Poisson-Lie analogues of spin Sutherland models revisited

Abstract

Some generalizations of spin Sutherland models descend from `master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson--Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the reduced systems possess the property of degenerate integrability on the dense open subset of the Poisson quotient space corresponding to the principal orbit type for the pertinent group action. After restriction to a smaller dense open subset, degenerate integrability on the generic symplectic leaves is demonstrated as well. The paper also contains a novel description of the reduced Poisson structure and a careful elaboration of the scaling limit whereby our reduced systems turn into the spin Sutherland models.

arXiv comment32 pages, v2: slightly modified the abstract and corrected small typos

References in corpus (22)

• Physics and Mathematics of Calogero particles
• Classical and Quantum Superintegrability with Applications
• Symplectic structures associated to Lie-Poisson groups
• Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
• Relativistic Calogero-Moser model as gauged WZW theory
• Poisson-Lie interpretation of trigonometric Ruijsenaars duality
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• A class of Calogero type reductions of free motion on a simple Lie group
• Symmetries and Integrability
• Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group
• Poisson-Lie analogues of spin Sutherland models
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
• Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types spins
• On moment maps associated to a twisted Heisenberg double
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• Trigonometric real form of the spin RS model of Krichever and Zabrodin
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of compact Lie groups
• On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars--Sutherland hierarchy