{
  "abstract": "This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\\mathbb R^N,\\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_\\mathbb R,S^{N-1}_\\mathbb C$ do have free analogues $S^{N-1}_{\\mathbb R,+},S^{N-1}_{\\mathbb C,+}$. There are many examples of real algebraic submanifolds $X\\subset S^{N-1}_{\\mathbb R,+},S^{N-1}_{\\mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $\\int:C(X)\\to\\mathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Teo Banica"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8pxfdw5esgqh.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "c8pxfdw5esgqh",
  "issns": [],
  "landing_url": "https://hal.science/hal-03312800v7",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-oa"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-03312800/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8pxfdw5esgqh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:45Z",
      "native_id": "hal-03312800",
      "source": "hal",
      "url": "https://hal.science/hal-03312800v7"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-oa",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Affine noncommutative geometry",
  "title_text": "Affine noncommutative geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:45Z",
  "url": "https://research.draconitic.com/paper/c8pxfdw5esgqh",
  "venue": "HAL",
  "versions": []
}