Folding of cluster algebras and quantum toroidal algebras

Abstract

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 ${\varphi }_{2n}$ 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 ${\varphi }_{2n}$, 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.