Bi-Hamiltonian structure of Sutherland models coupled to two ${\mathfrak u}(n)^*$-valued spins from Poisson reduction

Feher, L.

doi:10.48550/arxiv.2109.07391

Cited by 3 publications

(2 citation statements)

References 47 publications

(72 reference statements)

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“…Notable successes of the method include the reduction treatment of integrable many-body models of Calogero-Moser-Sutherland and Ruijsenaars-Schneider type [5,19,20,22,27,37,38,41]. The study of this celebrated family of integrable systems and their extensions by so-called spin variables started decades ago [8,35,52,48,23,30,33,56] and still attracts considerable attention [3,13,15,16,17,28,29,44,53]. The prototype of the spin-particle models was introduced by Gibbons and Hermsen [23] using Hamiltonian reduction in a complex holomorphic setting.…”

Section: Introductionmentioning

confidence: 99%

Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $\operatorname{U}(n)$

Fairon¹,

Feher²

2023

Preprint

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We construct a master dynamical system on a U(n) quasi-Poisson manifold, M d , built from the double U(n) × U(n) and d ≥ 2 open balls in C n , whose quasi-Poisson structures are obtained from T * R n by exponentiation. A pencil of quasi-Poisson bivectors P z is defined on M d that depends on d(d − 1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the U(n)-invariant functions. The master system on M d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T * U(n) × C n×d . Its commuting Hamiltonians are pullbacks of the class functions on one of the U(n) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space M d / U(n) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors P z . The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of U(n) provide a new real form of the complex, trigonometric spin Ruijsenaars-Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the d = 1 case.

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Section: Introductionmentioning

confidence: 99%

Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $\operatorname{U}(n)$

Fairon¹,

Feher²

2023

Preprint

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“…Our observation is that T * G carries also a quadratic Poisson bracket that descends to the relevant quadratic bracket on G via the same reduction procedure which works in the linear case. The idea arises from [3,4], where bi-Hamiltonian structures for spin Sutherland models were obtained by reducing bi-Hamiltonian structures on the cotangent bundle of GL(n, C).…”

Section: Introductionmentioning

confidence: 99%

A note on quadratic Poisson brackets on $$\mathrm {gl}(n,{\mathbb {R}})$$ related to Toda lattices

Fehér

Juhász

2022

Lett Math Phys

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It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic r-matrix Poisson brackets on the associative algebra $$\mathrm {gl}(n,{\mathbb {R}})$$ gl ( n , R ) . We here show that the quadratic bracket on $$\mathrm {gl}(n,{\mathbb {R}})$$ gl ( n , R ) , corresponding to the r-matrix defined by the splitting of $$\mathrm {gl}(n,{\mathbb {R}})$$ gl ( n , R ) into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $$T^* \mathrm {GL}(n,{\mathbb {R}})$$ T ∗ GL ( n , R ) . This complements the interpretation of the linear r-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.

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A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices

Feher,

Juhasz

2022

Preprint

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It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic r-matrix Poisson brackets on the associative algebra gl(n, R). We here show that the quadratic bracket on gl(n, R), corresponding to the r-matrix defined by the splitting of gl(n, R) into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle T * GL(n, R). This complements the interpretation of the linear r-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.

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Bi-Hamiltonian structure of Sutherland models coupled to two ${\mathfrak u}(n)^*$-valued spins from Poisson reduction

Cited by 3 publications

References 47 publications

Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $\operatorname{U}(n)$

Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $\operatorname{U}(n)$

A note on quadratic Poisson brackets on $$\mathrm {gl}(n,{\mathbb {R}})$$ related to Toda lattices

A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices

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