{
  "abstract": "We study the real three-dimensional unital algebra with basis $\\{1,i,j\\}$ and products $i^2=j^2=-1,\\quad ij=0,\\quad ji=i+j$. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is unique. Every nonzero subalgebra contains the unit, and every element outside these three planes generates the whole algebra. We classify one-sided inverses, including the singular cases, and show that two-sided invertibility does not imply cancellation. The multiplication algebra is $M_3(\\mathbb{R})$, so the algebra is absolutely central simple. Its derivations vanish, its automorphism group is trivial, and its commutator Lie algebra is $\\mathbb{R}\\oplus\\mathfrak{aff}(1)$. Its unique antiautomorphism is involutive, and its complexification admits no complex-linear structurable involution. Finally, replacing $ji=i+j$ by $ji=t(i+j)$ gives real algebras $\\mathcal{A}_t$ with $\\mathcal{A}_s\\cong\\mathcal{A}_t$ exactly when $s^2=t^2$. We describe the exceptional parameters and thereby exhibit nontrivial deformations of the original algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12483",
  "author_total": 1,
  "authors": [
    {
      "name": "Alejandro Jesús Bermejo Valdés"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0dn2rbc9xx2sr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12483",
  "fulltext_url": null,
  "id": "0dn2rbc9xx2sr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12483",
  "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": "W7221554831",
  "pdf_url": "https://arxiv.org/pdf/2610.12483",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0dn2rbc9xx2sr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12483",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12483"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Complex slices, inverses, and deformations of a three-dimensional real algebra",
  "title_text": "Complex slices, inverses, and deformations of a three-dimensional real algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/0dn2rbc9xx2sr",
  "venue": "arXiv",
  "versions": []
}