Universal Lax pairs for spin Calogero-Moser models and spin exchange models

For any root system Δ and a set of vectors R which form a single orbit of the reflection (Weyl) group GΔ generated by Δ, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member μ of R, to be called a 'site', we associate a vector space Vμ whose element is called a 'spin'. Its dynamical variables are the canonical coordinates {qj, pj} of a particle in Rr (r = rank of Δ) and spin exchange operators {P-circumflexρ} (ρ is an element of Δ) which exchange the spins at the sites μ and s ρ(μ). Here sρ is the reflection generated by p. For each Δ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by 'freezing' the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For Δ=Ar and R = set of vector weights it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials. (author)