Poisson involutions, spin Calogero-Moser systems associated with symmetric Lie subalgebras and the symmetric space spin Ruijsenaars-Schneider models

Abstract

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars- Schneider equations which correspond to the $N$-soliton solutions of $A_n^{(1)}$ affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.

arXiv comment48 pages

References in corpus (7)

• Quantum Solitons in Affine Toda Field Theories
• 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
• Affine Toda Solitons and Vertex Operators
• Stokes Matrices and Poisson Lie Groups
• Integrable spin Calogero-Moser systems
• Affine Toda Solitons and Systems of Calogero-Moser Type

Cited by in corpus (7)

• Poisson-Lie analogues of spin Sutherland models
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
• Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone
• Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction