Related papers: Rational Calogero-Moser Model: Explicit Form and r…
We construct the $r$-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the $r$-matrix for the $O(N)$ Euler-Calogero-Moser model and for the standard $A_N$…
We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. Our analysis is based on a…
In this paper, we construct a new Lax operator for the elliptic Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied . The relation between our Lax operator and the Lax operator given by…
We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with…
The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax…
We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the…
We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a…
The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental…
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…
We construct the classical Poisson structure and $r$-matrix for some finite dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows one to produce new kinds…
We demonstrate that in a certain gauge the Lax matrices of the rational and hyperbolic Ruijsenaars--Schneider models have a quadratic $r$-matrix Poisson bracket which is an exact quadratization of the linear $r$--matrix Poisson bracket of…
We describe the $R$-matrix structure associated with integrable extensions, containing both one-body and two-body potentials, of the $A_N$ Calogero-Moser $N$-body systems. We construct non-linear, finite dimensional Poisson algebras of…
We present a general scheme for constructing the Poisson structure of super-integrable dynamical systems of which the rational Calogero-Moser system is one of the most interesting one. This dynamical system is $2N$ dimensional with $2N- 1$…
The classical dynamical $r$-matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study…
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 apply Dirac's gauge fixing procedure to Chern-Simons theory with gauge group ISO(2,1) on manifolds RxS, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural…
We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable…
We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular…
Using a Lax pair based on twisted affine $sl(2,R)$ Kac-Moody and Virasoro algebras, we deduce a r-matrix formulation of two dimensional reduced vacuum Einstein's equations. Whereas the fundamental Poisson brackets are non-ultralocal, they…
In this paper we consider the Poisson algebraic structure associated with a classical $r$-matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the $r$-matrix…