{
  "abstract": "Let $G$ be a connected, simply connected complex simple algebraic group and $\\mathscr{A}_G=G/U^+$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags $\\mathcal{O}_q(\\mathrm{Conf}_K \\mathscr{A}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra $\\mathcal{O}_q(\\mathrm{Conf}_K \\mathscr{A}_G)$ gives a quantum analogue of the configuration space $\\mathrm{Conf}_K \\mathscr{A}_G$ of $K$ decorated flags, which provides local building blocks for the Fock--Goncharov moduli space $\\mathscr{A}_{G,Σ}$ of decorated twisted $G$-local systems on a marked surface $Σ$. We establish basic algebraic properties of $\\mathcal{O}_q(\\mathrm{Conf}_K \\mathscr{A}_G)$ such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of $\\mathscr{A}_{G,Σ}$. Moreover, we construct quantum seeds for $\\mathcal{O}_q(\\mathrm{Conf}_4 \\mathscr{A}_G)$ by transporting the Berenstein--Zelevinsky quantum cluster structure on $\\mathcal{O}_q(G)$ via quantum Wilson lines, and prove that $\\mathcal{O}_q(\\mathrm{Conf}_4 \\mathscr{A}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for $\\mathcal{O}_q(\\mathrm{Conf}_4 \\mathscr{A}_G)$ restrict to those for $\\mathcal{O}_q(\\mathrm{Conf}_3 \\mathscr{A}_G)$. Finally, we construct quantum seeds for $\\mathcal{O}_q(\\mathrm{Conf}_K \\mathscr{A}_G)$, $K\\geq 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space $\\mathcal{O}_q(\\mathrm{Conf}_K \\mathscr{A}_G)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02509",
  "author_total": 2,
  "authors": [
    {
      "name": "Tsukasa Ishibashi"
    },
    {
      "name": "Hironori Oya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7724t2wg3avkg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02509",
  "fulltext_url": null,
  "id": "7724t2wg3avkg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02509",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7207709562",
  "pdf_url": "https://arxiv.org/pdf/2609.02509",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7724t2wg3avkg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.02509",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02509"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Algebraic study of quantum configuration spaces of decorated flags",
  "title_text": "Algebraic study of quantum configuration spaces of decorated flags",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/7724t2wg3avkg",
  "venue": "arXiv",
  "versions": []
}