Affine noncommutative geometry — Draconitic Research

Abstract

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces ℝ^N,ℂ^N have no free analogues in the operator algebra sense, but the corresponding unit spheres S^N−1 R,S^N−1 C do have free analogues S^N−1_ℝ,+,S^N−1_ℂ,+. There are many examples of real algebraic submanifolds X⊂ S^N−1_ℝ,+,S^N−1_ℂ,+, some of which are of Riemannian flavor, coming with a Haar integration functional ∫:C(X)→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.

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BibTeX RIS

BibTeX

@misc{banica2026affine,
  title = {{Affine noncommutative geometry}},
  author = {Teo Banica},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-03312800v7},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Affine noncommutative geometry
AU  - Teo Banica
PY  - 2026
DA  - 2026/10/01
PB  - HAL
UR  - https://hal.science/hal-03312800v7
AB  - This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces ℝ^N,ℂ^N have no free analogues in the operator algebra sense, but the corresponding unit spheres S^N−1_ R,S^N−1_ C do have free analogues S^N−1_ℝ,+,S^N−1_ℂ,+. There are many examples of real algebraic submanifolds X⊂ S^N−1_ℝ,+,S^N−1_ℂ,+, some of which are of Riemannian flavor, coming with a Haar integration functional ∫:C(X)→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.
ER  -

Abstract

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