Classification of real and imaginary modules of quantum affine algebras in monoidal categorifications of affine cluster algebras
Heizo Sakamoto
TL;DR
The work constructs monoidal categorifications of affine-type cluster algebras for ADE quantum affine algebras via ADE subcategories $\mathscr{C}_{\mathfrak{g}}^{[a,b],\mathfrak{s}}$, establishing explicit affine-type categorifications and linking them to affine cluster combinatorics. It then provides a complete classification of real and imaginary simple modules in these categories using $\mathbf{d}$-vectors and $c$-cluster expansions, proving that real simple modules correspond to cluster monomials in these affine settings. A key outcome is that the null root $\delta$ gives a prime imaginary module, and the real-vs-cluster-monomial conjecture holds in these contexts despite the presence of imaginary modules. The analysis also integrates Q-datum replacements and ties to quiver Hecke algebras and dual PBW bases, highlighting the broader framework connecting representation theory of quantum affine algebras, categorification, and affine cluster algebras with practical criteria for realness and cluster-monomial identification.
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.