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abstract: 'We define the cluster algebra associated with the $Q$-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\widehat{{{\mathfrak{g}}}})$ for any simple Lie algebra ${{\mathfrak{g}}}$, 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.'
address:
- 'PDF: Institut de Physique Théorique de Saclay, CEA/DSM/IPhT, URA 2306 du CNRS, F-91191 Gif sur Yvette Cedex France. E-mail: philippe.di-francesco@cea.fr'
- 'RK: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61821. e-mail: rinat@uiuc.edu'
author:
- 'P. Di Francesco and R. Kedem'
title: '$Q$-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property'
---

Introduction
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The $Q$-system is a recursion relation for the characters of 