Quantum affine algebras at roots of unity and generalised cluster algebras

Abstract

Let $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_2)$ be the restricted integral form of the quantum loop algebra $U_q(L\mathfrak{sl}_2)$ specialised at a root of unity $\varepsilon$. We prove that the Grothendieck ring of a tensor subcategory of representations of $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l-1}$, where $l$ is the order of $\varepsilon^2$. Moreover, we show that the classes of simple objects in the Grothendieck ring essentially coincide with the cluster monomials. We also state a conjecture for $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_3)$, and we prove it for $l=2$.