Yu Li (University of Toronto)
CIRGET Algebraic Geometry Seminar
Title: Polynomial integrable systems from cluster structures
Abstract: We present a general framework for constructing polynomial integrable systems with respect to linearizations of Poisson varieties that admit log-canonical coordinate systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures. As special cases, we consider an arbitrary standard complex semisimple Poisson Lie group $G$ with the Berenstein-Fomin-Zelevinsky cluster structure, nilpotent Lie subgroups of $G$, identified with Schubert cells in the flag variety of $G$, with the standard cluster structure, and the dual Poisson Lie group of ${\rm GL}(n, \mathbb C)$ with the Gekhtman-Shapiro-Vainshtein generalized cluster structure. In each of the three cases, we show that every extended cluster in the respective cluster structure gives rise to a polynomial integrable system on the respective Lie algebra with respect to the linearization of the Poisson structure at the identity element. In the third example, we obtain an integrable system on $(\mathfrak{gl}(n, \mathbb C)^*, \pi_{KKS})$ which is different from the well-known Gelfand-Zeitlin system. For some of the integrable systems thus obtained, we give Lie theoretic interpretations of their Hamiltonians, and we further show that all their Hamiltonian flows are complete.
No prior knowledge of cluster algebras is required.
This is joint work with Yanpeng Li and Jiang-Hua Lu.
In person:聽Location: UQAM PK-5675