{
  "abstract": "This paper studies the category $\\mathcal{C}$ of grading-restricted $C_1$-cofinite generalized modules for the vertex operator algebra associated to the $W$-algebra $W(2,2)$. We first show that $\\mathcal{C}$ is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight $W(2,2)$--modules $L[r]$ of highest weight $(\\frac{1-r}{2}, 0)$ for $r \\in \\mathbb{Z}_{> 0}$, and hence $\\mathcal{C}$ carries a braided tensor category structure. Then we prove the fusion rules for the simple objects $L[r]$ are governed by the $\\mathfrak{sl}_2$ Clebsch--Gordan rule. In particular, we prove \\[ L[r]\\boxtimes L[s]\\cong \\bigoplus_{i=0}^{\\min\\{r,s\\}-1} L[r+s-1-2i]. \\] Using the fusion rules and a recent result of Etingof--Penneys, we establish the rigidity of $\\mathcal{C}$. We also show the semisimple subcategory generated by the simple objects $L[r]$ is tensor equivalent to the category Rep $\\mathfrak{sl}_2$ of finite dimensional $\\mathfrak{sl}_2$-modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02202",
  "author_total": 4,
  "authors": [
    {
      "name": "Drazen Adamovic"
    },
    {
      "name": "Mingjie Peng"
    },
    {
      "name": "Gordan Radobolja"
    },
    {
      "name": "Jinwei Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bpsre75xpxn2e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02202",
  "fulltext_url": null,
  "id": "bpsre75xpxn2e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02202",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7207654422",
  "pdf_url": "https://arxiv.org/pdf/2609.02202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bpsre75xpxn2e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:16Z",
      "native_id": "2609.02202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The tensor category for W(2,2)-vertex algebra",
  "title_text": "The tensor category for W(2,2)-vertex algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/bpsre75xpxn2e",
  "venue": "arXiv",
  "versions": []
}