Degenerately Integrable Systems

Reshetikhin, Nicolai

doi:10.1007/s10958-016-2738-9

Cited by 31 publications

(37 citation statements)

References 29 publications

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“…These spin Sutherland models can be interpreted as Hamiltonian reductions of free motion on G, relying on the cotangent lift of the conjugation action of G on itself. The reduction can be utilized to show their integrability, and to analyze their quantum mechanics with the aid of representation theory [11,17,18,41,42,43]. Spinless models can be obtained in this way only for G = SU(n), using a minimal coadjoint orbit, for which the T-action on O 0 is transitive.…”

Section: Introductionmentioning

confidence: 99%

Poisson-Lie analogues of spin Sutherland models

Fehér

2019

Nuclear Physics B

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We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of 'free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the 'reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [5], Sutherland [51] and Moser [35]. Thanks to their fascinating mathematics and diverse applications [10,37,38,44,52], the interest in these models shows no sign of diminishing. New connections to mathematics and new applications are still coming to light in the current literature, see e.g. [6,7,22,24,46,53].The richness of these models is also due to their many generalizations and deformations. These are associated with different interaction potentials (from rational to elliptic), root systems and extensions with internal degrees of freedom. We call 'Sutherland models' the systems defined by trigonometric or hyperbolic potentials. For all these systems, classical and quantum mechanical versions are studied separately, and one needs to pay attention to the distinct features of the systems with real particle positions and their complexifications. The investigations of Ruijsenaars-Schneider (RS) type deformations [44,45] is motivated, for example, by relations to solitons, spin chains, special functions and double affine Hecke algebras.The internal degrees of freedom are colloquially called 'spin', and can be of two rather different kinds. First, the point particles can carry spins varying in a vector space, as is the case for the Gibbons-Hermsen models [20] and their RS type generalizations introduced by Krichever and Zabrodin [28]. Second, the models can involve a collective spin variable that typically belongs to a coadjoint orbit, and is not assigned separately to the particles. An example of this second type is the trigonometric spin Sutherland model defined classically by a Hamiltonian of the following form:

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Section: Introductionmentioning

confidence: 99%

Poisson-Lie analogues of spin Sutherland models

Fehér

2019

Nuclear Physics B

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“…It should be mentioned that the Poisson structure (and the classical r-matrix structure) for the spin elliptic Ruijsenaars-Schneider model is unknown yet. At the same time for the trigonometric and rational models the Poisson structures and the group-theoretical description are known [1,15,5,4,2,14].…”

Section: Introductionmentioning

confidence: 99%

Relativistic Interacting Integrable Elliptic Tops

Zotov

2019

Theor Math Phys

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We propose relativistic generalization of integrable systems describing M interacting elliptic gl(N ) tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic GL(M ) Ruijsenaars-Schneider model for N = 1 case, while in the M = 1 case they turn into relativistic integrable GL(N ) elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed. M k:k =iand for its non-diagonal part (1.4) yieldṡ( 1.7) These equations can be viewed as relativistic deformation [16] (the deformation parameter is η ∈ C) of the spin elliptic Calogero-Moser model [3]:(1.8)The system (1.8) admits anisotropic gl(NM) generalization, where the spin variables S ij are replaced by matrix-valued variables S ij ∈ Mat(N, C). Equations of motion are of the form:(1.9)2 Spin Ruijsenaars-Schneider modelIn this Section we derive equations of motion. It will help to simplify the proof of the more complicated statement related to interacting tops. More precisely, we prove the following Proposition 2.1 Equations of motioṅ

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“…Maximal superintegrability for a quantum system of phase-space dimension 2n has been characterized in the literature; for a recent account see [18] where it is termed 'maximal degenerate integrability'. The Hamiltonian H is part of a rank-n abelian algebra of Liouville charges (commuting Hamiltonians), but this is embedded in a larger commutant of H, which is spanned by the Liouville charges plus n−1 additional integrals of motion (Wojciechowski charges).…”

Section: Introductionmentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

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Abstract:The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

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Degenerately Integrable Systems

Cited by 31 publications

References 29 publications

Poisson-Lie analogues of spin Sutherland models

Poisson-Lie analogues of spin Sutherland models

Relativistic Interacting Integrable Elliptic Tops

The tetrahexahedric angular Calogero model

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