{
  "id": "2011.13003",
  "version": "v1",
  "published": "2020-11-25T20:09:11.000Z",
  "updated": "2020-11-25T20:09:11.000Z",
  "title": "Faithfulness of simple 2-representations of $\\mathfrak{sl}_2$",
  "authors": [
    "Laurent Vera"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathcal U$ be the 2-category associated with $\\mathfrak{sl}_2$. We prove that a complex of 1-morphisms of $\\mathcal U$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild boundedness assumptions, we prove that it actually suffices for the image in the simple 2-representations to be acyclic. We apply this result to the study of the Rickard complex $\\Theta$ categorifying the action of the simple reflection of $\\mathrm{SL}_2$. We prove that $\\Theta$ is invertible in the homotopy category of $\\U$, and that there is a homotopy equivalence $\\Theta E \\simeq F\\Theta[-1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-25T20:09:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "faithfulness",
      "mild boundedness assumptions",
      "homotopy equivalence",
      "null-homotopic",
      "rickard complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}