Deformed Double Yangian Structures

Scaling limits when q tends to 1 of the elliptic vertex algebras A_qp(sl(N)) are defined for any N, extending the previously known case of N=2. They realise deformed, centrally extended double Yangian structures DY_r(sl(N)). As in the quantum affine algebras U_q(sl(N)), and quantum elliptic affine algebras A_qp(sl(N)), these algebras contain subalgebras at critical values of the central charge c=-N-Mr (M integer, 2r=ln p/ln q), which become Abelian when c=-N or 2r=Nh for h integer. Poisson structures and quantum exchange relations are derived for their abstract generators.Comment: 16 pages, LaTeX2e Document - packages amsfonts,amssymb,subeqnarra

Integrable extensions of the rational and trigonometric ANA_N Calogero Moser potentials

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 observables. Their $N \rightarrow \infty$ limit realize the infinite Lie algebras Sdiff$({\Bbb R} \times S_1 )$ in the trigonometric case and Sdiff$({\Bbb R }^2)$ in the rational case. It is then isomorphic to the algebra of observables constructed in the two-dimensional collective string field theory.Comment: 15 pages; LaTeX; PAR LPTHE 93-23 Revised version including extensive modifications in the demonstrations and the reference

Classical R-matrix structure for the Calogero model

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 integrability of the Type I, II and III potentials, and the algebraic structures associated with the Type V potential.Comment: Latex file 9 page

Explicit solutions of the classical Calogero & Sutherland systems for any root system

Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems.Comment: 18 pages, LaTeX, no figur

The Classical rr-Matrix for the Relativistic Ruijsenaars-Schneider System

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 Calogero-Moser $r$-matrix $(\lambda \rightarrow -1)$ and the $\lambda = 1$ sine-Gordon soliton limit.Comment: LaTeX file, no figures, 8 page

Deformed Virasoro algebras from elliptic quantum algebras

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the $gl(2)$ case) a Liouville formula.Comment: 24 page