Affine noncommutative geometry
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916344778194944 |
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| author | Banica, Teo |
| author_facet | Banica, Teo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_10973 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Affine noncommutative geometry Banica, Teo Quantum Algebra Operator Algebras 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. |
| title | Affine noncommutative geometry |
| topic | Quantum Algebra Operator Algebras |
| url | https://arxiv.org/abs/2012.10973 |