We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum ane algebra of type , , and at any level. We also prove the dilogarithm identities for these Y-systems 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.
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Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, Tomoki Nakanishi, Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras II: Types , , and . Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 43–85