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We define the cluster algebra associated with the Q-system for theKirillov-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, andexplain its relation to the deformed Q-systems which appeared on our proof ofthe combinatorial-KR conjecture. We prove that the polynomiality of the clustervariables in terms of the ``initial cluster seeds'', including solutions of theQ-system, is a consequence of the Laurent phenomenon and the boundaryconditions. We also give a formulation of both Q-systems and generalizedT-systems as cluster algebras with coefficients. This provides a proof of thepolynomiality of solutions of generalized T-systems with appropriate boundaryconditions.

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