$R$-matrix-valued Dunkl operators and spin Calogero–Moser system

The Calogero-Moser model is a celebrated example of a completely integrable system, with numerous connections to several areas of mathematics and physics. It describes a system of $n$ of identical particles scattering on the line with inverse-square potential. There are also trigonometric, hyperbolic and elliptic version of this model. The integrability of the system can be shown in different ways, for example, constructing the higher Hamiltonans via Dunkl operators.

We propose an $R$-matrix generalization of the quantum elliptic Calogero-Moser system, based on the Baxter-Belavin elliptic $R$-matrix. This is achieved by introducing $R$-matrix-valued Dunkl operators so that commuting quantum spin Hamiltonians can be obtained from symmetric combinations of those. Using the freezing procedure, we construct integrable long-range spin chains. The talk is based on the joint work with Oleg Chalykh arXiv:2509.18989.