{
  "id": "math/0604096",
  "version": "v4",
  "published": "2006-04-05T18:58:24.000Z",
  "updated": "2006-08-31T16:09:15.000Z",
  "title": "A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra",
  "authors": [
    "Nikolai Durov",
    "Stjepan Meljanac",
    "Andjelo Samsarov",
    "Zoran Škoda"
  ],
  "comment": "v2: expositional improvements (significant in sections 5,6); v3: minor expositional improvements (including in notation, and in introduction); v4: final version, to appear in Journal of Algebra (4 minor differences from v3 due wrong uploaded file in v3)",
  "journal": "J.Algebra 309 (2007) 318-359",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP",
    "math.QA",
    "math.RA"
  ],
  "abstract": "Given a $n$-dimensional Lie algebra $g$ over a field $k \\supset \\mathbb Q$, together with its vector space basis $X^0_1,..., X^0_n$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i = X^0_i t$ in $g\\otimes_k k [[t]]$, where $t$ is a formal variable, as a formal power series in $t$ with coefficients in the Weyl algebra $A_n$. Actually, the theorem is proved for Lie algebras over arbitrary rings $k\\supset Q$. We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2006-08-31T16:09:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B40",
      "14D15",
      "14L05"
    ],
    "keywords": [
      "representing lie algebra generators",
      "formal power series",
      "weyl algebra",
      "universal formula",
      "coefficients"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jalgebra.2006.08.025"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 736186,
      "adsabs": "2006math......4096D"
    }
  }
}