Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras II: Types $C_r$, $F_4$, and $G_2$

Rei Inoue; Osamu Iyama; Bernhard Keller; Atsuo Kuniba; Tomoki Nakanishi

doi:10.4171/prims/96

JournalsprimsVol. 49, No. 1pp. 43–85

Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras II: Types $C_{r}$ , $F_{4}$ , and $G_{2}$

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

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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 $C_{r}$ , $F_{4}$ , and $G_{2}$ at any level. We also prove the dilogarithm identities for these Y-systems 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.

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 II: Types $C_{r}$ , $F_{4}$ , and $G_{2}$ . Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 43–85

DOI 10.4171/PRIMS/96