{
  "abstract": "This paper studies commutators in finite-dimensional unimodular Hopf algebras and invariants derived from them. For a unimodular Hopf algebra $H$ with the distinguished group-like element $g$, the commutator of $a,b\\in H$ is defined by $\\{a,b\\}=a_{1}S^{2}(b_{1})S(a_{2})S^{-1}(b_{2})g^{-1}$. We show that the span of all commutators is a left coideal and generates a normal left coideal subalgebra $N$ of $H$ for which $H/HN^{+}$ is commutative. The main result asserts that for any central element $a$ and any integral $Λ$ of $H$, the commutator $\\{a,Λ\\}$ and the value $λ(\\{a,Λ\\})$ for a right integral $λ$ of $H^*$ are invariant under twisting. This yields several gauge invariants, including ratios of characters of iterated commutators with $Λ$. In the semisimple characteristic-zero case, explicit descriptions of these invariants are given. As applications, we compute some invariants for the group algebra of a finite group, its Drinfeld double, and the restricted quantum group $\\overline{U}_{q}(sl_{2})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24523",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhihua Wang"
    },
    {
      "name": "Libin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ydv14gkavdfb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24523",
  "fulltext_url": null,
  "id": "2ydv14gkavdfb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24523",
  "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.RT"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7214019412",
  "pdf_url": "https://arxiv.org/pdf/2609.24523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ydv14gkavdfb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Gauge invariants raised from commutators of unimodular Hopf algebras",
  "title_text": "Gauge invariants raised from commutators of unimodular Hopf algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2ydv14gkavdfb",
  "venue": "arXiv",
  "versions": []
}