Monoidal categorification and quantum affine algebras II

TL;DR

This paper introduces affine determinantial modules over quantum affine algebras, generalizes T-systems, and demonstrates their role in monoidal categorification of cluster algebras, expanding understanding of quantum affine representations.

Contribution

It introduces affine determinantial modules, establishes generalized T-systems, and develops combinatorial tools, advancing monoidal categorification of cluster algebras.

Findings

Affine determinantial modules include KR-modules as a subfamily.

Proved generalized T-systems among these modules.

Established monoidal categorifications of cluster algebras.

Abstract

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.