Related papers: Spin Calogero models and dynamical r-matrices

We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete…

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,…

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 develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary Poisson groupoids. The method is based on the successive use of Dirac reduction…

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 consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum…

We construct the Hamiltonians and symmetry generators of Calogero-oscillator and Calogero-Coulomb models on the N-dimensional sphere within the matrix-model reduction approach. Our method also produces the integrable Calogero-Coulomb-Stark…

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…

For any root system $\Delta$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_\Delta$ generated by $\Delta$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic,…

We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra…

We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a classical Calogero model by a subset of its discrete symmetries. Such reductions reproduce all known variants of these systems, including some…

Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP…

We consider the classical Calogero-Sutherland system with two types of interacting spin variables. It can be reduced to the standard Calogero-Sutherland system, when one of the spin variables vanishes. We describe the model in the Hitchin…

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the…

Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of…

We introduce a family of classical integrable systems describing dynamics of $M$ interacting ${\rm gl}_N$ integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the ${\rm GL}_N$…

Non linear sigma models on Riemannian symmetric spaces constitute the most general class of classical non-linear sigma models which are known to be integrable. Using the current algebra structure of these models their canonical structure is…

The Hamiltonian of the $N$-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian…

In this article we consider two particular examples of general construction proposed in arXiv:2012.15529. We consider the integrable extensions of the classical elliptic Calogero-Moser model of N particles with spin and the integrable…