Integrable extensions of the rational and trigonometric $A_N$ Calogero Moser potentials

TL;DR

This paper explores the algebraic structures underlying extended integrable Calogero-Moser systems, revealing connections to infinite-dimensional Lie algebras and string field theory.

Contribution

It introduces new $R$-matrix structures and Poisson algebras for extended Calogero-Moser systems, linking finite systems to infinite-dimensional Lie algebras.

Findings

Constructed non-linear Poisson algebras of observables.

Connected the $N o \infty$ limit to Sdiff algebras.

Linked algebraic structures to two-dimensional string field theory.

Abstract

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.