Based cluster algebras of infinite rank

TL;DR

This work develops a general method to extend based cluster algebras from finite to infinite rank by taking colimits of finite-rank seeds, enabling a robust framework for studying cluster-structures linked to representations of (shifted) quantum affine algebras. It shows that both ordinary and upper cluster algebras behave well under these colimits and provides a quantization mechanism in the infinite-rank setting. The authors prove that, when built from double Bott–Samelson cells, the infinite-rank cluster algebras satisfy A = U and admit braid-group actions that compute fundamental variables in finite type, yielding a direct path from signed-word seeds to canonical cluster data. They further apply the framework to cluster algebras from shifted quantum affine algebras, establishing seeds, their quantizations, and monoidal categorifications in the ADE setting, and connect these constructions to virtual quantum Grothendieck rings and KR-polynomials. Overall, the paper advances the understanding of infinite-rank cluster algebra structures and their connections to geometric, representation-theoretic, and categorification contexts, with concrete quantization results and braid-group computational tools.

Abstract

We extend based cluster algebras from the finite rank case to the infinite rank case. By extending (quantum) cluster algebras whose initial seeds are associated with signed words (arising from double Bott--Samelson cells), we recover infinite rank cluster algebras arising from representations of (shifted) quantum affine algebras. As a main application, we show that the fundamental variables of the cluster algebras arising from double Bott--Samelson cells can be computed via a braid group action when the Cartan matrix is of finite type. We also obtain the equality A=U for the associated infinite rank (quantum) cluster algebras. Additionally, several conjectures regarding quantum virtual Grothendieck rings due to Jang--Lee--Oh and Oh--Park follow as consequences. Finally, we show that the cluster algebras arising from representations of shifted quantum affine algebras, discovered by Geiss--Hernandez--Leclerc, admit natural quantizations.

Figures (12)

• Figure 3.1: A quiver for $\ddot{{\bf t}}(1,-1,2,-2,1,-1)$
• Figure 3.2: Fundamental variables for $\dot{{\bf t}}(1,2,1,2,1,2)$
• Figure 3.3: The quiver for $\dot{{\bf t}}(1,2,1,2,1,2)$
• Figure 5.1: The quiver for ${\bf t}_{4}=\dot{{\bf t}}({\underline{c}}^{4})$
• Figure 5.2: Quiver and initial cluster variables

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4: Theorem \ref{['thm:braid-formula-fundamental']}
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7: Proposition \ref{['prop:identify-seed-GHL']}, Proposition \ref{['prop:quantize-seed-GHL']}, Theorem \ref{['thm:basis-seed-GHL']}
• Lemma 2.1
• proof
• Definition 2.2