{
  "id": "2001.03858",
  "version": "v1",
  "published": "2020-01-12T05:30:46.000Z",
  "updated": "2020-01-12T05:30:46.000Z",
  "title": "Classification of primitive ideals of $U(\\mathfrak{o}(\\infty))$ and $U(\\mathfrak{sp}(\\infty))$",
  "authors": [
    "Aleksandr Fadeev"
  ],
  "comment": "PhD thesis",
  "categories": [
    "math.RT"
  ],
  "abstract": "The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras $U(\\mathfrak{o}(\\infty))$ and $U(\\mathfrak{sp}(\\infty))$. Let $\\mathfrak{g}(\\infty)$ denote any of the Lie algebras $\\mathfrak{o}(\\infty)$ or $\\mathfrak{sp}(\\infty)$. Then\\break $\\mathfrak{g}(\\infty)=\\bigcup_{n\\geq 2} \\mathfrak{g}(2n)$ for $\\mathfrak{g}(2n)=\\mathfrak{o}(2n)$ or $\\mathfrak{g}(2n)=\\mathfrak{sp}(2n)$, respectively. We show that each primitive ideal $I$ of $U(\\mathfrak{g}(\\infty))$ is weakly bounded, i.e., $I\\cap U(\\mathfrak{g}(2n))$ equals the intersection of annihilators of bounded weight $\\mathfrak{g}(2n)$-modules. To every primitive ideal $I$ of $\\mathfrak{g}(\\infty)$ we attach a unique irreducible coherent local system of bounded ideals, which is an analog of a coherent local system of finite-dimensional modules, as introduced earlier by A. Zhilinskii. As a result, primitive ideals of $U(\\mathfrak{g}(\\infty))$ are parametrized by triples $(x,y,Z)$ where $x$ is a nonnegative integer, $y$ is a nonnegative integer or half-integer, and $Z$ is a Young diagram. In the case of $\\mathfrak{o}(\\infty)$, each primitive ideal is integrable, and our classification reduces to a classification of integrable ideals going back to A. Zhilinskii, A. Penkov and I. Petukhov. In the case of $\\mathfrak{sp}(\\infty)$, only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples $(x,y,Z)$ where $y$ is a half-integer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-12T05:30:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "primitive ideal",
      "classification",
      "unique irreducible coherent local system",
      "nonnegative integer",
      "finite-dimensional modules"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}