We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of
BCn type. In the elliptic case, two sets of generators are represented in terms of the Baxter--Belavin quantum
R-matrix in the fundamental representation of
GLN, while the remaining generators are represented by the elliptic
K-matrix found by Komori and Hikami. We revisit its derivation using the Felder--Pasquier procedure and correct the unitarity relation for odd
N. We then use these matrix representations to construct examples of long-range quantum spin chains from
R-matrix-valued Lax pairs for the classical Calogero--Inozemtsev system associated with the
BCn root system. We also construct
R-matrix-valued Lax pairs for Calogero--Moser models in an external field, leading to Haldane--Shastry and Polychronakos--Frahm chains with modified spin positions.
We construct Lax pairs with a spectral parameter for a family of models of interacting tops associated with the
BCn root system. The construction uses the parametric
BCn associative Yang--Baxter equations, which involve both quantum
R-matrices and boundary
K-matrices satisfying the reflection equation. Special cases yield spin generalizations of the elliptic Calogero--Inozemtsev model. We also show how quantum Hamiltonians of interacting tops, combined with the Polychronakos freezing prescription, reproduce recently introduced anisotropic quantum long-range spin chains of Haldane--Shastry--Inozemtsev type.
In this paper we discuss R-matrix-valued Lax pairs for slN 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.