{
  "annotation_id": "ef7aa4d6-0d24-4061-9f02-e3d6e613fbfe",
  "date_created": "2026-03-02T18:01:28.531000Z",
  "date_modified": "2026-03-02T18:01:28.531000Z",
  "file_hash": "a566900d9ce954f3aef1f2f9b48e603253839ca5f9fbdf09d4e357259a3351ec",
  "private": false,
  "record": {
    "abstract": "The underlying algebra for a noncommutative geometry is taken to be a matrix\nalgebra, and the set of derivatives the adjoint of a subset of traceless\nmatrices. This is sufficient to calculate the dual 1-forms, and show that the\nspace of 1-forms is a free module over the algebra of matrices. The concept of\na generalised algebra is defined and it is shown that this is required in order\nfor the space of 2-forms to exist. The exterior derivative is generalised for\nhigher order forms and these are also shown to be free modules over the matrix\nalgebra. Examples of mappings that preserve the differential structure are\ngiven. Also given are four examples of matrix generalised algebras, and the\ncorresponding noncommutative geometries, including the cases where the\ngeneralised algebra corresponds to a representation of a Lie algebra or a\n$q$-deformed algebra.",
    "arxiv_id": "q-alg/9703034",
    "authors": [
      "Jonathan Gratus"
    ],
    "categories": [
      "q-alg",
      "math.QA"
    ],
    "doi": "10.1016/S0393-0440(97)00027-2",
    "title": "Non Commutative Differential Geometry, and the Matrix Representations of   Generalised Algebras",
    "url": "https://arxiv.org/abs/q-alg/9703034"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "bd41be84-d078-4af5-be8e-297781d52d28",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}