Atomic basis of quantum cluster algebra of type $\widetilde{A}_{2n-1,1}$

TL;DR

This paper establishes the atomic basis for the quantum cluster algebra of affine type A_{2n-1,1}, providing explicit bases and multiplication formulas that deepen understanding of its algebraic structure.

Contribution

It constructs two bar-invariant bases for the quantum cluster algebra of affine type A_{2n-1,1}, proving the atomic basis property and deriving quantum analogues of linear relations.

Findings

Constructed two bar-invariant a3[q^{\u22122}]a3-bases of the algebra.

Proved the atomic basis property of one of the bases.

Derived quantum cluster multiplication formulas and linear relations.

Abstract

Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the cluster variables associated with vertices of $Q$ satisfy a quantum analogue of the constant coefficient linear relations. We then construct two bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases $\mathcal{B}$ and $\mathcal{S}$ of $\mathcal{A}_{q}(Q)$ consisting of positive elements, and prove that $\mathcal{B}$ is an atomic basis.