Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type $B_r$
Publications of the Research Institute for Mathematical Sciences2013Vol. 49(1), pp. 1–42
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Abstract
We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type B_r at any level. We also prove the dilogarithm identities for the Y-systems of type B_r at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplified proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.