Superintegrable Systems on Moduli Spaces of Flat Connections

Arthamonov, Semeon; Reshetikhin, Nicolai

doi:10.1007/s00220-021-04128-5

Cited by 9 publications

(23 citation statements)

References 44 publications

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“…This model has a natural interpretation in terms of moduli spaces of flat G-connections on a punctured torus. It also has a natural generalization to moduli spaces of flat connections on surfaces [1]. We expect that the relativistic version of the two-sided Calogero-Moser system is related to the moduli space of flat connections on a torus with two punctures.…”

Section: Discussionmentioning

confidence: 99%

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Spin Calogero-Moser models on symmetric spaces

Reshetikhin¹

2019

Preprint

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In this paper we construct and prove superintegrability of spin Calogero-Moser type systems on symplectic leaves of K 1 \T * G/K 2 where K 1 , K 2 ⊂ G are subgroups. We call them two sided spin Calogero-Moser systems. One important type of such systems correspond towhere K is a subgroup of fixed points of Chevalley involution θ : G → G. The other important series of examples come from pair G ⊂ G × G with the diagonal embedding. We explicitly describe examples of such systems corresponding to symplectic leaves of rank one when G = SL n .4. This paper is the following organization. In the first section, we construct spin Calogero-Moser type systems on symplectic leaves of K 1 \T * G/K 2 and Poisson projections which prove their superintegrability. In the second section, we analyze the example of symmetric pairswhere G θ is the subgroup of fixed points of the Chevalley automorphism θ of G. In the third section, we consider an example of the symmetric pair G ∈ G × G where G is a simple Lie group embedded in G × G diagonally. In this case K 1 = K 2 = G. In section 4, we describe in details the two-sided spin Calogero-Moser system corresponding to two rank 1 orbits. In the Appendices, we recall some basic and useful facts on action of Lie grounds on their contangent bundles and Poisson brackets between matrix element functions. In the Conclusion, we outline some further problems.

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Section: Discussionmentioning

confidence: 99%

“…1. Calogero-Moser models are integrable Hamiltonian systems describing interacting onedimensional many particle systems 1 [2] [10].…”

Section: Introductionmentioning

confidence: 99%

Spin Calogero-Moser models on symmetric spaces

Reshetikhin¹

2019

Preprint

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“…This provides additional conserved quantities to the associated quantum integrable system, thereby turning it into a quantum superintegrable system. The latter can be regarded as a quantization of the classical superintegrable system on moduli spaces of flat connections over surfaces, as constructed in [2] in the case where the surface is a punctured torus.…”

Section: Introductionmentioning

confidence: 99%

“…as identities in U (2) . The element R (see (1.15)) is Drinfeld's [8] universal R-matrix of U q , considered as element in the completion U (2) of U ⊗2 q .…”

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Graphical calculus for quantum vertex operators, I: The dynamical fusion operator

Clercq¹,

Reshetikhin²,

Stokman³

2022

Preprint

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This paper is the first in a series on graphical calculus for quantum vertex operators. We establish in great detail the foundations of graphical calculus for ribbon categories and braided monoidal categories with twist. We illustrate the potential of this approach by applying it to various categories of quantum group modules, in particular to derive an extension of the linear operator equation for dynamical fusion operators, due to Arnaudon, Buffenoir, Ragoucy and Roche, to a system of linear operator equations of q-KZ type.Contents 1.5. The ribbon element and twisting in category M 1.6. Dual representations 2. Graphical calculus 2.1. Ribbon graphs and ribbon graph diagrams 2.2. Ribbon-braid graphs and ribbon-braid graph diagrams 2.3. Strictifications 2.4. Graphical calculus for D-colored ribbon-braid graphs 2.5. The Reshetikhin-Turaev functor 2.6. Colored ribbon graph subdiagrams in B D 2.7. Fusion morphisms 2.8. Bundling of strands 3. q-KZ equations for k-point dynamical fusion operators 3.1. k-point quantum vertex operators and their graphical representation 3.2. The topological operator q-KZ equations 3.3. The topological q-KZ equations for the k-point dynamical fusion operator Conclusion References

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

Fehér

2023

Ann. Henri Poincaré

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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.

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Superintegrable Systems on Moduli Spaces of Flat Connections

Cited by 9 publications

References 44 publications

Spin Calogero-Moser models on symmetric spaces

Spin Calogero-Moser models on symmetric spaces

Graphical calculus for quantum vertex operators, I: The dynamical fusion operator

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

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