Related papers: $R$-matrices for Elliptic Calogero-Moser Models
The classical (dynamical) $R$-matrices for the 2- and 3-body Calogero-Moser models with elliptic potentials are given. The 3-body case has an interesting nontrivial structure that goes beyond the known ansatz for momentum independent…
A new spectral parameter independent R-matrix (that depends on all of the dynamical variables) is proposed for the elliptic Calogero-Moser models. Necessary and sufficient conditions for this R-matrix to exist reduce to an equality between…
For the integrable $N$-particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical $r$-matrix structure. The $r$-matrix is a natural generalisation of the matrix found…
A classical R-matrix structure is described for the Lax representation of the integrable n-particle chains of Calogero-Olshanetski-Perelo\-mov. This R-matrix is dynamical, non antisymmetric and non-invertible. It immediately triggers the…
A complete description of the non-dynamical r-matrices of the degenerate Calogero-Moser models based on $gl_n$ is presented. First the most general momentum independent r-matrices are given for the standard Lax representation of these…
The most general momentum independent dynamical r-matrices are described for the standard Lax representation of the degenerate Calogero-Moser models based on $gl_n$ and those r-matrices whose dynamical dependence can be gauged away are…
In this paper, we construct a new Lax operator for the elliptic $A_{n-1}$ Calogero-Moser model with general $n(2\leq n$) from the classical dynamical twisting,in which the corresponding r-matrix is purely numeric (nondynamical one). The…
We use the definition of the Calogero-Moser models as Hamiltonian reductions of geodesic motions on a group manifold to construct their $R$-matrices. In the Toda case, the analogous construction yields constant $R$-matrices. By contrast,…
We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was…
Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of $ad L$. For generic $L$ a…
From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars'). The corresponding r-matrix is shown to be the classical $Z_n$-symmetric elliptic r-matrix,…
We compute the classical $r$-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter $\lambda$. We connect it with the non-relativistic…
We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate the theory of the classical r-matrix for integrable systems. In this picture the r-matrix is related to a fourth rank tensor, named the r-tensor, on the…
We study classical integrable systems based on the Alekseev-Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, ${\cal G}$. We prove that these r-matrices are uniquely characterized by a non-degeneracy…
I present a new formalism of the R-matrix theory where the formal parameters for the resonance energies and widths are identical to the observed values. By allowing the boundary condition parameters to vary from level to level, the freedom…
We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\rm GL}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$…
The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.
We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider model with N=2 admits a nondynamical r-matrix structure and the corresponding classical r-matrix is the same as that of its non-relativistic counterpart…
We present a derivation of the dynamical r-matrices of the Calogero-Moser models using the Hamiltonian reduction procedure to get general formulae. We describe the dynamical r-matrices thus found for spin Calogero-Moser models and…
The article is devoted to the study of $R$-matrix-valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde N=1$ the…