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arXiv 2011.073502020-11-14math.RTmath.QAmath.RA

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

Ming Ding, Fan Xu, Xueqing Chen

备注：: 21 pages

英文摘要：

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.

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