Laszlo Feher
Integrable Hamiltonian systems from reductions of doubles of compact Lie groups
We deal with reductions of integrable `master systems' living on the `classical doubles' of any semisimple, connected and simply connected compact Lie group $G$. The doubles in question are the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson/quasi-Hamiltonian double, each of which carries two natural integrable systems. In the cotangent bundle case, one of the integrable systems is generated by the class functions of $G$ and the other one by the invariant functions of its Lie algebra. The reduction is defined by taking quotientby the cotangent lift of the conjugation action of $G$ on itself, and this generalizes to the other two doubles. The quotient space of the internally fused double represents the moduli space of flat principal $G$-connections on the torus with a hole. We explain that degenerate integrability of the master systems is inherited on the smooth component of the Poisson quotient corresponding to the principal orbit type for the pertinent $G$-action, and present explicit formulas for the reduced Poisson structure and equations of motion in terms of dynamical $r$-matrices after further restriction to a dense open subset.
- Lecture 1. The integrable master systems on the classical doubles and the definition of their Poisson reductions. The warm up case of the cotangent bundle.
- Lecture 2. Degenerate integrability on the Poisson quotient of the Heisenberg double corresponding to the principal orbit type and the interpretation of the reduced systems as Ruijsenaars–Schneider (RS) type many-body models extended by `spin' degrees of freedom. A detour towards the spin RS models of Krichever and Zabrodin.
- Lecture 3. The case of the quasi-Poisson double. Specific examples on small symplectic leaves for $G=SU(n)$. Ruijsenaars duality in the framework of Hamiltonian reduction. Open problems.
