{
  "abstract": "We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle $σ$ on $K$ and a unital right-colinear algebra map $Γ:A\\to K$, we show that $Γ:A_σ\\to K^σ$ is an algebra map precisely when $σ$ is trivial on $\\operatorname{im}Γ\\otimes\\operatorname{im}Γ$. We describe comodule-algebra morphisms $A_σ\\to K^σ$ by characters and establish injectivity criteria using the coradical and extensions of characters to convolution-invertible functionals. For coactions on Hopf quotients projecting to the coproduct, character-evaluation maps are injective, and equality of their images is determined by a unique algebra character. For the $\\imath$-Hopf algebras $H^\\imath_τ$ of Chen--Lu--Pan--Ruan--Wang, with $H$ having a bijective antipode and a Hopf pairing, we obtain a Hopf bi-Galois realization $H^\\imath_τ\\hookrightarrow F_σ(H\\otimes H^{\\mathrm{cop}})$. We identify algebra characters with nonzero $τ$-twisted compatible maps $χ$ and prove that the corresponding morphisms $ξ_{τ,χ}$ of Chen--Lu--Pan--Ruan--Wang are injective, yielding right coideal realizations in $(H\\otimes H)^\\imath$. We classify the $H^\\imath_τ$ up to comodule-algebra isomorphism and determine when these realizations have the same image.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19503",
  "author_total": 1,
  "authors": [
    {
      "name": "Rongchuan Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7skph8kdfnbdd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19503",
  "fulltext_url": null,
  "id": "7skph8kdfnbdd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19503",
  "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.QA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7222133297",
  "pdf_url": "https://arxiv.org/pdf/2610.19503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7skph8kdfnbdd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Right Coideal Subalgebras, Hopf Galois Objects, and $\\imath$-Hopf Algebras",
  "title_text": "Right Coideal Subalgebras, Hopf Galois Objects, and imath-Hopf Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/7skph8kdfnbdd",
  "venue": "arXiv",
  "versions": []
}