{
  "id": "1504.05174",
  "version": "v1",
  "published": "2015-04-20T19:48:54.000Z",
  "updated": "2015-04-20T19:48:54.000Z",
  "title": "Closed Form of the Baker-Campbell-Hausdorff Formula for Semisimple Complex Lie Algebras",
  "authors": [
    "Marco Matone"
  ],
  "comment": "11 pages",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.RT",
    "quant-ph"
  ],
  "abstract": "In arXiv:1502.06589 it has been introduced an algorithm extending the Van-Brunt and Visser result, leading to new closed forms of the Baker-Campbell-Hausdorff formula. In particular, there are closed forms even when the commutator $[X,Y]$ also contains elements of the algebra different from $X$ and $Y$. In arXiv:1503.08198 it has been shown that there are {\\it 13 types} of such commutator algebras. We show, by providing the explicit solutions, that these include the semisimple complex Lie algebras. In particular, if $X$, $Y$ and $Z$ are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, $W$, linear combination of $X$, $Y$ and $Z$, such that $$ e^X e^Y e^Z=e^W $$ It turns out that the relevant commutator algebras are {\\it type 1c-i}, {\\it type 4} and {\\it type 5}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-20T19:48:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semisimple complex lie algebras",
      "closed form",
      "baker-campbell-hausdorff formula",
      "relevant commutator algebras",
      "explicit solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150405174M",
      "inspire": 1362206
    }
  }
}