Triangular bases in quantum cluster algebras and monoidal categorification conjectures

TL;DR

This paper introduces a unique triangular basis in quantum cluster algebras that aligns with the Fock-Goncharov conjecture and confirms monoidal categorification conjectures for certain algebra classes.

Contribution

It establishes the existence and uniqueness of a common triangular basis in quantum cluster algebras under specific conditions, linking it to tropical points and categorification conjectures.

Findings

Existence of a unique common triangular basis for injective-reachable quantum cluster algebras.

Confirmation of monoidal categorification conjectures for quantum affine algebra representations.

Validation of the basis parametrization by tropical points as predicted by the Fock-Goncharov conjecture.

Abstract

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture. As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.