{
  "id": "0706.4195",
  "version": "v1",
  "published": "2007-06-28T10:57:14.000Z",
  "updated": "2007-06-28T10:57:14.000Z",
  "title": "Flag Partial Differential Equations and Representations of Lie Algebras",
  "authors": [
    "Xiaoping Xu"
  ],
  "comment": "30pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.AP",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "In this paper, we solve the initial value problems of variable-coefficient generalized wave equations associated with trees and a large family of linear constant-coefficient partial differential equation by algebraic methods. Moreover, we find all the polynomial solutions for a 3-dimensional variable-coefficient flag partial differential equation of any order, the linear wave equation with dissipation and the generalized anisymmetrical Laplace equation. Furthermore, the polynomial-trigonometric solutions of a generalized Klein-Gordan equation associated with 3-dimensional generalized Tricomi operator $\\ptl_x^2+x\\ptl_y^2+y\\ptl_z^2$ are also given. As applications to representations of Lie algebras, we find certain irreducible polynomial representations of the Lie algebras $sl(n,\\mbb{F}), so(n,\\mbb{F})$ and the simple Lie algebra of type $G_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-28T10:57:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "representations",
      "generalized wave equations",
      "linear constant-coefficient partial differential equation",
      "variable-coefficient flag partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.4195X"
    }
  }
}