{
  "id": "2212.10369",
  "version": "v1",
  "published": "2022-12-20T15:51:19.000Z",
  "updated": "2022-12-20T15:51:19.000Z",
  "title": "Two geometric models for graded skew-gentle algebras",
  "authors": [
    "Yu Qiu",
    "Chao Zhang",
    "Yu Zhou"
  ],
  "comment": "114 pages, many figures. Any comments are welcome",
  "categories": [
    "math.RT",
    "math.CT",
    "math.GT"
  ],
  "abstract": "In Part 1, we classify (indecomposable) objects in the perfect derived category $\\mathrm{per}\\Lambda$ of a graded skew-gentle algebra $\\Lambda$, generalizing technique/results of Burban-Drozd and Deng to the graded setting. We also use the usual punctured marked surface $\\mathbf{S}^\\lambda$ with grading (and a full formal arc system) to give a geometric model for this classification. In Part2, we introduce a new surface $\\mathbf{S}^\\lambda_*$ with binaries from $\\mathbf{S}^\\lambda$ by replacing each puncture $P$ by a boundary component $*_P$ (called a binary) with one marked point, and composing an equivalent relation $D_{*_P}^2=\\mathrm{id}$, where $D_{*_p}$ is the Dehn twist along $*_P$. Certain indecomposable objects in $\\mathrm{per}\\Lambda$ can be also classified by graded unknotted arcs on $\\mathbf{S}^\\lambda_*$. Moreover, using this new geometric model, we show that the intersections between any two unknotted arcs provide a basis of the morphisms between the corresponding arc objects, i.e. formula $\\mathrm{Int}=\\mathrm{dim}\\mathrm{Hom}$ holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-20T15:51:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "graded skew-gentle algebra",
      "geometric model",
      "full formal arc system",
      "unknotted arcs",
      "boundary component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 114,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}