# Questions tagged [noncommutative-geometry]

Noncommutative geometry is a study of noncomutative algebras from geometrical point of view. The motivation of this approach is Gelfand representation theorem, which shows that every commutative C*-algebra is *-isomorphic to the space of continuous functions on some locally compact Hausdorff space.

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### Constructing Cyclic Division Algebras

I'm studying the construction of cyclic division algebras but I don't see how a given example holds. According to the literature, we start with a finite extension $L$ of a number field $K$ such that ...
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### Index of Callias operator and application in physics

In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$...
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### Why do Kontsevich and Rosenberg use algebra epimorphisms rather than surjections?

In the article Noncommutative spaces Kontsevich and Rosenberg make the following definition: 2.6. The Q-category of infinitesimal algebra epimorphisms. Let $A$ be the category $Alg_k$ of associative ...
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### Technical question about computing the left ideals of the ring $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$

Computing the left ideals of $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$. This is the same that computing the ideals of $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, ...
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### Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
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### Definition of Fredholm modules

I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] ...
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### Lattices and noncommutative algebras in noncommutative geometry

I am interested in the relation between lattices and noncommutative algebras in the context of noncommutative geometry. In the commutative case, a lattice is a discrete subgroup of a locally compact ...
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1 vote
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### The map $C^\infty(M,U(A_F))\to C^\infty(M,U(A_F)/\mathfrak{H}(F))$ is an homomorphism.

I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have problems to identify one map as homomorphism. Let $M\times F$ be an almost-commutative manifold. The ...
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### Quantum''/ non-commutating extension of polytopes

Are there non-commutative (ie. quantum) extensions of polytopes? More specifically I was wondering if there are some deformation, say $\hbar$, to polytopes which when is taken to be zero, one gets ...
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### Showing the existence of a right-inverse in a von Neumann algebra

In this paper, specifically Theorem 4.1 on page 10, one of the last steps in the proof involves saying that a particular right-inverse exists. I'll try to restate what I think are the relevant pieces ...
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### Geometry vs Topology

I am a beginner in functional analysis and operator algebras. Recently I stumbled upon the two subjects, non-commutative topology and non-commutative geometry. What are the similarities and ...
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### symmetry group for noncommutative manifold

(This post is cross-post in MathOverflow https://mathoverflow.net/questions/381731/symmetry-group-for-noncommutative-manifold-from-spectral-triple) Is there any notion of symmetry group for ...
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### Is the KO dimension of commutative real spectral triple agree with it's dimension of the manifold?

Given a spectral triple with some conditions(such as the algebra is commutative), Connes's reconstruction theorem states a we can recover a Riemannian manifold with spin structure.(see here) Now I ...
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### what is noncommutative geometry studying if "noncommutative space" does not exist?

(If this is not a suitable question in here, I will delete it.) I am new to noncommutative geometry. I know the motivation is to try to generalize the Gelfrand duality to noncommutative $C^{*}$ ...
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### supersymmetry and quantum groups

I have some background in non-commutative geometry (in particular, I am doing research in quantum groups) and I have of course heard several times about the concept of supergeometry and supergroups. ...
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### Grassman variable and Grassmannian?

Grassman variables are anticommuting number or supernumber, is an element of the exterior algebra over the complex numbers. Grassmannian $Gr(k, V)$ is a space that parameterizes all $k$-dimensional ...
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### Hilbert bundle and set of sections.

Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity ...
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### Is there any reference treating explicitly Lie-Rinehart pairs over non-commutative base algebras?

Wondering around in the literature, any reference to Lie-Rinehart algebras define them as pairs $(A,L)$ where $A$ is a commutative algebra over some field $\Bbbk$ (or even commutative ring) and $L$ is ...
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### How to read Alain Connes's Noncommutative Geometry [closed]

I'm a undergraduate student interested in noncommutative geometry and quantum field theory. And I have learned basic functional analysis,theoretical mechanics,basic quantum mechanics, now, I'm ...
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### $f(t) = \min\{1,t\}$ not operator monotone
I want to show that the function on $\mathbb{R}^+$ $f(t) = \min \{1,t\}$ is not operator monotone on the complex $2\times 2$ matrices. My plan is to find matrices $A$ and $B$, $B\geq A$, such that the ...