Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models

Abstract

For any root system $\Delta$ and an irreducible representation $\mathcal{R}$ of the reflection (Weyl) group $G_{\Delta }$ generated by $\Delta$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $\mu$ of $\mathcal{R}$, to be called a "site", we associate a vector space ${\mathbf{V}}_{\mu }$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\mathbf{R}}^r$, ($r=$ rank of $\Delta$), and spin exchange operators $\{{\hat{\mathcal{P}}}_{\rho }\}$ ($\rho \in \Delta$) which exchange the spins at the sites $\mu$ and $s_{\rho }(\mu )$. Here $s_{\rho }$ is the reflection generated by $\rho$. For each $\Delta$ and $\mathcal{R}$ 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 $\Delta =A_r$ and $\mathcal{R}=$ 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.

18 pages, LaTeX2e, no figures

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• On R-matrix valued Lax pairs for Calogero-Moser models
• Geometric construction of elliptic integrable systems and N=1^* superpotentials
• Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Hierarchies of Spin Models related to Calogero-Moser Models
• Exactly solvable potentials of Calogero type for q-deformed Coxeter groups