{
  "id": "2011.07350",
  "version": "v1",
  "published": "2020-11-14T17:51:25.000Z",
  "updated": "2020-11-14T17:51:25.000Z",
  "title": "Atomic basis of quantum cluster algebra of type $\\widetilde{A}_{2n-1,1}$",
  "authors": [
    "Ming Ding",
    "Fan Xu",
    "Xueqing Chen"
  ],
  "comment": "21 pages",
  "categories": [
    "math.RT",
    "math.QA",
    "math.RA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-14T17:51:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "13F60"
    ],
    "keywords": [
      "quantum cluster algebra",
      "atomic basis",
      "constant coefficient linear relations",
      "cluster multiplication formulas",
      "affine quiver"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}