Work

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

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.33913

Authors: Matushko, M., Mostovskii, A., Zotov, A.

Institution: Lomonosov Moscow State University

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Ministry of Science and Higher Education of the Russian Federation, Foundation for the Advancement of Theoretical Physics and Mathematics

Awards: G6444257319, G7301595098