New Approach to the Integrability of the Calogero Models

Abstract

We develop a new, systematic approach towards studying the integrability of the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero models associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras. It is based on the introduction of a function F, defined on the relevant root system and with values in the respective Cartan subalgebra, satisfying a certain set of combinatoric identities that ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, given by completely explicit formulas. It is shown that among the simple Lie algebras, only those belonging to the A-series admit such a function F, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p) x U(q)), allows non-trivial solutions when |p-q| <= 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our approach provides new ones, namely for the ordinary models when |p-q| = 1 and for the elliptic models when |p-q| = 1 or p = q.