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Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras. I: Type \(B_r\). (English) Zbl 1273.13041
In the paper under review, the authors prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type \(B_r\) at any level. They also prove the dilogarithm identities for the Y-systems of type \(B_r\) at any level. Their proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by P.-G. Plamondon [Adv. Math. 227, No. 1, 1–39 (2011; Zbl 1288.13016)]. Using this new method, they also give an alternative and simplified proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.

13F60 Cluster algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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