Monoidal categorification of cluster algebras II

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 up to a power of q.

Contribution

It demonstrates the existence of a quantum monoidal seed with initial mutations, confirming a key conjecture in the theory.

Findings

Proved the monoidal categorification of quantum unipotent coordinate algebras.

Confirmed that all cluster monomials are in the upper global basis up to a power of q.

Established the existence of a quantum monoidal seed with first-step mutations in all directions.

Abstract

We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of our earlier work, we achieve it by showing 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 solve the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$.