Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type $B_r$

Abstract

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum ane 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 categorication of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplied proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.