{
  "id": "1904.00894",
  "version": "v1",
  "published": "2019-04-01T14:52:55.000Z",
  "updated": "2019-04-01T14:52:55.000Z",
  "title": "Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theorem",
  "authors": [
    "François Chapon",
    "Reda Chhaibi"
  ],
  "comment": "28 pages ; Draft version",
  "categories": [
    "math.PR",
    "math.QA",
    "math.RT",
    "math.SG"
  ],
  "abstract": "It is understood that Pitman's theorem in probability theory is intimately related to the representation theory of $\\mathcal{U}_{q}(\\mathfrak{sl}_2)$, in the so-called crystal regime $q \\rightarrow 0$. This relationship has been explored by Biane and then Biane-Bougerol-O'Connell at several levels. On the other hand, Bougerol and Jeulin showed the appearance of the Pitman transform in the infinite curvature limit $r \\rightarrow \\infty$ of Brownian motion on the symmetric space $SL_2(\\mathbb{C})/SU_2$. In order to understanding the phenomenon, we exhibit a presentation of the Jimbo-Drinfeld quantum group $\\mathcal{U}_q^\\hbar(\\mathfrak{sl}_2)$ which isolates the role $r$ of curvature and that of the Planck constant $\\hbar$. The simple relationship between parameters is $q=e^{-r}$. The semi-classical limits $\\hbar \\rightarrow 0$ are the Poisson-Lie groups dual to $SL_2(\\mathbb{C})$ with varying curvatures $r \\in \\mathbb{R}_+$. We also construct classical and quantum random walks, drawing a full picture which includes Biane's quantum walks and the construction of Bougerol-Jeulin. The curvature parameter $r$ leads to both to the crystal regime at the level of representation theory ($\\hbar>0$) and to the Bougerol-Jeulin construction in the classical world ($\\hbar=0$). All these results are neatly in accordance with the philosophy of Kirillov's orbit method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-01T14:52:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58B32",
      "60B99"
    ],
    "keywords": [
      "pitmans 2m-x theorem",
      "representation theory",
      "crystal regime",
      "kirillovs orbit method",
      "infinite curvature limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}