Greedy bases in rank 2 quantum cluster algebras

TL;DR

This paper constructs a quantum version of the greedy basis for rank 2 cluster algebras, showing it is independent of initial choices, includes all cluster monomials, and is bar-invariant, with several conjectures proposed.

Contribution

It introduces a quantum lift of the greedy basis for rank 2 cluster algebras, demonstrating its independence from initial clusters and its comprehensive inclusion of cluster monomials.

Findings

Constructed a quantum greedy basis for rank 2 cluster algebras

Proved the basis is independent of initial cluster choice

Produced bar-invariant elements within the basis

Abstract

We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.