{
  "id": "1506.06792",
  "version": "v1",
  "published": "2015-06-22T21:10:51.000Z",
  "updated": "2015-06-22T21:10:51.000Z",
  "title": "The images of Lie polynomials evaluated on $2\\times 2$ matrices over an algebraically closed field",
  "authors": [
    "Alexei Kanel-Belov",
    "Sergey Malev",
    "Louis Rowen"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $f$ be an arbitrary polynomial in several non commutative variables. Kaplansky asked about the possible images of $f$. In this note we let $f$ be a Lie polynomial in several non-commuting variables with constant term $0$ and coefficients in an algebraically closed field $K$. We describe all possible images of $f$ and provide an example of $f$ whose image is the set of trace zero matrices without nilpotent non zero matrices. We provide an arithmetic criterion for this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-22T21:10:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraically closed field",
      "lie polynomial",
      "nilpotent non zero matrices",
      "trace zero matrices",
      "non commutative variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606792K"
    }
  }
}