Title: Integrable systems on the dual space of Lie algebras arising from log-canonical cluster structures
| Speaker: | Li Yu |
| Affiliation: | University of Toronto |
| Location: | MC 6029 |
There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.
Abstract: Let $(X, \{~,~\})$ be an (affine) Poisson variety. A log-canonical cluster structure on $X$ is a cluster structure on the coordinate ring of $X$ such that $\{\phi, \psi\} = \text{const} \cdot \phi \psi$ whenever $\phi, \psi$ are cluster variables which belong to the same cluster. When the Poisson bivector vanishes at some point $x \in X$, the tangent space $T_x X$ comes equipped with a Poisson bracket $\{~,~\}^{\text{lin}}$, the linearization of $\{~,~\}$. Given a function $\phi$ on $X$, we propose a way of linearizing it to get a function $\phi^{\text{lin}}$ on $T_x X$. Very often, when $\{\phi_1, \cdots, \phi_n\}$ is a cluster in a log-canonical cluster structure on $(X, \{~,~\})$, $\{\phi_1^{\text{lin}}, \cdots, \phi_n^{\text{lin}}\}$ is an integrable system on $(T_x X, \{~,~\}^{\text{lin}})$. We present two scenarios where this is the case.
\begin{enumerate}
\item Any cluster in the CGL cluster structure on the Bruhat cells in the flag variety;
\item The initial cluster in a generalized cluster structure (due to Gekhtman-Shapiro-Vainshtein) on the Poisson dual group of $GL_n$.
\end{enumerate}
In the second scenario, the resulting integrable system on $(\mathfrak {gl}_n^{\ast}, \{~,~\}_{\text{KKS}})$ is different than the well-known Gelfand-Zeitlin integrable system. Time permitting, we will explain some geometric and combinatorial properties of these integrable systems. This is based on joint work in progress with Yanpeng Li and Jiang-Hua Lu.