{
  "id": "2410.04973",
  "version": "v1",
  "published": "2024-10-07T12:16:30.000Z",
  "updated": "2024-10-07T12:16:30.000Z",
  "title": "Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation",
  "authors": [
    "Yunhe Sheng",
    "Rong Tang",
    "Chenchang Zhu"
  ],
  "comment": "30 pages",
  "categories": [
    "math-ph",
    "math.DG",
    "math.MP",
    "math.RA"
  ],
  "abstract": "The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-07T12:16:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "yang-baxter equation",
      "post-groupoid",
      "quiver-theoretical solutions",
      "relative rota-baxter operator",
      "skew-left brace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}