T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras

TL;DR

This paper extends the framework of T-systems and Y-systems to quantum affinizations of quantum Kac-Moody algebras, establishing relations between these systems and their connection to cluster algebras.

Contribution

It introduces Y-systems for quantum affinizations of quantum Kac-Moody algebras and links T and Y-systems with cluster algebra structures.

Findings

Established Y-systems for quantum affinizations.

Connected T and Y-systems through algebraic relations.

Linked T and Y-systems to cluster algebra frameworks.

Abstract

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently the T-systems were generalized to quantum affinizations of a wide class of quantum Kac-Moody algebras by Hernandez. In this note we introduce the corresponding Y-systems and establish a relation between T and Y-systems. We also introduce the T and Y-systems associated with a class of cluster algebras, which include the former T and Y-systems of simply laced type as special cases.