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 $G = S U (n)$ . Ruijsenaars duality in the framework of Hamiltonian reduction. Open problems.