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

Understand

We prove the periodicities of the restricted T 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 and Y-systems associated with pairs of simply laced Dynkin diagrams.