5 is a discussion of the results. Appendix A is a reformulation of the $Q$-system as a cluster algebra with coefficients, addressing the technical point of subtraction-free expressions. In the body of the paper, this is done through a renormalization of the cluster variables, which is not always generalizable. Appendix B is a discussion of the formulation of generalized $T$-systems as cluster algebras, with two main examples, both of which have the polynomiality property.

.2in [**Acknowledgements.**]{} We thank Bernhard Keller, Hugh Thomas and especially Sergei Fomin for their valuable input. RK thanks CEA-Saclay IPhT for their hospitality. We thank the organizers of the MSRI program on “Combinatorial Representation Theory” for their hospitality. RK is supported by NSF grant DMS-05-00759. PDF acknowledges the support of European Marie Curie Research Training Networks ENIGMA MRT-CT-2004-5652, ENRAGE MRTN-CT-2004-005616, ESF program MISGAM, and of ANR program GIMP ANR-05-BLAN-0029-01.

Definitions
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The $Q$-system
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### ${{{\rm KR}}}$-modules

Let ${{\mathfrak{g}}}$ be a simple Lie algebra of rank $r$ and let $I_r=\{1,...,r\}$ be the parameterizing set for the simple roots of ${{\mathfrak{g}}}$. Let $C$ be the Cartan matrix of ${{\mathfrak{g}}}$.

For any such algebra, there is a corresponding quantum affine algebra $U_q(\widehat{{{\mathfrak{g}}}})$ and a Yangian algebra $Y({{\mathfrak{g}}})$. In 