameters µ and ν. Next, let us consider the smooth function V : P R → S defined by the equation and also introduce the product manifold where U(1) * stands for the diagonal embedding of U(1) in the product group K × K. Having equipped with the above objects, now we are in a position to provide a convenient parametrization of the level set L 0 (2.78). Indeed, in [25] we proved that the map [25] we also proved that Υ R gives rise to a diffeomorphism from M R onto the embedded submanifold L 0 . In other words, the pair (M R , Υ R ) provides a model for the level set L 0 (2.78).
To complete the Marsden-Weinstein reduction, notice that the (residual) K × K-action on the model space M R (2.83) takes the form thus the orbit space M R /(K × K) can be naturally identified with the base manifold of the trivial principal (K × K)/U(1) * -bundle Now, the crux of the matter is the relationship that we proved in [25] by applying a chain of delicate arguments. Therefore, for the symplectic quotient in question we obtain the identification Finally, note that the K × K-invariant function survives the reduction, and by applying straightforward algebraic manipulations one can verify that the corresponding reduced function coincides with the Hamiltonian of the rational C n RSvD system with two independent coupling parameters µ and ν, that can be obtained from the BC n -type Hamiltonian (1.4) by setting κ = 0.
3 Dynamical r-matrix for the C n -type model Building on the symplectic reduction picture outlined in the previous subsection, our goal is to construct a classical r-matrix structure for the C n -type rational RSvD system with two independent coupling parameters. In the context of the CMS models, this geometric approach goes back to the work of Avan, Babelon, and Talon [31]. Eventually, in our paper [32], we succeeded to construct a dynamical r-matrix for the most general hyperbolic BC n Sutherland model with three independent coupling constants, too. It is worth mentioning that the surrounding ideas proves to be fruitful in the broader context of integrable field theories as well. For a systematic review see e.g. [33].
As we have seen in [25], the eigenvalues of the Lax matrix A (2.81) do commute, whence it follows from general principles that A obeys an r-matrix Poisson bracket (for proof, see e.g. [34,35]). However, we wish to make this r-matrix structure as explicit as possible. For this reason, Subsection 3.1 is devoted to the study of certain local extensions for the Lax matrix of the rational C n RSvD model. As it turns out, these local extensions are at the heart of the construction of the dynamical r-matrix structure for the RSvD system, that we elaborate in Subsection 3.2.

Local extensions of the Lax matrix A
The backbone of our reduction approach is the construction of the so-called local extensions of the Lax operator A (2.81), that we wish to describe below. For this reason, take an arbitrary point (λ (0) , θ (0) ) ∈ P R (3.1) and keep it fixed. Clearly the point is one of the representatives of (λ (0) , θ (0) ) in M R (2.83), that is, π R (s (0) ) = (λ (0) , θ (0) ). Moreover, let us introduce the shorthand notations together with Corresponding to s (0) (3.2), in the extended phase space we also introduce the reference point Now, associated with the elements given in (3.3-3.4), let us choose a local section (e, σ) of π (2.35), and also a local section W of ξ (2.57), as described in Propositions 1 and 2, respectively.
Upon defining the open subsetǧ it is clear that is a well-defined smooth function. Due to the conditions imposed in the equations (2.38) and (2.63), at the point u (0) (3.5) for the first n components of Ψ we have Since these components are strictly positive, there is an open subsetP ext ⊆ G×ǧ×Ǒ containing the distinguished point u (0) , such that for all a ∈ N n the map is well-defined and smooth. Let us keep in mind that by construction m a (u (0) ) = 1. Now we are in a position to define those group-valued functions that play the most important role in the construction of a dynamical r-matrix for the rational C n RSvD system. First, making use of the functions m a (3.9), we build up the M-valued function m :P ext → M, u → diag(m 1 (u), . . . , m n (u), m 1 (u), . . . , m n (u)), (3.10) which satisfies m(u (0) ) = 1. Next, we introduce the K-valued functions
Lemma 3. The G-valued smooth function A (3.14) is a local extension of the Lax matrix A (2.81) around the point u (0) in the sense that A(u (0) ) = A(λ (0) , θ (0) ) and Proof. It is enough to verify (3.15). For, take an arbitrary point with some λ ∈ c, θ ∈ R n and η L , η R ∈ K. Also, for brevity we define On the other hand, since Y ∈ǧ, we have Y − ∈p reg . Thus, making use of the local section (e, σ) introduced in (2.37), we see that Next, remembering the parametrization (2.85), we can write However, utilizing the local section W introduced in (2.62), we also have ξ(W (ρ)) = ρ, whence by (2.56) we can write that with some constant ψ ∈ R. From the above observations it readily follows that from where we get e iψ Ψ(u) =mF (λ, θ). Componentwise, for each a ∈ N n we can write e iψ Ψ a (u) =m a F a (λ, θ), (3.25) thus the relationship |Ψ a (u)| = F a (λ, θ) and alsõ m a = e iψ Ψ a (u) F a (λ, θ) = e iψ Ψ a (u) |Ψ a (u)| = e iψ m a (u) (3.26) are evident. In other words,m = e iψ m(u), whence from (3.21) and (3.12) we conclude that Now, turning to the functions (3.13) and (3.14), notice that Since s (3.16) is an arbitrary element of (Υ R ) −1 (P ext ), the Lemma follows.

