{
  "id": "math/9803159",
  "version": "v1",
  "published": "1998-03-12T00:00:00.000Z",
  "updated": "1998-03-12T00:00:00.000Z",
  "title": "Down-up Algebras",
  "authors": [
    "Georgia Benkart",
    "Tom Roby"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra $U(\\fsl)$ of the Lie algebra $\\fsl$ including a Poincar\\'e-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category $\\mathcal O$ modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-03-12T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "down-up algebras",
      "highest weight representations",
      "poincare-birkhoff-witt type basis",
      "exact expressions",
      "encodes essential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......3159B"
    }
  }
}