Polynomial integrable systems from cluster structures (Yanpeng Li, Sichuan University)

17.12.2024 15:30 – 16:30

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 examples, we consider an arbitrary standard complex semi-simple Poisson Lie group G with the Berenstein-Fomin-Zelevinsky cluster structure, nilpotent Lie subgroups of G associated to elements of the Weyl group of G, identified with Schubert cells in the flag variety of G and equipped with the standard cluster structure (defined by Geiss-Leclerc-Schröer in the simply-laced case), and the dual Poisson Lie group of GL(n, 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. This is joint work with Yu Li and Jiang-Hua Lu.

Lieu

Bâtiment: Conseil Général 7-9

Room 6-13, Séminaire "Groupes de Lie et espaces de modules"

entrée libre