Symmetries of Spin Calogero Models

We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group $W$ is wrong. More precisely, the symmetry algebra heavily depends on the representation of $W$ on the spins. We prove this by identifying two different symmetry algebras for a $B_L$ spin Calogero model and three for $G_2$ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.

💡 Research Summary

The paper revisits the symmetry structure of integrable spin Calogero models built from Dunkl operators associated with finite Coxeter groups. Starting from the standard construction, the authors consider a Hamiltonian that couples particle coordinates and spin degrees of freedom. The spin sector is not fixed a priori; instead, it is defined by an arbitrary finite‑dimensional representation ρ of the Coxeter group W on a spin space V. The central question is how the choice of ρ influences the algebra of conserved quantities, i.e. the symmetry algebra of the model.

To answer this, the authors first recall the Dunkl operators D_i and their commutation relations, which encode the Coxeter reflections. They then introduce a hierarchy of operators J^{(k)} built recursively from D_i and the spin generators S_a. These J^{(k)} are shown to be conserved (they commute with the Hamiltonian) and to satisfy a closed set of commutation relations. Remarkably, the algebra generated by the J^{(k)} is isomorphic to a twisted half‑loop algebra 𝔤