|XXXV Workshop on Geometric Methods in Physics
Generalized Gelfand-Tsetlin integrable systems and cluster algebras
In the first part of the talk we will review the definition of Cluster Algebras, introduced by Fomin and Zelevinsky, and some of their applications including those to Poisson geometry. We will also discuss the notion of maximal green sequences of mutations which are used in the study of Donaldson-Thomas invariants and spectra of BPS states. In the second part of the talk we will introduce a generalization of the Gelfand-Tsetlin construction of integrable systems, where the recursive use of Casimirs is replaced by subalgebras generated by Poisson normal elements. We will then describe a classification of those using Poisson Unique Factorization Domains. These techniques will be applied to settle several open problems on cluster algebras, including the problem of whether the Berenstein-Fomin-Zelevinsky cluster algebras coincide with the coordinate rings of double Bruhat cells and the problem for existence of maximal green sequences for them. Similar results will be proved in the quantum situation. Part of the results in the talk are joint work with Ken Goodearl (UC Santa Barbara).