Computing the r-matrix
The natural idea impregnated by Lemma 3 is that the Poisson brackets of the components of the Lax matrix A (2.81) can be computed by inspecting the Poisson brackets of the components of the locally defined function A (3.14). Indeed, since we reduce the symplectic manifold P ext (2.72) at the zero value of the K ×K-equivariant momentum map J ext (2.77), and since the local extension A is (locally) K × K-invariant on the level set L 0 (2.78), using the St. Petersburg tensorial notation we can simply write However, for the function A (0) (3.13) we clearly have that is, A (0) obeys an r-matrix bracket with the trivial zero r-matrix. Therefore, due to the relationship A = ϕ −1 A (0) ϕ (3.14), it is clear that A also obeys a linear r-matrix bracket with the transformed r-matrix Now, recalling that ϕ(u (0) ) = 1, from the relationships (3.30) and (3.33) we infer that for the Lax matrix A we can write with the r-matrix However, since ϕ = km (3.12), Leibniz rule yields together with Thus, in order to provide an explicit formula for the above r-matrix whereas the remaining gradients of Re(tr(vA (0) )) and Im(tr(vA (0) )) are trivial, i.e.
Having the necessary gradients at our disposal, now we are ready to work out the tensorial Poisson brackets appearing in (3.36) and (3.37).

89)
where for each a ∈ N n we have Proof. Working with the basis {v I } (2.27) of gl(N, C), from Proposition 5 we see that ∇ g (Re(tr(v I k)))(u (0) ) ∈ a ⊥ ⊆ p.
with the coefficients Proof. Using the antisymmetry of the Poisson bracket, we find that {m a , m b } ext (u (0) )(e a,a + e n+a,n+a ) ⊗ (e b,b + e n+b,n+b ) (3.96) To proceed further, let us choose arbitrary a, b ∈ N n satisfying a < b. Notice that the Poisson bracket formula (2.75) naturally leads to the expression {m a , m b } ext (u (0) ) = − (∇ G (Im(m a )))(u (0) ), (∇ g (Im(m b )))(u (0) ) (Im(m a )))(u (0) ), (∇ O (Im(m b )))(u (0) )). (3.97) However, by utilizing Proposition 6, each term on the right hand side of the above equation can be cast into a fairly explicit form. Starting with the first term, the application of (3.61) gives rise to the relationship (∇ G (Im(m a )))(u (0) ), (∇ g (Im(m b )))(u (0) ) = 1 Keeping in mind that a < b, a similar argument provides (3.99) Now, let us turn to the third appearing in (3.97 (3.100) Now, by simply putting together the above equations, the Lemma follows at once.
At this point we are in a position to provide an explicit formula for the r-matrix (3.35). Remembering (3.36), let us notice that Lemma 7 itself implies that r is in fact linear in A, having the form Recalling (3.37), the above expressions can be further expanded. Indeed, by simply plugging the formulae displayed in Lemmas 8 and 9 into (3.102), we may obtain explicit expressions for both p ± 12 and r. However, since r is linear in A as dictated by (3.101), the linear r-matrix Poisson bracket (3.34) can be cast into a quadratic form. Also, since the point (λ (0) , θ (0) ) (3.1) we fixed at the beginning of Subsection 3.1 was an arbitrary element of P R , the zero superscripts become superfluous and can be safely omitted. With the usual conventions for the symmetric and the antisymmetric tensor products, we end up with the following result.
