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

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.

35 pages; minor changes, references updated

References in corpus (5)

• Cluster algebras via cluster categories with infinite-dimensional morphism spaces
• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2
• T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras
• Dilogarithm identities for conformal field theories and cluster algebras: simply laced case

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• The classification of Zamolodchikov periodic quivers
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