{
  "id": "2210.08598",
  "version": "v1",
  "published": "2022-10-16T17:51:56.000Z",
  "updated": "2022-10-16T17:51:56.000Z",
  "title": "On derived-indecomposable solutions of the Yang--Baxter equation",
  "authors": [
    "Ilaria Colazzo",
    "Maria Ferrara",
    "Marco Trombetti"
  ],
  "comment": "24 pages",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-16T17:51:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16T25",
      "16Nxx",
      "81R50",
      "20F24",
      "08A05"
    ],
    "keywords": [
      "yang-baxter equation",
      "derived-indecomposable solutions",
      "finite non-degenerate set-theoretic solution",
      "structure skew brace",
      "brace-theoretic analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}