{
  "id": "2107.01204",
  "version": "v1",
  "published": "2021-07-02T17:59:21.000Z",
  "updated": "2021-07-02T17:59:21.000Z",
  "title": "Closed forms of Zassenhaus formula",
  "authors": [
    "Léonce Dupays"
  ],
  "comment": "8 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Zassenhaus formula is used in a wide range of fields in physics and mathematics, from fluid dynamics to differential geometry. The non commutativity of the elements of the algebra implies that the exponential of a sum of operators cannot be splitted in the product of exponential of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which form is generally an infinite product of exponentials. However, for some special commutators, closed forms can be found. We propose a closed form for Zassenhaus formula when the commutator of the operators $X$ and $Y$ satisfy the relation $$[X,Y]=uX+vY+c\\mathbb{1}.$$ Zassenhaus formula then reduces to the closed form \\begin{eqnarray} e^{X+Y}=e^{X}e^{Y}e^{g_{r}(u,v)[X,Y]}=e^{X}e^{g_{c}(u,v)[X,Y]}e^{Y}=e^{g_{l}(u,v)[X,Y]}e^{X}e^{Y}, \\end{eqnarray} a left-sided, centered and right-sided formula are found, with respective arguments, \\begin{eqnarray} g_{r}(u,v)=g_{c}(v,u)e^{u}=g_{l}(v,u)=\\frac{u\\left(e^{u-v}-e^{u}\\right)+v\\left(e^{u}-1\\right)}{vu(u-v)}. \\end{eqnarray}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-02T17:59:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zassenhaus formula",
      "closed form",
      "exponential",
      "wide range",
      "fluid dynamics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}