Bases of the quantum cluster algebra of the Kronecker quiver

TL;DR

This paper constructs specific bases for the quantum cluster algebra of the Kronecker quiver, extending classical bases to the quantum setting and proving their positivity.

Contribution

It introduces bar-invariant quantum analogues of canonical, semicanonical, and dual semicanonical bases for the Kronecker quiver's quantum cluster algebra.

Findings

Constructed quantum bases analogous to classical bases.

Proved positivity of the basis elements.

Extended classical cluster algebra results to the quantum case.

Abstract

We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.