{
  "id": "1704.07853",
  "version": "v1",
  "published": "2017-04-25T18:21:07.000Z",
  "updated": "2017-04-25T18:21:07.000Z",
  "title": "Undecidability of the first order theories of free non-commutative Lie algebras",
  "authors": [
    "Olga Kharlampovich",
    "Alexei Myasnikov"
  ],
  "categories": [
    "math.LO",
    "math.RA"
  ],
  "abstract": "Let $R$ be a commutative integral unital domain and $L$ a free non-commutative Lie algebra over $R$. In this paper we show that the ring $R$ and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the standard ring language $+, \\cdot,0$. Furthermore, if $R$ has characteristic zero then we prove that the elementary theory $Th(L)$ of $L$ in the standard ring language is undecidable. To do so we show that the arithmetic ${\\bf N} = \\langle{\\bf N}, +,\\cdot,0 \\rangle$ is 0-interpretable in $L$. This implies that the theory of $Th(L)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-25T18:21:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C60"
    ],
    "keywords": [
      "free non-commutative lie algebra",
      "first order theories",
      "standard ring language",
      "undecidability",
      "commutative integral unital domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}