JournalsprimsVol. 49, No. 1pp. 1–42

Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type

  • Rei Inoue

    Suzuka University of Medical Science, Japan

  • Osamu Iyama

    Nagoya University, Japan

  • Bernhard Keller

    Université Paris Diderot - Paris 7, France

  • Atsuo Kuniba

    University of Tokyo, Japan

  • Tomoki Nakanishi

    Nagoya University, Nagoya, Chikusa-Ku, Japan
Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type $B_r$ cover
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Abstract

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type at any level. We also prove the dilogarithm identities for the Y-systems of type 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-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.

Cite this article

Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, Tomoki Nakanishi, Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type . Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 1–42

DOI 10.4171/PRIMS/95