Classification of real and imaginary modules of quantum affine algebras in monoidal categorifications of affine cluster algebras

TL;DR

This paper classifies real and imaginary simple modules in monoidal categories of quantum affine algebras, confirming a conjecture that real modules correspond to cluster monomials in affine types.

Contribution

It provides a complete classification of simple modules in certain monoidal categories and verifies a key conjecture relating real modules to cluster monomials.

Findings

Classification of real and imaginary simple modules in affine types

Verification that real simple modules correspond to cluster monomials

Extension of categorification results to affine type cluster algebras

Abstract

Recently, Kashiwara-Kim-Oh-Park introduced a wide family of monoidal categories of finite-dimensional representations of quantum affine algebras, which provide monoidal categorifications of cluster algebras. In this paper, we prove that, for types $ADE$, some of these categories provide monoidal categorifications of cluster algebras of affine type. Moreover, by means of the combinatorial theory of affine type cluster algebras, we give a complete classification of real and imaginary simple modules in these categories. In particular, we show that, in these cases, the conjecture asserting that real simple modules correspond exactly to cluster monomials holds.