  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.

Quantum Algebra;Representation Theory