{
  "abstract": "Let $Q$ be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with $Q$, which exhibits certain symmetries similar to 2-Calabi--Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra $\\mathcal {D}\\mathcal {H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$ via a certain quotient of a derived Hall subalgebra associated with $Q$, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with $Q$ is isomorphic to the exceptional subalgebra of $\\mathcal{D}\\mathcal{H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$, namely, the subalgebra generated by exceptional objects. Furthermore, when $Q$ is of Dynkin type, the quantum cluster algebra is naturally isomorphic to $\\mathcal{D}\\mathcal{H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$ itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of $\\mathcal{D}\\mathcal{H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$ generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander--Reiten translation of cluster categories induces an algebra automorphism of $\\mathcal {D}\\mathcal {H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$. Finally, we apply the BGP-reflections to $\\mathcal {D}\\mathcal {H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$, and prove that the Hall algebra $\\mathcal {D}\\mathcal {H}_Λ^{cl}(\\widetilde{\\mathcal{A}})$ is invariant up to isomorphism under the mutations at sink vertices of $Q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14388",
  "author_total": 2,
  "authors": [
    {
      "name": "Changjian Fu"
    },
    {
      "name": "Haicheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18nzajkery9yp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14388",
  "fulltext_url": null,
  "id": "18nzajkery9yp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14388",
  "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.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7213234593",
  "pdf_url": "https://arxiv.org/pdf/2609.14388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18nzajkery9yp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Acyclic quantum cluster algebras via Hall algebras associated to cluster categories",
  "title_text": "Acyclic quantum cluster algebras via Hall algebras associated to cluster categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/18nzajkery9yp",
  "venue": "arXiv",
  "versions": []
}