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.

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

References in corpus (8)

• Physics and Mathematics of Calogero particles
• Symmetries and Integrability
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• 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