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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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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36 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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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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62 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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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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45 views

two disjoint closed sets in a locally compact Hausdorff space

Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the $C^{*}$ algebra of all continuous complex functions on $X$ which vanish at infinity. For a closed subset $F$ of $X$, denote by ...
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60 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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49 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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37 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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32 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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90 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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33 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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65 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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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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96 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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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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A question about C* probability spaces.

If I have $C(\Omega)$: the commutative C*-Algebra of all continuous random variables then the C*Algebra $C(\Omega)$ acts on the Hilbert space $L^2(\Omega)$ by left multiplication, and so we have 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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131 views

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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178 views

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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166 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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199 views

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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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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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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150 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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193 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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56 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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154 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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65 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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70 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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131 views

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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179 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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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 ...
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312 views

About PhD in non-commutative topology

I am a 3rd year undergraduate student. What are the things that would be good to know if I apply for PhD? I want to do a PhD in non-commuative topology. I am fascinated by this non-commutative ...
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144 views

Free probability background requirements

I wish to learn free probability, and looking for a kind of program to learn it. Where should I start? Where do I continue? Which is the bibliography? and finally where do I start to learn free ...
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quasiperiodic tilings: inverse semigroup — non-commutative geometry connect?

A Connes' Noncommutative Geometry (1994) and M Lawson's Inverse Semigroups (1998) contain sections on quasiperiodic tilings, yet as far as I can tell neither seems to refer to the other field of ...
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Dixmier-Douady class: computations

As far as I know, Dixmier-Douady classes represent obstrucions to spin$^c$ structures. Questions: Could somebody prove or give a reference: manifolds of dimension lower than $5$ always have a ...
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140 views

Compact resolvent VS certain boundedness condition

The following question is motivated by the definition of spectral triples in noncommutative geometry. This question was split in the following parts: First: Could somebody give diverse examples of ...
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202 views

Does this notion of morphism of noncommutative rings appear in the ring theory literature?

Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
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437 views

8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries

Periodicity modulo 8 appears in the classification of real Clifford algebras $C\ell_{p,q}(\mathbb{R})$ (usualy refered to as the "Clifford Clock"), in real Bott periodicity and in the definition of a ...
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406 views

How much algebra is there in Noncommutative Geometry?

My Professor of Homological Algebra got me into some Hochschild (co)homology and then suggested to continue with formally smooth algebras, noncommutative differential forms and so forth. Now, my ...
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An example of functions on a groupoid

There are two $C^\ast$-algebras associated to the $\ast$-algebra (under a convolution and the usual involution) $$C_c(G) := \{ f:G\longrightarrow \mathbb C :\:f \text{ has compact support}\}$$ of ...
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Could Motives aid in the study of the Navier-Stokes equations?

Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these ...
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Why study Hopf Algebras?

I'm looking for reasons that motivate the study of Hopf Algebra, like its applications in other branches of mathematics or maybe with physics. The first I've got is that they're interesting by ...
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177 views

Dense *-subalgebras of C*-algebras and their intersections with sub-C*-algebras

Consider the following question: Let $A$ be a normed space containing a closed subset $B\subseteq A$ and a dense subset $D\subseteq A$. Is $B \cap D$ necessarily a dense subset of $B$? My conclusion ...
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162 views

The crossed product of a non unital C*-algebra

Let $X$ be a locally compact space, and let $\mathbb{Z}$ act on $C_0(X)$ by an automorphism $\alpha$. Is the resulting crossed product unital?