Related papers: Many-body integrable systems implied by WLZZ model…
A new series of integrable cases of the many-body problem in many-dimensional spaces is found. That series appears as a part of the larger series of integrable problems, which are in 1-1 correspondence with Krichever-Novikov algebras of…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
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…
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…
The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the…
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 a previous paper, we introduce a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter. Here the main purpose is to give explicit solutions of several factorization…
Recently, a new generalized family of infinite-dimensional $ \widetilde{W} $ algebras, each associated with a particular element of a commutative subalgebra of the $ W_{1+\infty} $ algebra, was described. This paper provides a comprehensive…
A new class of infinite-dimensional Lie algebras given a name of Lax operator algebras, and the related unifying approach to finite-dimensional integrable systems with spectral parameter on a Riemann surface, such as Calogero--Moser and…
We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero-Moser (CM) system. In the discrete level, the Lax pairs with a parameter are introduced and, of course, the discrete-time equations of…
The Calogero type matrix discretization scheme is applied to constructing the Lax type integrable discretizations of one wide enough class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and…
The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A…
A review of the appearence of integrable structures in the matrix model description of $2d$-gravity is presented. Most of ideas are demonstrated at the technically simple but ideologically important examples. Matrix models are considered as…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the…
We formulate one dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of…
Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are…
It is shown that spin Calogero-Moser systems are completely integrable in a sense of degenerate integrability. Their Liouville tori have dimension less then half of the dimension of the phase space. It is also shown that rational spin…
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 ${\rm GL}_N$…
The present article is an extended version of the paper {\it Phys. Rev.} {\bf B 59}, R2490 (1999), where, we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach…
The subject of this paper is degenerate integrability in Hamiltonian mechanics. It starts with a short survey of degenerate integrability. The first section contains basic notions. It is followed by a number of examples which include the…