Monoidal categorification of cluster algebras (merged version)

TL;DR

This paper establishes a monoidal categorification of quantum cluster algebras associated with symmetric Kac-Moody algebras, proving that cluster monomials belong to the upper global basis and confirming Leclerc's conjecture on basis element products.

Contribution

It introduces a criterion for monoidal categorification of quantum cluster algebras and proves the existence of suitable seeds, advancing the understanding of their algebraic and categorical structures.

Findings

Proved the monoidal categorification of quantum cluster algebras for symmetric Kac-Moody cases.

Confirmed that all cluster monomials are in the upper global basis up to a power of q^{1/2}.

Provided a proof of Leclerc's conjecture on the product of upper global basis elements.

Abstract

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.