{
  "id": "1011.1528",
  "version": "v1",
  "published": "2010-11-05T23:55:23.000Z",
  "updated": "2010-11-05T23:55:23.000Z",
  "title": "On twin and anti-twin words in the support of the free Lie algebra",
  "authors": [
    "Ioannis C. Michos"
  ],
  "comment": "24 pages (10pt) LaTeX file",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $L_{K}(A)$ be the free Lie algebra on a finite alphabet $A$ over a commutative ring $K$ with unity. For a word $u$ in the free monoid $A^{*}$ let $\\tilde{u}$ denote its reversal. Two words in $A^{*}$ are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let $l$ denote the left-normed Lie bracketing and $\\lambda$ be its adjoint map with respect to the canonical scalar product on the corresponding free associative algebra. Studying the kernel of $\\lambda$ and using several techniques from combinatorics on words and the shuffle algebra, we show that when $K$ is of characteristic zero two words $u$ and $v$ of common length $n$ that lie in the support of ${\\mathcal L}_{K}(A)$ - i.e., they are neither powers $a^{n}$ of letters $a \\in A$ with exponent $n > 1$ nor palindromes of even length - are twin (resp. anti-twin) if and only if $u = v$ or $u = \\tilde{v}$ and $n$ is odd (resp. $u = \\tilde{v}$ and $n$ is even).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-05T23:55:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B01",
      "68R15",
      "05E10"
    ],
    "keywords": [
      "free lie algebra",
      "anti-twin words",
      "common length",
      "characteristic zero",
      "adjoint map"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.1528M"
    }
  }
}