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

Abstract

Let $Q$ be the affine quiver of type ${\tilde{A}}_{2n-1,1}$ 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.

21 pages