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

Abstract

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type Br at any level. We also prove the dilogarithm identities for the Y-systems of type Br 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. © 2013 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.