Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property

TL;DR

This paper establishes a connection between Q-systems, cluster algebras, and polynomial properties for quantum affine algebras, extending previous work to all simple Lie algebras and proving polynomiality of solutions.

Contribution

It generalizes the cluster algebra framework for Q-systems to all simple Lie algebras and proves polynomiality of solutions using the Laurent phenomenon.

Findings

Cluster algebra structure for Q-systems of all simple Lie algebras.

Polynomiality of cluster variables from initial seeds.

Relation between Q-systems, T-systems, and cluster algebras.

Abstract

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the ``initial cluster seeds'', including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of generalized T-systems with appropriate boundary conditions.