Quantum affine algebras at roots of unity and generalised cluster algebras

Abstract

Let $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_2)$ be the restricted integral form of the quantum loop algebra $U_q(L{\mathfrak{sl}}_2)$ specialised at a root of unity $\epsilon$. We prove that the Grothendieck ring of a tensor subcategory of representations of $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_2)$ is a generalised cluster algebra of type $C_{l-1}$, where $l$ is the order of ${\epsilon }^2$. 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 $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_3)$, and we prove it for $l=2$.

26 pages, 9 figures

Cited by in corpus (1)

• A quantum analogue of generalized cluster algebras