Poisson-Lie Dual and Langlands Dual via Cluster Theory and Tropicalization

Yanpeng Li

For the Poisson-Lie dual K^* of a compact semisimple Lie group K, we construct a Poisson manifold PT(K^*) = C×T with a constant Poisson structure (here C is a certain polyhedral cone and T is a torus). The manifold PT(K^*) carries natural completely integrable systems with action-angle variables. In the case of K=SU(n), one of these integrable systems is the Gelfand-Cetlin completely integrable system. Our main result is a one-to-one correspondence between generic symplectic leaves in PT(K^*) and generic coadjoint orbits in Lie(K)^* preserving symplectic volumes of the leaves. This observation gives hope to construct a dense Darboux chart in Lie(K)^* modeled on PT(K^*).

http://scgp.stonybrook.edu/video_portal/video.php?id=3693

Simons- Program: Poisson geometry of moduli spaces, associators and quantum field theory