Integrable systems on multiplicative quiver varieties from cyclic quivers

Fairon, Maxime

doi:10.48550/arxiv.2108.02496

Cited by 3 publications

(4 citation statements)

References 60 publications

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“…so it suffices to check that (2.6) with F = g ij evaluated on v k verifies the first identity in (A. 13).…”

Section: A An Alternative Approach To the Quasi-poisson Ballmentioning

confidence: 96%

“…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%

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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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“…so it suffices to check that (2.6) with F = g ij evaluated on v k verifies the first identity in (A. 13).…”

Section: A An Alternative Approach To the Quasi-poisson Ballmentioning

confidence: 96%

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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“…Remarkably, we are still witnessing intense research in this area, regarding especially the 'relativistic' deformations [48] of the originally studied particle systems and their spin extensions [17,26,30,34]. Recent enquiries on these models are concerned, for example, with their derivation by Hamiltonian reduction [4,9,14], relations to moduli spaces of flat connections [2], harmonic analysis and special functions [10, 31,46], pole dynamics in the matrix KP hierarchy [43] and quiver varieties [8,13].…”

Section: Introductionmentioning

confidence: 99%

Bi-Hamiltonian structure of Sutherland models coupled to two u(n)* -valued spins from Poisson reduction

Fehér

2022

Nonlinearity

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We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group GL ( n , C ) , and study its Poisson reduction with respect to the action of the product group U(n) × U(n) arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of GL ( n , C ) . When taking the quotient of T * GL ( n , C ) we focus on the dense open subset of GL ( n , C ) whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the U(n) × U(n) invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two u ( n ) * -valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.

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“…Remarkably, we are still witnessing intense research in this area, regarding especially the 'relativistic' deformations [48] of the originally studied particle systems and their spin extensions [17,26,30,34]. Recent enquiries on these models are concerned, for example, with their derivation by Hamiltonian reduction [4,8,14], relations to moduli spaces of flat connections [2], harmonic analysis and special functions [10,31,46], pole dynamics in the matrix KP hierarchy [43] and quiver varieties [8,13].…”

mentioning

confidence: 99%

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

Feher

2021

Preprint

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We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group GL(n, C), and study its Poisson reduction with respect to the action of the product group U(n) × U(n) arising from left-and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of GL(n, C). When taking the quotient of T * GL(n, C) we focus on the dense open subset of GL(n, C) whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the U(n) × U(n) invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two u(n) * -valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.

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Integrable systems on multiplicative quiver varieties from cyclic quivers

Cited by 3 publications

References 60 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 $\operatorname{U}(n)$

Bi-Hamiltonian structure of Sutherland models coupled to two u(n)* -valued spins from Poisson reduction

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

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