Compare triangular bases of acyclic quantum cluster algebras

TL;DR

This paper proves that for acyclic quivers, two different constructions of triangular bases in quantum cluster algebras coincide, confirming that Berenstein-Zelevinsky's basis includes all quantum cluster monomials.

Contribution

It establishes the equivalence of two triangular bases in quantum cluster algebras for acyclic quivers and bipartite skew-symmetrizable matrices, clarifying their relationship.

Findings

Triangular bases coincide for acyclic quivers.

Berenstein-Zelevinsky's basis contains all quantum cluster monomials.

The bases are also the same for bipartite skew-symmetrizable matrices.

Abstract

Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with acyclic quivers. This result implies that the Berenstein-Zelevinsky's basis contains all the quantum cluster monomials. We also give an easy proof that the two bases are the same for the seeds associated with bipartite skew-symmetrizable matrices.