{
  "abstract": "The behavior of representations under restriction is a central theme in Lie theory. We study wide regular subalgebras of symmetrizable Kac--Moody algebras, extending work of Douglas and Repka on semisimple Lie algebras. A subalgebra is \\emph{wide} if every irreducible integrable highest weight module remains indecomposable upon restriction. Let $\\mathfrak{g}$ be a symmetrizable Kac--Moody algebra with Cartan subalgebra $\\mathfrak{h}$, root system $Φ$, simple roots $Π$, and root space decomposition $\\mathfrak{g}=\\mathfrak{h}\\oplus\\bigoplus_{α\\inΦ}\\mathfrak{g}_α$. Denote by $Φ_{\\operatorname{re}}$ the set of real roots. To a regular subalgebra $\\mathfrak{s}$ normalized by $\\mathfrak{h}$, we associate a subset $T\\subseteq Φ$ by declaring $α\\in T$ if $\\mathfrak{s}\\cap \\mathfrak{g}_α\\ne \\{0\\}$. One of our main results is an extension of Schur's lemma: if $\\mathfrak{h}\\subseteq \\mathfrak{s}$ and the real closure of $(T\\cup(-T))\\cap Φ_{\\operatorname{re}}$ contains $Π$, then $(\\operatorname{End} V)^{\\mathfrak{s}}=\\mathbb{C}\\operatorname{Id}_V$ for every irreducible integrable highest weight module $V$. As a consequence, this real-root closure condition yields a sufficient condition for wideness. In the affine case, we establish a necessary condition for wideness: if $\\mathfrak{s}$ is wide, then the closure of $T\\cup(-T)$ in $Φ$ is all of $Φ$, and this implication holds without assuming that $\\mathfrak{h}\\subseteq \\mathfrak{s}$. A key ingredient is a structural result showing that closed subsets of affine root systems are closed under arbitrary finite root sums that remain roots.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.01371",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrew Douglas"
    },
    {
      "name": "Abid Ali"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2x205gvh352g5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01371",
  "fulltext_url": null,
  "id": "2x205gvh352g5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01371",
  "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": "W7163256716",
  "pdf_url": "https://arxiv.org/pdf/2606.01371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2x205gvh352g5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2606.01371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma",
  "title_text": "Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/2x205gvh352g5",
  "venue": "arXiv",
  "versions": []
}