R-matrix-valued Lax pairs and long-range spin chains
In this paper we discuss R -matrix-valued Lax pairs for sl N Calogero-Moser model and their relation to integrable quantum long-range spin chains of the Haldane-Shastry-Inozemtsev type. First, we construct the R -matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M -matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of the R -matrix these Hamiltonians reproduce those for the Inozemtsev chain. In the general case related to the Baxter's elliptic R -matrix we obtain a natural anisotropic extension of the Inozemtsev chain. Commutativity of the Hamiltonians is verified numerically. Trigonometric limits lead to the Haldane-Shastry chains and their anisotropic generalizations.