Theorem 10. The Lax matrix A (2.81) of the rational C n RSvD model with two independent coupling parameters obeys the quadratic Poisson bracket with the g ⊗ g-valued dynamical structure coefficients where the constituent objects are defined in Lemmas 7, 8 and 9.
We conclude this section with an important remark. Since the quadratic structure matrices (3.105-3.108) are derived from an r-matrix linear in A as described in (3.101), from the relationships (3.109) it follows immediately that they satisfy the consistency conditions a 21 = −a 12 , d 21 = −d 12 , b 21 = c 12 , a 12 + b 12 = c 12 + d 12 . (3.110) The above observation can be paraphrased as follows. If a Lax matrix A obeys a tensorial Poisson bracket (3.34), and if the governing r-matrix is itself linear in A as in (3.101), then the tensorial Poisson bracket can be rewritten as a quadratic bracket (3.104) with quadratic structure matrices obeying the consistency conditions (3.110) automatically. It is a nice, but essentially trivial algebraic fact that the converse of this statement is also true. Indeed, suppose that a Lax matrix A obeys a quadratic Poisson bracket (3.104) with coefficients satisfying (3.110). Under these assumptions the quadratic bracket can be cast into a linear form (3.34). More precisely, the governing r-matrix can be written in the form of (3.101) with where u 12 is an arbitrary g ∨ g-valued function on the phase space, i.e. it obeys the symmetry condition u 21 = u 12 . This observation plays a crucial role in the developments of the next section.
4 Classical r-matrix structure of the BC n -type model Utilizing a symplectic reduction framework, so far we have studied the classical r-matrix structure for the rational C n RSvD model with two independent coupling parameters µ and ν. However, to handle the BC n -type model as well, in this section we slightly change our point of view. Switching to a purely algebraic approach, we shall generalize Theorem 10 to cover the most general rational BC n RSvD model with three independent coupling constants. As an added bonus, at the end of this section we will provide a Lax representation of the dynamics, too.
To describe the Lax matrix of the rational BC n RSvD system with the additional third real parameter κ, we need the functions where x ∈ (0, ∞). Also, with each λ = (λ 1 , . . . , λ n ) ∈ c we associate the group element h(λ) = diag(α(λ 1 ), . . . , α(λ n )) diag(β(λ 1 ), . . . , β(λ n )) −diag(β(λ 1 ), . . . , β(λ n )) diag(α(λ 1 ), . . . , α(λ n )) ∈ G. In [26] we proved that the smooth functionÃ : P R → G defined by the formulã provides a Lax matrix for the rational BC n RSvD model (1.4) with the independent coupling parameters µ, ν and κ. Our first goal in this section to construct a quadratic algebra relation for the Lax matrixÃ with structure coefficients satisfying the consistency conditions analogous to (3.110 Therefore, upon introducing the g ⊗ g-valued function we can write the tensorial Poisson bracket Now, by simply applying the Leibniz rule, from (3.104) we get that with the dynamical coefficients Since the decorations coming from h are 'equally distributed' among these new functions, we expect that likewise they satisfy the consistency conditions analogous to (3.110). Somewhat surprisingly, this naive idea is fully confirmed by the following result. In other words, the Lax matrixÃ (4.3) of the rational BC n RSvD system satisfies a quadratic Poisson bracket (4.7) characterized by the consistent dynamical structure coefficients (4.8-4.11).
