On Moment Maps Associated to a Twisted Heisenberg Double

Klimčı́k, Ctirad

doi:10.1142/s0129055x06002796

Cited by 10 publications

(24 citation statements)

References 19 publications

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“…The material collected below is well known to experts (see e.g. [16,27,22,23,40]), except perhaps the presentation of the quasi-adjoint action that we shall give. We start by recalling that the Heisenberg double of the standard Poisson-Lie group U(n) is the real Lie group GL(n, C) equipped with a certain Poisson structure.…”

Section: The Rudiments Of the Heisenberg Doublementioning

confidence: 99%

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

Fehér

2019

Lett Math Phys

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We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group U(n) in such a way that the invariant functions of the u(n) * -valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson-Lie group U(n) into T * U(n), and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on T * U(n) using the conjugation action of U(n), for which the ring of invariant functions is closed under both Poisson brackets. We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars-Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment.

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Section: The Rudiments Of the Heisenberg Doublementioning

confidence: 99%

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

Fehér

2019

Lett Math Phys

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“…The symplectic reduction now can be performed and the question arises whether we can identify the orbits of a LS action on L u which would coincide with the integrated surfaces of the integrable distribution. The answer is affirmative [4] and it reads:…”

Section: Symplectic Reductionmentioning

confidence: 99%

“…The theory of Poisson-Lie symmetric deformations of the standard WZW models [6] was developed in [2,3,4] and it is based on the concept of the twisted Heisenberg double [5]. This contribution is a review of a part of our work [4].…”

Section: Introductionmentioning

confidence: 99%

“…This contribution is a review of a part of our work [4]. It is intended to the attention of those readers who are interested just in the direct description and gauging of one particular example of the Poisson-Lie WZW deformation and do not wish to go through the general theory of the twisted Heisenberg doubles exposed in [4].…”

Section: Introductionmentioning

confidence: 99%

“…It was established in [4] that these actions can infinitesimally be expressed via the Poisson bivector Π u , corresponding to the symplectic form ω u :…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

u-Deformed WZW Model and Its Gauging

Klimčı́k¹

2006

SIGMA

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

Fairon

Fehér

Marshall

2020

Ann. Henri Poincaré

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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.

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On Moment Maps Associated to a Twisted Heisenberg Double

Cited by 10 publications

References 19 publications

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

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

u-Deformed WZW Model and Its Gauging

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

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