Integrable system of generalized relativistic interacting tops

Сечин, Иван Андреевич; Zotov, A.

doi:10.1134/s0040577920100049

Cited by 6 publications

(2 citation statements)

References 29 publications

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“…The calculations for the terms with a = b are performed in the same way as was described in [33]. So that, here we consider the case a = b only.…”

Section: Generalized Model Through R-matrix Formulationmentioning

confidence: 99%

See 1 more Smart Citation

Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

Trunina¹,

Zotov²

2022

J. Phys. A: Math. Theor.

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We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ interacting relativistic ${\rm GL}_N$ tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on ${\rm GL}$-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars-Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general ${\rm GL}_{NM}$ model is also described through $R$-matrices satisfying associative Yang-Baxter equation. This description provides the trigonometric and rational analogues of ${\rm GL}_{NM}$ models.

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“…The calculations for the terms with a = b are performed in the same way as was described in [33]. So that, here we consider the case a = b only.…”

Section: Generalized Model Through R-matrix Formulationmentioning

confidence: 99%

“…Finally, the third family consists of the mixed type GL NM models similarly to its nonrelativistic analogue from the scheme 1. The models 4 and 7 on the scheme 2 were described in [33,46]. The model 1 is on the top of the scheme 2, and this is the subject of this article.…”

Section: Introduction: Classification Schemementioning

confidence: 99%

Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

Trunina¹,

Zotov²

2022

J. Phys. A: Math. Theor.

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Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

Matushko,

Zotov

2024

Nuclear Physics B

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Elliptic generalisation of integrable q-deformed anisotropic Haldane–Shastry long-range spin chain

Matushko

Zotov

2022

Nonlinearity

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We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars–Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin G L M R -matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars–Schneider model. Trigonometric degenerations are studied as well. For example, in M = 2 case our construction provides q-deformation for anisotropic XXZ Haldane–Shastry model. The standard Haldane–Shastry model and its Uglov’s q-deformation based on U q ( g l ˆ M ) XXZ R-matrix are included into consideration by separate verification.

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Integrable system of generalized relativistic interacting tops

Cited by 6 publications

References 29 publications

Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

Elliptic generalisation of integrable q-deformed anisotropic Haldane–Shastry long-range spin chain

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