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arXiv:2610.33913

Matrix representations of ${\rm BC}_n$ associative Yang--Baxter equations and related long-range spin chains

M. Matushko, A. Mostovskii, A. Zotov

Published Oct 24, 2026• 0 citations
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Abstract

We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of ${\rm BC}_n$ 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 ${\rm GL}_N$, 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 ${\rm BC}_n$ 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.

DOI
10.48550/arxiv.2610.33913

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