{
  "id": "2310.12163",
  "version": "v1",
  "published": "2023-10-01T12:01:32.000Z",
  "updated": "2023-10-01T12:01:32.000Z",
  "title": "Computation of Gelfand-Kirillov dimension for $B$-type structures",
  "authors": [
    "Akshay Bhuva",
    "Bipul Saurabh"
  ],
  "comment": "26 pages",
  "categories": [
    "math.RT",
    "math.OA"
  ],
  "abstract": "Let $\\mathcal{O}(\\mbox{Spin}_{q^{1/2}}(2n+1))$ and $\\mathcal{O}(SO_q(2n+1))$ be the quantized algebras of regular functions on the Lie groups $\\mbox{Spin}(2n+1)$ and $SO(2n+1)$, respectively. In this article, we prove that the Gelfand-Kirillov dimension of a simple unitarizable $\\mathcal{O}(\\mbox{Spin}_{q^{1/2}}(2n+1))$-module $V_{t,w}^{\\mbox{Spin}}$ is the same as the length of the Weyl word $w$. We show that the same result holds for the $\\mathcal{O}(SO_q(2n+1))$-module $V_{t,w}$, which is obtained from $V_{t,w}^{\\mbox{Spin}}$ by restricting the algebra action to the subalgebra $\\mathcal{O}(SO_q(2n+1))$ of $\\mathcal{O}(\\mbox{Spin}_{q^{1/2}}(2n+1))$. Moreover, we consider the quantized algebras of regular functions on certain homogeneous spaces of $SO(2n+1)$ and $\\mbox{Spin}(2n+1)$ and show that its Gelfand-Kirillov dimension is equal to the dimension of the homogeneous space as a real differentiable manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-01T12:01:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gelfand-kirillov dimension",
      "type structures",
      "computation",
      "regular functions",
      "homogeneous space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}