Folding of cluster algebras and quantum toroidal algebras

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 $ϕ_{2n}$ 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 $ϕ_{2n}$, 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.

💡 Research Summary

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The paper investigates the deep relationship between the representation theory of the infinite‑rank quantum affine algebra (U_q(\widehat{\mathfrak{sl}\infty})) and that of the quantum toroidal algebra (U_q(\mathfrak{sl}{2n,\mathrm{tor}})). The central object of study is the folding map (\varphi_{2n}) on (q)-characters, originally introduced by Hernandez, which sends (q)-characters of integrable (U_q(\widehat{\mathfrak{sl}\infty}))-modules to (virtual) (q)-characters of integrable (U_q(\mathfrak{sl}{2n,\mathrm{tor}}))-modules. The authors provide a cluster‑theoretic interpretation of this map, thereby upgrading it from a mere projection to a structure‑preserving morphism between cluster algebras associated with the two quantum groups.

The work proceeds in several stages. First, the authors define an infinite ice quiver (\Gamma_\infty) whose principal part is of type (A_\infty) with alternating orientation. They introduce the notion of strong global foldability for infinite quivers equipped with a finite group action. Roughly, a quiver is strongly globally foldable with respect to a group (G) if the orbit quiver obtained by identifying vertices along (G)‑orbits is again a finite quiver and the induced mutation dynamics on the orbit quiver coincide with the usual cluster mutations. Theorem 1.3 shows that when (G) is generated by a shift of length (n+1), the quiver (\Gamma_\infty) is strongly globally foldable precisely when the underlying finite quiver (Q) (of type (A^{(1)}n)) is not cyclically oriented. This establishes a precise combinatorial bridge between the infinite‑type cluster algebra (\mathcal A\infty) and the finite‑type cluster algebra (\mathcal A_{2n}) associated with the toroidal side.

Next, the authors employ the monoidal categorifications of Hernandez–Leclerc and Nakajima. They construct a monoidal category (\mathcal C_1) of (U_q(\widehat{\mathfrak{sl}\infty}))-modules whose Grothendieck ring is isomorphic to (\mathcal A\infty). In this categorification, cluster variables correspond to classes of real simple objects. Using the group action described above, they define orbit‑clusters, i.e., collections of cluster variables that lie in the same (G)‑orbit. The main result, Theorem 1.4, proves that (\Gamma_\infty) is strongly globally foldable with respect to the shift group, and that for any cluster variable (\chi_q(L)) belonging to an orbit‑cluster, the folded image (\varphi_{2n}(\chi_q(L))) is a genuine cluster variable of (\mathcal A_{2n}). Consequently, (\varphi_{2n}(\chi_q(L))) is the (q)-character of a simple (U_q(\mathfrak{sl}_{2n,\mathrm{tor}}))-module. This result verifies Hernandez’s Conjecture 1.2 for all modules whose classes lie in orbit‑clusters, thereby providing the first non‑Kirillov–Reshetikhin examples where the conjecture holds.

The paper then turns to modules whose (q)-characters are not cluster variables. By exploiting the cluster‑theoretic description of (\varphi_{2n}), the authors compute explicit (q)-characters for certain toroidal modules that fall outside the cluster monomial framework. In particular, they focus on a simple toroidal module denoted (L(m_1,2n)). They conjecture (Conjecture 1.5) that this module is imaginary, meaning that its self‑fusion product is not simple. The case (n=1) is already known from Nakajima’s work; the authors show that, assuming the validity of Conjecture 1.2, the imaginary nature of (L(m_1,2n)) follows for all (n).

Beyond the specific results, the paper introduces a new methodological tool—strong global foldability—for handling infinite quivers with group symmetries. This concept may be applicable to other infinite‑type cluster algebras (e.g., of types (D) or (E)) and could facilitate the study of further connections between infinite‑rank quantum groups and their finite‑rank “folded” counterparts. The authors also highlight that their approach yields new multiplicative formulas for (q)-characters of toroidal algebras by transporting known formulas from the affine side via the folding map.

In summary, the article achieves three major advances: (1) it provides a rigorous cluster‑theoretic interpretation of Hernandez’s folding map, (2) it proves the folding conjecture for a broad new class of modules, and (3) it identifies and conjecturally characterizes a family of non‑cluster toroidal modules as imaginary. The work deepens our understanding of the interplay between infinite‑dimensional quantum algebras, toroidal algebras, and cluster algebra combinatorics, and opens several promising directions for future research.