Cluster algebras and Poisson geometry

Abstract: Cluster algebras are commutative rings with distinguished sets of generators characterized by a remarkable combinatorial structure. Discovered by S. Fomin and A. Zelevinsky in the early 2000s, these algebraic structures have found applications across diverse mathematical fields, including integrable systems, total positivity, Teichmüller theory, Poisson geometry, knot theory and mathematical physics. Fomin and Zelevinsky conjectured that numerous varieties in Lie theory are equipped with a cluster structure. Early examples include double Bruhat cells, Grassmannians and simple complex algebraic groups. M. Gekhtman, M. Shapiro and A. Vainshtein observed that cluster algebras in these examples are compatible with certain Poisson brackets. Specifically, for any given cluster $𝑥_1,𝑥_2,...,𝑥_n$, there exist constants 𝟂ij such that $\{𝑥_i,𝑥_j\} = \omega_{ij} 𝑥_i 𝑥_j$. This observation led to a program aiming to construct cluster algebras by addressing the inverse problem: given a Poisson bracket in the coordinate ring of an algebraic variety and a collection of regular functions $(𝑥_1,𝑥_2,...,𝑥_n)$ satisfying $\{𝑥_i,𝑥_j\} = \omega_{ij} 𝑥_i 𝑥_j$, does there exist a compatible cluster algebra? The research initiative led to the formulation of the GSV conjecture: for a given simple complex algebraic group and a Poisson bracket from the Belavin-Drinfeld class, there exists a compatible cluster structure. The plan for the talk is as follows. First, we will discuss an example of a cluster structure on ${\rm GL}_3(\mathbb{C})$. Then we will explore the connection between cluster algebras and Poisson geometry, as well as discuss how to construct a cluster structure compatible with a Poisson bracket. After that, we will discuss the recent results on the three main families of objects: simple connected simple complex algebraic groups, their Drinfeld doubles and their Poisson duals.

mathematical physicsalgebraic geometrygeometric topologyquantum algebra

Audience: researchers in the discipline

Geometry, Algebra and Physics at KIAS