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Abstract:Hernandez and Leclerc introduced the notion of monoidal categorification of cluster algebras. We define similarly the notion of monoidal categorifications of generalized cluster algebras: an abelian monoidal category $\mathcal M$ is said to be a monoidal categorification of a generalized cluster algebra $\mathcal A$ if the Grothendieck ring of $\mathcal M$ is isomorphic to the upper generalized cluster algebra $\mathcal A^{\mathrm{up}}$, and if cluster monomials (resp. cluster variables) of $\mathcal A$ correspond to classes of real simple (resp. real prime simple) objects of $\mathcal M$.
Let $\varepsilon$ be a root of unity such that $\varepsilon^{2\ell}=1$ for some $\ell\in\mathbb{Z}_{\geq 2}$. Denote by $\mathcal{C}_{\varepsilon}$ the category of finite-dimensional modules of the restricted quantum loop algebra $U_\varepsilon^{\res}(L\mathfrak{sl}_k)$ at root $\varepsilon$ of unity, and let $\mathcal{C}_{\varepsilon, \xi}$ be a full subcategory of $\mathcal{C}_{\varepsilon}$ determined by a bipartition $\xi: I \to \{0,1\}$ of the Dynkin diagram. For $k=3$, Gleitz conjectured that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$, and that generalized cluster monomials correspond to classes of simple modules. This conjecture is a special case of a more general conjecture of Fraser.
In this paper, we prove the first part of Gleitz's conjecture. More precisely, for $k=3$ and arbitrary $\ell\ge2$, we prove that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$. We also classify the real Kirillov--Reshetikhin modules of $U^{\mathrm{res}}_\varepsilon(L\mathfrak{sl}_3)$ and obtain mutation sequences for the real Kirillov--Reshetikhin modules from the initial seed of the generalized cluster algebra.

Submission history

From: Jian-Rong Li [view email]
[v1] Mon, 15 Jun 2026 07:56:39 UTC (34 KB)