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 $l=2$.
Categories: math.RT