Based cluster algebras of infinite rank

TL;DR

This paper extends based cluster algebras to infinite rank, connecting them with quantum affine algebra representations and providing computational methods and conjectures related to their structure.

Contribution

It introduces a framework for infinite rank based cluster algebras linked to quantum affine algebra representations and proves key properties and conjectures.

Findings

Recovery of infinite rank cluster algebras from double Bott--Samelson cells

Computation of fundamental variables via braid group actions

Proof of the equality A=U for certain quantum cluster algebras

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.