Proof. A moment of reflection reveals thatã 21 = −ã 12 ,d 21 = −d 12 , andb 21 =c 12 , whence it is enough to prove thatã 12 +b 12 =c 12 +d 12 . Since the verification of this last equation is basically an involved algebraic computation, in the following we wish to highlight only the key steps. First, we introduce the functions Remembering (4.1), we see that (4.14) To make the presentation a slightly simpler, we also introduce the G-valued function Now, recalling (4.1), (4.2) and (4.14), with the notations together with the relations Having completed the proof, now we offer a few remarks on the result. First, since the Lax matrixÃ obeys the quadratic bracket (4.7) with the dynamical objects (4.8-4.11) satisfying the consistency conditions (4.12), the quadratic bracket (4.7) can be rewritten as (4.41) Indeed, recalling our discussion at the end of the previous section, an appropriate r-matrix is provided by the formular with an arbitrary g ∨ g-valued dynamical objectũ 12 . Second, one may raise the objection that the formulae (4.8-4.11) for the quadratic structure matrices in the BC n case are 'less explicit' than the analogous objects (3.105-3.108) in the C n case. The trouble is mainly caused by the derivatives of h −1 appearing in the definition of Γ 12 (4.5). Though these derivatives can be worked out rather easily, we propose an alternative approach to cure the problem. Namely, let us apply the gauge transformation are built up from the explicitly given functions (3.105-3.108), (4.17) and (4.20). Furthermore, by construction, they satisfy the consistency conditionŝ a 21 = −â 12 ,d 21 = −d 12 ,b 21 =ĉ 12 ,â 12 +b 12 =ĉ 12 +d 12 . (4.52) Theorem 13. With the aid of the g-valued function B = 1 2 α,ǫ tr(X +,ǫ α (Â −Ǎ))X −,ǫ α − tr(X −,ǫ α (Â +Ǎ))X +,ǫ α α(λ)

Discussion
One of the most important objects in the algebraic formulation of the theory of classical integrable systems is undoubtedly the r-matrix structure encoding the tensorial Poisson bracket of the Lax matrix. In the context of the A n -type CMS and RSvD models the underlying dynamical r-matrix structure is under complete control, even in the elliptic case (see e.g. [36,37,38,39]). In sharp contrast, for the models associated with the non-A n -type root systems the theory is far less developed. By generalizing the ideas of Avan, Babelon and Talon [31], in our earlier paper [32] we constructed a dynamical r-matrix structure for the most general hyperbolic BC n Sutherland system with three independent coupling constants. However, for the elliptic case only partial results are available [40]. For the non-A n -type RSvD systems the situation is even more delicate. Prior to our present paper, the r-matrix structure of the BC n RSvD systems was studied only in [41], based on the special one-parameter family of Lax matrices coming from Z 2 -folding of the A 2n root system. Nevertheless, in the present paper we succeeded in constructing a quadratic r-matrix structure for the rational BC n RSvD systems with the maximal number of three coupling parameters, as formulated in Theorems 11 and 12. It is also clear that by applying a standard analytic continuation argument on our formulae, one can easily derive a dynamical r-matrix structure for the rational RSvD system appearing in [42]. Regarding the hyperbolic, trigonometric and elliptic variants of the non-A n -type RSvD systems we also face many interesting questions. Indeed, except from some very special cases [24,43,44], even the construction of Lax matrices for these models is a wide open problem. However, let us note that in the last couple of years many results for the A n -type models have been reinterpreted in a more geometric context using advanced techniques from the theory of reductions (see e.g. [45,46]). Relatedly, it would be of considerable interest to see whether the underlying classical r-matrix structures can be explored from these geometric pictures along the line of our present paper. We also expect that the various reduction approaches eventually may lead to a progress in the rigorous geometric theory of the non-A n -type trigonometric, hyperbolic and elliptic RSvD systems as well. As a starting point, it is worth mentioning the recent paper [47], in which a Hamiltonian reduction approach based on the Heisenberg double of SU(n, n) gives rise to a new integrable particle system, that in the cotangent bundle limit gives back the familiar hyperbolic BC n Sutherland model with three independent coupling parameters.
Turning back to our quadratic r-matrix algebra (4.47), let us observe that the structure matricesâ 12 ,b 12 ,ĉ 12 andd 12 are fully dynamical, i.e. they depend on all variables of the phase space P R (1.2) in an essential way. It is in contrast with the CMS models, where the naturally appearing dynamical r-matrices usually depend only on the configuration space variables. Moreover, in many variants of the CMS models the r-matrices can be related to the dynamical Yang-Baxter equation, as first realized in [48]. However, in the A n case Suris [49] observed that in some special choice of gauge the CMS and the RSvD models can be characterized by the same dynamical r-matrices. Working in this gauge, Nagy, Avan and Rollet proved that the quadratic structure matrices of the hyperbolic A n RSvD system do obey certain dynamical quadratic Yang-Baxter equations (see Proposition 1 in [50], and relatedly also [51]). As a natural next step, we find it an important question whether such claims can be made about the quadratic algebra relation (4.47) in an appropriate gauge. Also, it would be of considerable interest to investigate whether the non-A n -type RSvD models can be characterized by numerical, i.e. non-dynamical r-matrices. In the A n case the answer is in the affirmative (see [52]), but in the BC n case the analogous tasks seem to be quite challenging even for the rational models. Nevertheless, we wish to come back to these problems in later publications.