Poisson reductions of master integrable systems on doubles of compact Lie groups
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. This yields a unified treatment of a large family of reduced systems, which contains new models as well as well familiar spin Sutherland and Ruijsenaars–Schneider models. It is proved that the reduced systems restricted on a dense open subset of the Poisson quotients are integrable in the degenerate sense. The talk is based on arXiv:2208.03728 complemented by more recent results.