## Lie Groups Seminar

Benjamin HoffmanCornell University
Positivity and potentials in Poisson geometry
Friday, September 15, 2017 - 3:30pm
Malott 406

The set of Hermitian matrices has a natural Poisson structure and is a completely integrable system, with action variables given by the Gelfand-Zeitlin cone. The Gelfand-Zeitlin cone also parametrizes crystal bases of irreducible highest weight modules of $U_q(\mathfrak{gl}_n)$. Our goal is to construct completely integrable systems on $\mathfrak{g}^*$, for compact simple $\mathfrak{g}$ of any type, using ideas from the study of crystal bases. I will speak about the first step toward that goal. This is joint with A. Alekseev, A. Berenstein, and Y. Li.