{
  "abstract": "In this paper, we present a geometric construction of contragredient Lie superalgebras, with the exceptions of $A(m,n)$ $(m,n>0)$ and $G(3)$. Let $\\mathfrak{g}$ be a complex simple Lie algebra with an order 4 automorphism $σ$, the eigenspace decomposition being $\\mathfrak{g}=\\mathfrak{g}(1)+\\mathfrak{g}(i)+\\mathfrak{g}(-1)+\\mathfrak{g}(-1)$. When $\\mathfrak{g}(i)$ admits a $\\mathfrak{g}(1)$-invariant symplectic structure, $(\\mathfrak{g},σ)$ is called a \\emph{generalized symmetric pair with symplectic subspace}. In this paper, we classify all such pairs and construct a natural Lie superalgebra structure on $L= L_{\\bar{0}}+L_{\\bar{1}}$ with $L_{\\bar{0}}=\\mathfrak{g}(1)$ and $L_{\\bar{1}}=\\mathfrak{g}(i)$. Furthermore, we investigate the constants arising from the squares of two distinct Dirac operators: $D$ for $\\mathfrak{g}=\\mathfrak{g}^{σ^2}+\\mathfrak{g}^{-σ^2}$, and $D_s$ for $L=\\mathfrak{g}(1)+\\mathfrak{g}(i)$. We prove that the constant $C$ in $D^2$ admits a remarkable factorization \\[C=J^*(Ω_{L_{\\bar{0}}})\\cdot\\dim L_{\\bar{1}}\\cdot h^\\vee_{\\mathrm{sign}},\\] where $h^\\vee_{\\mathrm{sign}}$ is the signed dual Coxeter number of $L$ and $J^*(Ω_{L_{\\bar{0}}})$ is a mapped Casimir constant of its even part. This formula reveals an unexpected unification of symmetric space geometry and Lie superalgebra invariants. Finally, we prove that the Kostant constant $\\mathrm{K_s} = \\left.Ω_{L_{\\bar{0}}}\\right|_{L_{\\bar{1}}}=\\frac{1}{2}$ is universal for all contragredient Lie superalgebras with non-degenerate Killing form, which implies that the constant $C_s$ in $D_s^2$ satisfies $C_s=\\frac{1}{16}\\dim L_{\\bar{0}}$. We then establish a bijection between such superalgebras and solutions of the superadmissibility index equations $E_s^{I\\!I} =\\frac{1}{2}, E_s^I = \\frac{1}{2}$, recovering the classification from an invariant-theoretic perspective.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25226",
  "author_total": 2,
  "authors": [
    {
      "name": "Mingjing Zhang"
    },
    {
      "name": "Fuhai Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezm338220z47z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25226",
  "fulltext_url": null,
  "id": "ezm338220z47z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25226",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7223057253",
  "pdf_url": "https://arxiv.org/pdf/2610.25226",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezm338220z47z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25226",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25226"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants",
  "title_text": "Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/ezm338220z47z",
  "venue": "arXiv",
  "versions": []
}