{
  "abstract": "Let $W$ be an algebra with involution over a field of characteristic zero. We develop a theory of $W$-polynomial identities with involution for finite-dimensional $(W,*)$-algebras $A$, using the multiplier algebra with involution of $A$. We prove that the $(W,*)$-exponent of $A$ exists and coincides with the ordinary $*$-exponent. We then consider a four-dimensional subalgebra $M$ of the algebra of $4\\times4$ upper triangular matrices, endowed with the reflection involution, and study two non-equivalent $(W,*)$-algebra structures on it. For both structures, we determine the corresponding $T_W^*$-ideals of $W$-polynomial identities with involution and compute the $(W,*)$-codimension sequences explicitly; for one of them, we also determine the complete $(W,*)$-cocharacter sequence. Finally, we prove that these two $W$-algebras with involution generate distinct varieties of almost polynomial growth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10406",
  "author_total": 3,
  "authors": [
    {
      "name": "Ginevra Giordani"
    },
    {
      "name": "Antonio Ioppolo"
    },
    {
      "name": "Carla Rizzo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4v05cdrmmeaq5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10406",
  "fulltext_url": null,
  "id": "4v05cdrmmeaq5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10406",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7220917238",
  "pdf_url": "https://arxiv.org/pdf/2610.10406",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4v05cdrmmeaq5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10406",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10406"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$W$-algebras with involution: polynomial identities and asymptotic growth",
  "title_text": "W-algebras with involution: polynomial identities and asymptotic growth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/4v05cdrmmeaq5",
  "venue": "arXiv",
  "versions": []
}