{
  "abstract": "Let $\\mathfrak g$ be a finite-dimensional simple complex Lie algebra, and consider the loop algebra $L\\mathfrak g = \\mathfrak g[s, s^{-1}]$ and the affine Lie algebra $\\widehat{\\mathfrak g}$, which is an extension of $L\\mathfrak g$ by a central element $c$. It is known that the rings $\\operatorname{U}(\\widehat{\\mathfrak g})/(c-λ)$ are simple for any nonzero scalar $λ$. On the other hand, the universal enveloping algebra $\\mathrm{U}(L\\mathfrak g) \\cong \\mathrm{U}(\\widehat{\\mathfrak g})/(c)$ has many two-sided ideals, and in this paper we investigate their structure. We show that the two-sided structure of $\\mathrm{U}(L\\mathfrak g)$ is both intricate and well-behaved. Specifically, $\\mathrm{U}(L\\mathfrak g)$ does not satisfy the ascending chain condition on two-sided ideals, but the two-sided ideals still have a nice structure: there is a canonical nested collection of ideals $I_n$, parameterised by positive integers, so that any nonzero two-sided ideal of $\\mathrm{U}(L\\mathfrak g)$ contains all $I_n$ for $n \\gg 0$. The ideals $I_n$ can be thought of as universal annihilators of natural classes of finite-dimensional representations of $L\\mathfrak g$. We also study the Poisson structure of the symmetric algebra $\\mathrm{S}(L\\mathfrak g)$ and give applications to the representation theory of $L\\mathfrak g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22040",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexey V. Petukhov"
    },
    {
      "name": "Susan J. Sierra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffr2jjggjfkwf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22040",
  "fulltext_url": null,
  "id": "ffr2jjggjfkwf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22040",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7222573716",
  "pdf_url": "https://arxiv.org/pdf/2610.22040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffr2jjggjfkwf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ideals in Enveloping Algebras of Loop Algebras",
  "title_text": "Ideals in Enveloping Algebras of Loop Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/ffr2jjggjfkwf",
  "venue": "arXiv",
  "versions": []
}