arXiv2001
For any root system $Δ$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_Δ$ generated by $Δ$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $μ$ of ${\cal R}$, to be called a "site", we associate a vector space ${\bf V}_μ$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$, ($r=$ rank of $Δ$), and spin exchange operators $\{\hat{\cal P}_ρ\}$ ($ρ\inΔ$) which exchange the spins at the sites $μ$ and $s_ρ(μ)$. Here $s_ρ$ is the reflection generated by $ρ$. For each $Δ$ and ${\cal R}$ a {\em 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 $Δ=A_r$ and ${\cal R}=$ vector representation 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 {\em degenerate} potentials.