{
  "abstract": "Let $\\mathfrak{g}=\\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\\mathfrak{h}$ of $\\mathrm{g}$ and the classical cone of dominant integral weights $P^+\\subset \\mathfrak{h}^*$. The non-integrable simple highest weight $\\mathfrak{g}$-modules $L(μ)$'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for $μ$'s dot-linked to $P^+$-translates of sums $- \\sum_{j\\in J}α_j$ of mutually orthogonal and imaginary simple roots $α_j$'s. Recently, we computed weights of all highest weight $\\mathfrak{g}$-modules $V$'s (over all BKM LA's), and character of $L(ρ)$ for Weyl vector $ρ$. These needed a family of ``integrable'' $L(μ)$'s for $μ$'s inside our novel signed-dominant-integral cone $P^{\\pm}$ (which generalizes $P^+$). Pairings $μ(α_i^{\\vee})\\leq 0$ for $μ\\in P^{\\pm}$ are multiples of $\\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(μ)$ contains ``Chevalley-Serre relations'' $f_i^{\\frac{2}{A_{ii}}{μ(α_i^{\\vee})}+1}L(μ)_μ=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(μ)$'s of $L(μ)$'s for $μ\\in P^{\\pm}$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.20386",
  "author_total": 3,
  "authors": [
    {
      "name": "Souvik Pal"
    },
    {
      "name": "Supravat Sarkar"
    },
    {
      "name": "G. Krishna Teja"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5tpjvh7fwv4dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.20386",
  "fulltext_url": null,
  "id": "5tpjvh7fwv4dd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.20386",
  "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",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7165384633",
  "pdf_url": "https://arxiv.org/pdf/2606.20386",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5tpjvh7fwv4dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2606.20386",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.20386"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras",
  "title_text": "Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/5tpjvh7fwv4dd",
  "venue": "arXiv",
  "versions": []
}