{
  "id": "1904.04859",
  "version": "v1",
  "published": "2019-04-09T18:33:57.000Z",
  "updated": "2019-04-09T18:33:57.000Z",
  "title": "On auto-equivalences and complete derived invariants of gentle algebras",
  "authors": [
    "Sebastian Opper"
  ],
  "categories": [
    "math.RT",
    "math.SG"
  ],
  "abstract": "We study triangulated categories which can be modeled by an oriented marked surface $\\mathcal{S}$ and a line field $\\eta$ on $\\mathcal{S}$. This includes bounded derived categories of gentle algebras and -- conjecturally -- all partially wrapped Fukaya categories introduced by Haiden-Katzarkov-Kontsevich. We show that triangle equivalences between such categories induce diffeomorphisms of the associated surfaces preserving orientation, marked points and line fields up to homotopy. This shows that the pair $(\\mathcal{S}, \\eta)$ is a triangle invariant of such categories and prove that it is a complete derived invariant for gentle algebras of arbitrary global dimension. We deduce that the group of auto-equivalences of a gentle algebra is an extension of the stabilizer subgroup of $\\eta$ in the mapping class group and a group, which we describe explicitely in case of triangular gentle algebras. We show further that diffeomorphisms associated to spherical twists are Dehn twists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-09T18:33:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E35",
      "55M25"
    ],
    "keywords": [
      "complete derived invariant",
      "auto-equivalences",
      "line field",
      "categories induce diffeomorphisms",
      "arbitrary global dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}