We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum ane algebra of type at any level. We also prove the dilogarithm identities for the Y-systems of type at any level. Our proof is based on the tropical Y-systems and the categorication of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplied proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.
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Rei Inoue, Bernhard Keller, Osamu Iyama, Atsuo Kuniba, Tomoki Nakanishi, Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type . Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 1–42DOI 10.4171/PRIMS/95