R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type
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arXiv
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| Format: | Preprint |
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2025
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| author | Matushko, M. Mostovskii, A. Zotov, A. |
| author_facet | Matushko, M. Mostovskii, A. Zotov, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type Matushko, M. Mostovskii, A. Zotov, A. Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems 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. |
| title | R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type |
| topic | Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2503.22659 |