Mini course on topics in cluster algebras

2019/05/20 Mon - 2019/05/22 Wed

RIMS research project "Cluster Algebras"
Mini course on topics in cluster algebras

Bernard Leclerc (Caen)
Michael Gekhtman (Notre Dame)

May 20 (mon)
10:30-12:00 Leclerc 1
13:00-14:30 Leclerc 2

May 21 (tue)
10:30-12:00 Leclerc 3
13:00-14:30 Gekhtman 1

May 22 (wed)
10:30-12:00 Gekhtman 2
13:00-14:30 Gekhtman 3

Title and abstract

Bernard Leclerc (Caen)
Caldero-Chapoton formula for skew-symmetrizable cluster algebras of finite type

This minicourse is based on a joint work with C. Geiss and J. Schröer.
I will first review the classical Caldero-Chapoton formula for cluster algebras of finite ADE type, which expresses cluster expansions of cluster variables in terms of quiver Grassmannians of indecomposable representations of a quiver of the same type.
Path algebras of ADE quivers are particular cases of a class of Iwanaga-Gorenstein algebras that one can attach to any symmetrizable Cartan matrix. I will review the definition of these algebras and some features of their representation theory.
Then I will explain how one can obtain a generalization of the original Caldero-Chapoton formula valid for all Cartan-Killing types ABCDEFG, by replacing indecomposable representations of quivers by rigid indecomposable representations of these Iwanaga-Gorenstein algebras. I will conclude with a discussion of a (conjectural) variant of this formula yielding generalized cluster algebras (in the sense of Chekhov and Shapiro).

Michael Gekhtman (Notre Dame)
Cluster algebras and Poisson-Lie groups

We will discuss an interaction between Poisson geometry and the theory of cluster algebra focusing, on one hand, on Poisson-geometric approach to uncovering (generalized) cluster structures in coordinate rings of Poisson-Lie groups and Poisson homogeneous spaces and, on the other hand, on the way distinguished sequences of cluster transformations give rise to discrete completely integrable models.

To this end we will
- briefly recall the basic facts on Poisson brackets and Hamiltonian flows, review the notion of a Poisson bracket compatible with a cluster algebra structure and discuss its applications;
- provide a motivation for and various versions and examples of generalized cluster structures;
- review a construction of a generalized cluster structure in the Drinfeld double of GL(n) and of exotic cluster structures in GL(n) compatible with Poisson-Lie brackets covered by the Belavin-Drinfeld classification.