Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras

Abstract

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra G are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus τ and the Calogero-Moser couplings m to infinity, while keeping fixed the combination M = m e iδθτ for some exponent δ. Critical scaling limits arise when 1/δ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras G (1) and ( G (1)) V. The limits of the untwisted or twisted Calogero-Moser system, for δ less than these critical values, but non-zero, consists of the ordinary Toda system, while for δ = 0, it consists of the trigonometric Calogero-Moser systems for the algebras G and G V respectively.