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

TL;DR

This paper proves the periodicities and dilogarithm identities of T and Y-systems for type B_r quantum affine algebras at any level, using tropical Y-systems and cluster algebra categorification, providing new proofs and insights.

Contribution

It introduces a novel method leveraging tropical Y-systems and categorification to establish periodicities and identities for type B_r systems, also simplifying proofs for simply laced Dynkin diagrams.

Findings

Proved periodicities of T and Y-systems for type B_r at all levels.

Established dilogarithm identities for Y-systems of type B_r.

Provided an alternative proof for simply laced Dynkin diagram systems.

Abstract

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.