{
  "id": "1712.03801",
  "version": "v1",
  "published": "2017-12-11T14:56:21.000Z",
  "updated": "2017-12-11T14:56:21.000Z",
  "title": "On the Zariski topology of $Ω$-groups",
  "authors": [
    "Ruvim Lipyanski"
  ],
  "comment": "7 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "A number of geometric properties of algebras from a given variety can be characterized using the notions of anticommutativity and stability of algebras. Let $\\Theta$ be a variety and $F$ be a finitely generated free algebra in $\\Theta$. The stability of the $\\Omega$-group $H$ in the variety $\\Theta$ means that we can equip the space ${\\rm Hom}(F,H)$ with the Zariski topology, whose closed sets are precisely algebraic sets. In three classical cases of $\\Omega$-groups (Lie algebras, groups, and associative rings) necessary and sufficient conditions of their stability are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-11T14:56:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C40",
      "14E20",
      "46E25"
    ],
    "keywords": [
      "zariski topology",
      "geometric properties",
      "finitely generated free algebra",
      "precisely algebraic sets",
      "lie algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}