Monoidal categorification and quantum affine algebras

TL;DR

This paper introduces new invariants derived from R-matrices to determine when a monoidal category of quantum affine algebra modules categorifies a cluster algebra, advancing understanding of their algebraic structure.

Contribution

It presents novel invariants and a criterion for monoidal categorification of cluster algebras via quantum affine algebra modules.

Findings

New invariants based on R-matrices for module pairs

Criterion for monoidal categorification of cluster algebras

Enhanced understanding of module category structures

Abstract

We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q'(\mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.