Gleb E. Arutyunov, Enrico Olivucci, “Hyperbolic Spin Ruijsenaars–Schneider Model from Poisson Reduction”, Modern problems of mathematical and theoretical physics, Collected papers. On the occasion of the 80th birthday of Academician Andrei Alekseevich Slavnov, Trudy Mat. Inst. Steklova, 309, Steklov Math. Inst. RAS, Moscow, 2020, 38–53; Proc. Steklov Inst. Math., 309 (2020), 31

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Volume 309, Pages 38–53
DOI: https://doi.org/10.4213/tm4089 (Mi tm4089)

This article is cited in 12 scientific papers (total in 12 papers)

Hyperbolic Spin Ruijsenaars–Schneider Model from Poisson Reduction

Gleb E. Arutyunov^ab, Enrico Olivucci^ab

^a II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
^b Zentrum für Mathematische Physik, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany

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DOI: https://doi.org/10.4213/tm4089

Abstract: We derive a Hamiltonian structure for the

N

$N$ -particle hyperbolic spin Ruijsenaars–Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for

N ℓ

$N\ell$ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson–Lie symmetry of the spin group

$\mathrm {GL}_{\ell }(\mathbb C)$ , which explains its superintegrability. Our results are obtained in the formalism of the classical

$r$ -matrix, and they are compatible with the recent findings on the different Hamiltonian structure of the model established in the framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.

Funding agency	Grant number
Deutsche Forschungsgemeinschaft	EXC 2121 - 390833306 Research Training Group 1670
The work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—EXC 2121 “Quantum Universe”—390833306. The work of E.O. is also supported by the DFG under the Research Training Group 1670.

Received: August 23, 2019
Revised: October 18, 2019
Accepted: March 30, 2020

English version:
Proceedings of the Steklov Institute of Mathematics, 2020, Volume 309, Pages 31–45
DOI: https://doi.org/10.1134/S0081543820030037

Bibliographic databases:

Document Type: Article

UDC: 514.853+517.938

Language: Russian

Citation: Gleb E. Arutyunov, Enrico Olivucci, “Hyperbolic Spin Ruijsenaars–Schneider Model from Poisson Reduction”, Modern problems of mathematical and theoretical physics, Collected papers. On the occasion of the 80th birthday of Academician Andrei Alekseevich Slavnov, Trudy Mat. Inst. Steklova, 309, Steklov Math. Inst. RAS, Moscow, 2020, 38–53; Proc. Steklov Inst. Math., 309 (2020), 31–45

Citation in format AMSBIB

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\pages 38--53

\publ Steklov Math. Inst. RAS

\publaddr Moscow

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Linking options:

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https://doi.org/10.4213/tm4089

https://www.mathnet.ru/eng/tm/v309/p38

This publication is cited in the following 12 articles:

Maxime Fairon, “Integrable systems on multiplicative quiver varieties from cyclic quivers”, J. Phys. A: Math. Theor., 58:4 (2025), 045202
M. Matushko, A. Zotov, “Supersymmetric generalization of $q$ -deformed long-range spin chains of Haldane–Shastry type and trigonometric $\mathrm{GL}(N|M)$ solution of associative Yang–Baxter equation”, Nuclear Phys. B, 1001 (2024), 116499–14
L Fehér, “Poisson–Lie analogues of spin Sutherland models revisited”, J. Phys. A: Math. Theor., 57:20 (2024), 205202
A. V. Zotov, M. G. Matushko, “Anisotropic spin generalization of elliptic Macdonald–Ruijsenaars operators and $R$ -matrix identities”, Ann. Henri Poincaré, 24 (2023), 3373–3419
I. Burić, F. Russo, A. Vichi, “Spinning partial waves for scattering amplitudes in d dimensions”, J. High Energ. Phys., 2023:10 (2023), 90
M. Fairon, L. Fehér, “Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of ${\text {U}}(n)$ ”, Ann. Henri Poincaré, 24:10 (2023), 3461
L. Fehér, “Poisson reductions of master integrable systems on doubles of compact Lie groups”, Ann. Henri Poincaré, 24:6 (2023), 1823
L Fehér, “Bi-Hamiltonian structure of Sutherland models coupled to two $(n)*$ -valued spins from Poisson reduction”, Nonlinearity, 35:6 (2022), 2971
A. V. Zotov, E. S. Trunina, “Lax equations for relativistic $\mathrm{G}\mathrm{L}(NM,\mathbb{C})$ Gaudin models on elliptic curve”, J. Phys. A, 55:39 (2022), 395202, 31 pp.
Fairon M., Feher L., Marshall I., “Trigonometric Real Form of the Spin Rs Model of Krichever and Zabrodin”, Ann. Henri Poincare, 22:2 (2021), 615–675
M. Fairon, L. Feher, “A decoupling property of some Poisson structures on $\mathrm{Mat}_{n\times d}(\mathbb{C})\times\mathrm{Mat}_{d\times n}(\mathbb{C})$ supporting $\mathrm{GL}(n,\mathbb{C})\times \mathrm{GL}(d,\mathbb{C})$ Poisson–Lie symmetry”, J. Math. Phys., 62:3 (2021), 033512
I. A. Sechin, A. V. Zotov, “Integrable system of generalized relativistic interacting tops”, Theoret. and Math. Phys., 205:1 (2020), 1291–1302

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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