Periodicities of T-systems and Y-systems

Abstract

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

83 pages with 8 figures; references added, Prop. 3.4 added, proofs for Theorem 2.12 and Prop. 7.3 shortened + minor corrections

Cited by in corpus (4)

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• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
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• T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras