Action-angle coordinates are a type of coordinate chart on symplectic manifolds originating from the theory of commutative completely integrable systems. Symplectic toric manifolds are the prototypical example of symplectic manifolds with action-angle coordinates on a dense subset.

Multiplicity-free spaces are the natural non-abelian generalization of toric manifolds. For example, coadjoint orbits of compact Lie groups are multiplicity-free spaces. Unlike toric manifolds, multiplicity-free spaces do not come readily equipped with action-angle coordinates on a dense subset.

This talk will present a new construction of action-angle coordinates on big subsets of a large family of multiplicity-free spaces. The main tools are Poisson-Lie groups and tropicalization of double Bruhat cells.

This talk is based on collaboration with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li.