{
  "id": "1706.05950",
  "version": "v1",
  "published": "2017-06-19T13:58:32.000Z",
  "updated": "2017-06-19T13:58:32.000Z",
  "title": "Categories O for Dynkin Borel Subalgebras of Root-Reductive Lie Algebras",
  "authors": [
    "Thanasin Nampaisarn"
  ],
  "comment": "This is a dissertation in Mathematics for a doctoral degree at Jacobs University Bremen. It contains 111 pages and 1 figure",
  "categories": [
    "math.RT"
  ],
  "abstract": "The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\\mathcal{O}$ for the Lie algebra $\\mathfrak{g}$, where $\\mathfrak{g}$ is one of the finitary, infinite-dimensional Lie algebras $\\mathfrak{gl}_\\infty(\\mathbb{K})$, $\\mathfrak{sl}_\\infty(\\mathbb{K})$, $\\mathfrak{so}_\\infty(\\mathbb{K})$, and $\\mathfrak{sp}_\\infty(\\mathbb{K})$. Here, $\\mathbb{K}$ is an algebraically closed field of characteristic $0$. We call these categories \"extended categories $\\mathcal{O}$\" and use the notation $\\bar{\\mathcal{O}}$. While the categories $\\bar{\\mathcal{O}}$ are defined for all splitting Borel subalgebras of $\\mathfrak{g}$, this research focuses on the categories $\\bar{\\mathcal{O}}$ for very special Borel subalgebras of $\\mathfrak{g}$ which we call Dynkin Borel subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig multiplicities carry over from usual categories $\\mathcal{O}$ to our categories $\\bar{\\mathcal{O}}$. There are differences which we shall explore in detail, such as the lack of some injective hulls. In this connection, we study truncated categories $\\bar{\\mathcal{O}}$ and are able to establish an analogue of BGG reciprocity in the categories $\\bar{\\mathcal{O}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-19T13:58:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B22",
      "17B65"
    ],
    "keywords": [
      "dynkin borel subalgebras",
      "root-reductive lie algebras",
      "infinite-dimensional lie algebras",
      "special borel subalgebras",
      "results concerning block decomposition"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 111,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}