Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

TL;DR

This paper studies simple modules over simply-laced quantum affine algebras, introduces subcategories with cluster algebra structures, classifies real simple modules, and proposes conjectures supported by finite type cases.

Contribution

It introduces new subcategories with quantum cluster algebra structures and classifies real simple modules, including new families of type D and E.

Findings

Quantum Grothendieck rings admit cluster algebra structures.

Complete classification of real simple modules in certain subcategories.

Proposed and proved conjectures for finite type cluster algebra cases.

Abstract

Let $\mathscr{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $\xi$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain subcategories $\mathscr{C}^{\leq \xi}_\ell$ of $\mathscr{C}$, and prove that the quantum Grothendieck ring $K_t(\mathscr{C}^{\leq \xi}_\ell)$ of $\mathscr{C}^{\leq \xi}_\ell$ admits a quantum cluster algebra structure. Using $F$-polynomials and monoidal categorifications of cluster algebras, we classify all real simple modules in $\mathscr{C}^{\leq \xi}_1$ in terms of their highest $\ell$-weight monomials, among them the families of type $D$ and type $E$ are new. For any $\ell$, inspired by Hernandez and Leclerc's work, we propose two conjectures for the study of real simple modules, and prove them for the subcategories $\mathscr{C}^{\leq \xi}_\ell$ whose Grothendieck rings are cluster algebras of finite type.