Reduction of a bi-Hamiltonian hierarchy on $$T^*\mathrm{U}(n)$$ to spin Ruijsenaars–Sutherland models

2021

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We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Section: Introductionmentioning

confidence: 99%

Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

2021

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“…In this section, we present a quick overview of these structures, to be used in the subsequent sections. More details can be found in the references [16,19,29,30,35,52] and in "Appendix A. "…”

Section: Heisenberg Double Primary Spins and 'Free' Integrable Systemmentioning

confidence: 99%

“…to transfer the Heisenberg double Poisson bracket to C ∞ (U(n) × B(n)). The formula of the resulting Poisson bracket [16], called { , } 1 + , can be written as follows: 17) for any F, H ∈ C ∞ (U(n) × B(n)). The derivatives on the right-hand side are taken at (g, b) ∈ U(n) × B(n); the subscripts 1 and 2 refer to derivatives with respect to the first and second arguments.…”

Section: The Heisenberg Double and Its Modelsmentioning

confidence: 99%

“…(2.22) This definition makes sense since (L+tX) ∈ P(n) for small t. By using m 2 , one can transfer the Poisson bracket { , } + to C ∞ (U(n) × P(n)). The resulting Poisson bracket is called { , } 2 + and is given [16] by the following explicit formula:…”

Section: The Heisenberg Double and Its Modelsmentioning

confidence: 99%

“…The unbroken interest in integrable many-body systems of Calogero-Moser-Sutherland [8,38,54] and Ruijsenaars-Schneider (abbreviated RS) [50] types is due to their ubiquity in physical applications and rich web of connections to important areas of mathematics [3,12,40,49,55]. The same can be said about spin extensions of these models, which currently attract attention [5,6,10,[14][15][16]28,33,42,[45][46][47]51,57].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

Fairon

Marshall

2020

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We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the $${\mathrm{U}}(n)$$ U ( n ) Poisson–Lie group. The Poisson structure on the unreduced real phase space $${\mathrm{GL}}(n,{\mathbb {C}})\times {\mathbb {C}}^{nd}$$ GL ( n , C ) × C nd is the direct product of that of the Heisenberg double and $$d\ge 2$$ d ≥ 2 copies of a $${\mathrm{U}}(n)$$ U ( n ) covariant Poisson structure on $${\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}$$ C n ≃ R 2 n found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.

Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups

2023

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We consider three ‘classical doubles’ of any semisimple, connected and simply connected compact Lie group G: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of ‘master integrable systems’ and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical r-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars–Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.