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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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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Noncommutative manifold: Spectral triples on noncommutative quotients

I'm interested in taking the noncommutative quotient of a manifold, and endowing it with a kind of noncommutative smooth structure. More formally I'm interested in the question: is there a canonical ...
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Question about Noncommutative quotients

I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are ...
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How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
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Algebra of matrix valued functions on the sphere

I was watching a video of a lecture by Alain Connes, and at around 8:00 he very briefly mentions a way to think about the algebra of 2x2 matrix functions over the 2-sphere, i.e. the maps from $S^2 ...
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Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
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NCG with all noncommutativity in a nilpotent ideal

While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ...
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Can the Berezinian be used instead of the determinant of a block matrix?

Suppose we have a $2N \times 2N$ matrix $(N\ge 2)$. For example we can consider a block matrix: $$X= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]$$ with $A,B,C,D=$ $2\times 2$ ...
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pullback of global sections with respect to an automorphism of schemes

Let $X$ be a projective scheme and $\sigma:X\to X$ an automorphism of $X$. Is there a natural pullback of global sections map $H^0(X,\mathcal{F}) \to H^0(X,\sigma^*\mathcal{F})$ for ...
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There is only one interesting measure space

On page 52 of Noncommutative Geometry (available here: http://www.alainconnes.org/docs/book94bigpdf.pdf), Alain Connes states, "This wealth of transformations of a measure space $X$ is bound up in ...
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Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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1answer
40 views

Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
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95 views

Construction of noncommutative torus

In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, ...
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36 views

Example of a regular element with a commutative quotient

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit and $A/(x)$ is commutative?
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58 views

Example of a regular element in noncommutative rings

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit?
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40 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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Global dimension of translation algebra

What is the Hochschild cohomological dimension of the "translation algebra": $\mathbb{C}\langle x,y\rangle/(xy-yx-x)$? I expect it to be $2$, but I haven;t found a serious argument as to why this ...
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1answer
77 views

Isn't the center of a von Neumann algebra on a separable Hilbert space a hyperfinite von Neumann subalgebra?

this is a very quick, probably dumb, question, I was reading this chapter from "Hochschild cohomology of von Neumann algebras" by Allan Sinclair and Roger M. Smith and I came across this theorem on ...
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2answers
75 views

Roe algebra of a countably infinite set of points

First let me state some definitions. Let $X$ be a second countable, proper metric space. Let $H$ be a separable Hilbert space equipped with a nondegenerate $*$-representation $C_{0}(X)\rightarrow ...
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1answer
72 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
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63 views

Example of excision in Hochschild homology

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology). However, I cannot actually seem to think of a result ...
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45 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
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37 views

Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
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250 views

Obtaining rotation matrix from Euler angles if all three rotations happen at once. Does order of multiplication matter?

I'm having a problem getting my head around Euler Angles. Specifically if I wish to obtain a rotation matrix for a system where pitch, roll and yaw have all changed at once by various values... how ...
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44 views

Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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89 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
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1answer
32 views

Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
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140 views

Is there a discrete version of non-commutative geometry (yet)?

I wonder if mathematicians have developed a discrete version of non-commutative geometry, a bit like graphs, simplicial complexes etc may be seen as a discrete version of (Riemannian) geometry (of ...
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21 views

Reference for proof of homotopy invarance of Cyclic cohomolgy

I'm looking for a good reference for a proof of the homotopy invariance of cyclic (co)homology. I'm following a refernce book by Joachim Cuntz, the proofs are ommited therein, or only shown in the ...
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Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
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A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus ...
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247 views

Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
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Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
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2answers
127 views

Specific (algebraic) directions in NCG

Since my initial question (How much algebra is there in Noncommutative Geometry?), I got to study the basics of NCG and I now consider starting a PhD. program in NCG. I read Khalkhali's Basic NCG and ...
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45 views

Exactness of $\Gamma^\infty$ Functor

Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth ...
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1answer
170 views

Left-ratio and right-ratio in (not necessarily commutative) field

I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. The authors use terms right-ratio and left-ratio in section 13. Right-ratio is denoted as $[x_1, x_2, ... ...
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1answer
234 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
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1answer
61 views

Describing the algebra of functions on $S^2$

Chapter 2 of the book "Elements of Noncommutative Geometry" claims that the $C^*$-algebra of functions on $S^2$ can be described as an algebra with 3 generators a,b,c all with norm 1, where $a,b$ are ...
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167 views

Prize for textbook aesthetics

When browsing through Alain Connes' textbook on Noncommutative Geometry, whose illustrations must have been conceived as true works of love, I was wondering if there is a recognized prize for ...
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72 views

how to calculate the derivative of a plane wave in non commutative geometry

Shahn Majid and Eliezer Batista find the derivative of a wave plane in their paper: Non Commutative Geometry of Angular Momentum Space $U \mathfrak{su}(2)$. They obtained the non commutating ...
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1answer
77 views

Functional Interpretation of Variety?

I am simultaneously taking courses in functional analysis and commutative algebra. In doing so, I found that there is, at least heuristically, some similarity between the notion of an algebraic ...
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Analogs of group schemes over non-commutative rings

For a commutative ring $R$, I can consider $\operatorname{GL}_n(R)$ as a group scheme over $\operatorname{Spec} R$. Are there analogs of this notion when $R$ is non-commutative, say $R = ...
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Functions space of discrete space: how does taking quotients lead to noncommutativity?

It is pointed out in Geometry from the spectral point of view the following: If one considers a discrete space, say, the two-point space $\{1,2\}$, after identifying its points $X=\{1,2\}/\sim$, the ...
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Preparation for a foray into noncommutative geometry

I am applying for a short undergraduate/honours research project, and am trying to decide on preferences for a topic. One of the possible topics is noncommutative geometry. I like the sound of this, ...
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1answer
243 views

Localization of a non-commutative ring

Let $A$ be the non-commutative ring given by $$ A=\mathbb{C}\langle x,y,z \rangle /(xy=ayx,yz=bzy,zx=cxz) $$ for some $a,b,c\in \mathbb{C}$. What is the localization $A_{(x)}$ of A with respect to the ...
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1answer
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Showing that the universal enveloping algebra of some $\mathfrak{g}$ is isomorphic to $\mathbb{C}[x_i,\partial/\partial x_i]$

The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is defined to be $$ \dfrac{\mathbb{C}\oplus\mathfrak{g}\oplus ( \mathfrak{g}\otimes ...