R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type
M. Matushko, A. Mostovskii, A. Zotov
TL;DR
The paper develops an $R$-matrix valued Lax pair for the elliptic BC$_n$ Calogero–Inozemtsev system with five complex constants, extending Kirillov’s ${\rm B}$-type associative Yang–Baxter equations to a BC$_n$ setting with parameters via the Shibukawa–Ueno $R$-operator and Komori–Hikami $K$-operators. Through the Felder–Pasquier bridge, the authors express the Lax pair in terms of Baxter’s elliptic $R$-matrix and present a BC$_n$ long-range XYZ spin chain by applying a Polychronakos-like freezing trick, arguing for integrability. The work provides a blockwise verification of the Lax equation for both the scalar Takasaki form and the $R$-matrix generalization, leveraging intricate elliptic function identities (Fay-type) and a network of AYBE relations. Overall, the results unify elliptic integrable structures with BC$_n$ symmetry, offering a concrete path to BC$_n$ XYZ spin-chain Hamiltonians with an underlying Lax representation and suggesting further integrability proofs in the BC$_n$ context.
Abstract
We consider the elliptic Calogero-Inozemtsev system of ${\rm BC}_n$ type with five arbitrary constants and propose $R$-matrix valued generalization for $2n\times 2n$ Takasaki's Lax pair. For this purpose we extend the Kirillov's ${\rm B}$-type associative Yang-Baxter equations to the similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa-Ueno $R$-operator and the Komori-Hikami $K$-operators satisfying reflection equation. Then, using the Felder-Pasquier construction the answer for the Lax pair is also written in terms of the Baxter's 8-vertex $R$-matrix. As a by-product of the constructed Lax pair we also propose ${\rm BC}_n$ type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.