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We prove the periodicities of the restricted T and Y-systems associated withthe quantum affine algebra of type B_r at any level. We also prove thedilogarithm identities for the Y-systems of type B_r at any level. Our proof isbased on the tropical Y-systems and the categorification of the cluster algebraassociated with any skew-symmetric matrix by Plamondon. Using this new method,we also give an alternative and simplified proof of the periodicities of the Tand Y-systems associated with pairs of simply laced Dynkin diagrams.

Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r