{
  "id": "1803.09931",
  "version": "v2",
  "published": "2018-03-27T07:14:16.000Z",
  "updated": "2018-05-14T12:59:39.000Z",
  "title": "Classical $N$-Reflection Equation and Gaudin Models",
  "authors": [
    "Vincent Caudrelier",
    "Nicolas Crampe"
  ],
  "comment": "12 pages. References added. Explicit relation between our non-skew symmetric r-matrices and standard rational r-matrix given in the Gaudin models section",
  "categories": [
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We introduce the notion of $N$-reflection equation which provides a large generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the $N=2$ case. The basic theory is established and illustrated with several examples of solutions of the $N$-reflection equation associated to the rational and trigonometric $r$-matrices. A central result is the construction of a Poisson algebra associated to a non skew-symmetric $r$-matrix whose form is specified by a solution of the $N$-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of $BC_L$-type .",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-05-14T12:59:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaudin models",
      "poisson algebra",
      "usual classical reflection equation",
      "integrable boundary conditions",
      "special example"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}