# Cluster algebra structures on module categories over quantum affine algebras

## TL;DR

This paper explores the connection between cluster algebra structures and module categories over quantum affine algebras, revealing new monoidal categorifications and identifying modules corresponding to cluster monomials as real simple modules.

## Contribution

It introduces new monoidal categorifications of certain subcategories of quantum affine algebra modules using quiver Hecke algebras, especially for types A and B.

## Key findings

- Cluster algebra structures coincide with those from quiver Hecke algebra modules.
- Modules corresponding to cluster monomials are identified as real simple modules.
- The results unify and extend previous categorification frameworks for quantum affine algebras.

## Abstract

We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.