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Matrix representations of BCn associative Yang--Baxter equations and related long-range spin chains

  • M. Matushko
  • A. Mostovskii
  • A. Zotov
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{matushko2026matrix,
  title = {{Matrix representations of $\rm BC_n$ associative Yang--Baxter equations and related long-range spin chains}},
  author = {M. Matushko and A. Mostovskii and A. Zotov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33913},
  eprint = {2610.33913},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33913},
}
RIS
TY  - UNPB
TI  - Matrix representations of BC_n associative Yang--Baxter equations and related long-range spin chains
AU  - M. Matushko
AU  - A. Mostovskii
AU  - A. Zotov
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33913
UR  - https://arxiv.org/abs/2610.33913
AB  - We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of 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 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 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.
ER  -