T-systems, Y-systems, and cluster algebras: Tamely laced case

TL;DR

This paper extends the connection between T-systems, Y-systems, and cluster algebras to tamely laced Cartan matrices, including nonfinite, nonsimply laced types, broadening the algebraic framework.

Contribution

It generalizes the identification of T-systems and Y-systems with cluster algebra relations to all tamely laced Cartan matrices, including nonfinite types.

Findings

Established correspondence for tamely laced Cartan matrices.

Extended cluster algebra relations to nonfinite, nonsimply laced cases.

Unified framework for algebraic relations in quantum affine settings.

Abstract

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently they were generalized to quantum affinizations of quantum Kac-Moody algebras associated with a wide class of generalized Cartan matrices which we say tamely laced. Furthermore, in the simply laced case, and also in the nonsimply laced case of finite type, they were identified with relations arising from cluster algebras. In this note we generalize such an identification to any tamely laced Cartan matrices, especially to the nonsimply laced ones of nonfinite type.