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

TL;DR

This paper presents a unified scheme for constructing integrable systems using symmetric Lie algebroids and Poisson groupoids, applying Dirac and Poisson reductions to derive spin Calogero-Moser and Ruijsenaars-Schneider models.

Contribution

It introduces a novel framework connecting symmetric Lie structures with integrable models via reduction techniques, extending the understanding of spin Calogero-Moser and Ruijsenaars-Schneider systems.

Findings

Constructed integrable systems from symmetric coboundary structures.

Derived spin Calogero-Moser systems using Dirac reduction.

Connected spin Ruijsenaars-Schneider equations to affine Toda solutions.

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 $A^{(1)}_n$ 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.