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<p>We consider the elliptic Calogero–Inozemtsev system of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{BC}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>BC</mtext> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> type with five arbitrary constants and propose <i>R</i>-matrix valued generalization for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2n\times 2n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> Takasaki’s Lax pair. For this purpose, we extend the Kirillov’s <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>B</mtext> </math></EquationSource> </InlineEquation>-type associative Yang–Baxter equations to similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa–Ueno <i>R</i>-operator and the Komori–Hikami <i>K</i>-operators satisfying the 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 <i>R</i>-matrix. As a by-product of the constructed Lax pair we also propose a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\textrm{BC}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>BC</mtext> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.</p>

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R-matrix valued Lax pair for elliptic Calogero–Inozemtsev system and associative Yang–Baxter equations of \(\textrm{BC}_n\) type

  • 期刊论文

  • 发表日期：

    2026年1月17日

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  • M. Matushko,
  • A. Mostovskii,
  • A. Zotov

摘要

We consider the elliptic Calogero–Inozemtsev system of \(\textrm{BC}_n\) BC n type with five arbitrary constants and propose R-matrix valued generalization for \(2n\times 2n\) 2 n × 2 n Takasaki’s Lax pair. For this purpose, we extend the Kirillov’s \(\textrm{B}\) B -type associative Yang–Baxter equations to 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 the 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 a \(\textrm{BC}_n\) BC n type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.