Calogero-Moser Models: A New Formulation

A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax-pairs and the proof of the integrability applicable to all models based on the simply-laced algebras (ADE) are given for two types which we will call ‘root’ and ‘minimal’. The root type Lax pair is new; the matrices used in its construction bear a resemblance to the adjoint representation of the associated Lie algebra, and exist for all models, but they do not contain elements associated with the zero weights corresponding to the Cartan subalgebra. The root type provides a simple proof of the integrability of all models, including the one based on E8, whose integrability had been an unsolved problem for more than twenty years. The minimal types provide a unified description of all known examples of CalogeroMoser Lax pairs and add some more.