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

Abstract

We construct a quantum integrable model which is an $R$-matrix generalization of the Calogero-Moser system, based on the Baxter-Belavin elliptic $R$-matrix. This is achieved by introducing $R$-matrix Dunkl operators so that commuting quantum spin Hamiltonians can be obtained from symmetric combinations of those. We construct quantum and classical $R$-matrix Lax pairs for these systems. In particular, we recover in a conceptual way the classical $R$-matrix Lax pair of Levin, Olshanetsky, and Zotov, as well as the quantum Lax pair found by Grekov and Zotov. Finally, using the freezing procedure, we construct commuting conserved charges for the associated quantum spin chain proposed by Sechin and Zotov, and introduce its integrable deformation. Our results remain valid when the Baxter-Belavin $R$-matrix is replaced by any of the trigonometric $R$-matrices found by Schedler and Polishchuk in their study of the associative Yang-Baxter equation.