{
  "id": "1812.00330",
  "version": "v1",
  "published": "2018-12-02T05:17:02.000Z",
  "updated": "2018-12-02T05:17:02.000Z",
  "title": "On the module structure of the center of hyperelliptic Krichever-Novikov algebras II",
  "authors": [
    "Ben Cox",
    "Xiangqian Guo",
    "Mee Seong Im",
    "Kaiming Zhao"
  ],
  "comment": "24 pages, submitted",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $R := R_{2}(p)=\\mathbb{C}[t^{\\pm 1}, u : u^2 = t(t-\\alpha_1)\\cdots (t-\\alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $\\mathfrak{g}\\otimes R$ be the corresponding current Lie algebra. \\color{black} Here $\\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\\mathbb C$ and \\begin{equation*} p(t)= t(t-\\alpha_1)\\cdots (t-\\alpha_{2n})=\\sum_{k=1}^{2n+1}a_kt^k. \\end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $\\mathfrak{g}\\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $\\Omega_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(\\alpha_1,\\dots,\\alpha_{2n})$, which are the automorphism groups $\\mathbb G_n=\\text{Dic}_{n}$, $\\mathbb U_n\\cong D_n$ ($n$ odd), or $\\mathbb U_n$ ($n$ even) of the hyperelliptic curves \\begin{equation} S=\\mathbb{C}[t, u: u^2 = t(t-\\alpha_1)\\cdots (t-\\alpha_{2n})] \\end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $\\mathbb U_n=D_n$, where $n$ is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-02T05:17:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperelliptic krichever-novikov algebras",
      "module structure",
      "automorphism group",
      "finite dimensional simple lie algebra",
      "hyperelliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}