Q-systems as Cluster Algebras Ii: Cartan Matrix of Finite Type and the Polynomial Property

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(bg) 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.