Poisson reductions of master integrable systems on doubles of compact Lie groups

Abstract

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.

arXiv comment33 pages, minor edits in v2, correction of small typos in v3. v4: Corrected equation (3.27). Refined some remarks about averaging the unreduced constants of motion, after equation (1.10) and in Appendix C

References in corpus (19)

• Physics and Mathematics of Calogero particles
• Symplectic structures associated to Lie-Poisson groups
• Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
• Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra
• Duality in Integrable Systems and Gauge Theories
• Poisson-Lie interpretation of trigonometric Ruijsenaars duality
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• Integrable spin Calogero-Moser systems
• Symmetries and Integrability
• Spin Calogero models obtained from dynamical r-matrices and geodesic motion
• Poisson-Lie analogues of spin Sutherland models
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
• Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction
• New compact forms of the trigonometric Ruijsenaars-Schneider system
• Superintegrable Systems on Moduli Spaces of Flat Connections
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• On moment maps associated to a twisted Heisenberg double
• 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

Cited by in corpus (2)

• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Poisson-Lie analogues of spin Sutherland models revisited