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

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 tau and the Calogero-Moser couplings m to infinity, while keeping fixed the combination M = m e(i pi delta tau) for some exponent delta. Critical scaling limits arise when 1/delta 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)))(boolean OR). The limits of the untwisted or twisted Calogero-Moser system, for delta less than these critical values, but non-zero, consists of the ordinary Toda system, while for delta = 0, it consists of the trigonometric Calogero-Moser systems for the algebras G and G(boolean OR) respectively. (C) 1998 Published by Elsevier Science B.V.