Folding of cluster algebras and quantum toroidal algebras

In this paper, we study the relationship between the representation theory of the quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}_\infty})$ of infinite rank, and that of the quantum toroidal algebra $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$. Using monoidal categorifications due to Hernandez-Leclerc and Nakajima, we establish a cluster-theoretic interpretation of the folding map $\phi_{2n}$ of $q$-characters, introduced by Hernandez. To this end, we introduce a notion of foldability for cluster algebras arising from infinite quivers and study a specific case of cluster algebras of type $A_\infty$. Using this interpretation of $\phi_{2n}$, we prove a conjecture of Hernandez in new cases. Finally, we study a particular simple $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$-module whose $q$-character is not a cluster variable, and conjecture that it is imaginary.