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Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras. II: Types \(C_r, F_4\), and \(G_2\). (English) Zbl 1283.13021
In the paper under review, the authors complete their study on the periodicities of restricted T-systems and Y-systems associated to the quantum affine algebras of non-simply laced types (for part I, cf. [ibid. 49, No. 1, 1–42 (2013; Zbl 1273.13041)]. In particular, they show that the restricted T-systems and Y-systems associated to the quantum affine algebras of types \(C_{r}, F_{4}\) and \(G_{2}\) have periodicities at any level. They also establish the dilogarithm identities for these Y-systems at any level. They employ the approaches of tropical Y-systems and categorifications of cluster algebras. The statements and methods in this paper are analogous to the ones for the quantum affine algebra of type \(B_{r}\), which was treated in a previous paper by the same authors. In spite of a detailed study of the tropical Y-systems at level \(2\) which is a key ingredient of their approach, most of the proofs are skipped.

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