Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2

TL;DR

This paper proves the periodicities and dilogarithm identities for T and Y-systems related to quantum affine algebras of types C_r, F_4, and G_2 at any level, using tropical Y-systems and categorification techniques.

Contribution

It extends the proof of periodicities and dilogarithm identities to new algebra types C_r, F_4, and G_2, employing tropical Y-systems and cluster algebra categorification.

Findings

Proved periodicities for T and Y-systems of types C_r, F_4, G_2.

Established dilogarithm identities for these Y-systems.

Applied tropical Y-systems and categorification methods in the proofs.

Abstract

We prove the periodicities of the restricted T 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.