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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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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61 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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47 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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125 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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44 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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58 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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21 views
A good description of $M^{\vee\vee}$?
Let $R$ be a f.g. $\mathbb{N}$-graded non-commutative $\mathbb{C}$-algebra. Assume $R$ is connected, i.e. $R_0=\mathbb{C}$. Let $M$ be a f.g. torsion free graded right $R$-module of rank $1$. Is there ...
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26 views
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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55 views
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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89 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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89 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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83 views
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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1answer
227 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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2answers
99 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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25 views
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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80 views
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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98 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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171 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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1answer
216 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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2answers
271 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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72 views
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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97 views
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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521 views
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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139 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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1answer
151 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?
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90 views
What is the connection between Hilbert modules and tangent bundles in this paper?
A paper by Cipriani and Sauvageot, available at
http://dx.doi.org/10.1016/S0022-1236(03)00085-5
shows that for many Dirichlet forms on $C^*$-algebras there is a derivation $\delta$ from the domain ...
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3answers
301 views
Applications or uses of the Serre-Swan theorem
The Serre-Swan theorem states (at least in one form) that the category of real vector bundles over a compact Hausdorff space $M$ is equivalent to the category of finitely generated projective modules ...
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1answer
811 views
A fleshed-out version of the Noncommutative Geometry proof of the Gauss-Bonnet Theorem?
In Connes's book on noncommutative geometry, he outlines a rather short "algebraic" proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given ...
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181 views
Minimal spectrum of a commutative ring
Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
4
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1answer
144 views
The primitive spectrum of a unital ring
I'm trying to investigate the notion of primitive spectrum and its so-called Jacobson or hull-kernel topology, but I can only find references which define it for C*-algebras: see the Wikipedia page ...
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1answer
280 views
Fundamental group of noncommutative torus
Let $\tau \in \mathbb{S}^1$ be such that $\tau$ is not a root of unity. Let $E_\tau$ be the quotient space $S^1/\tau^{\mathbb Z}$. Consider it as a pointed space with basepoint the equivalence class ...
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312 views
What is the commutative analogue of a $C^*$-subalgebra?
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...
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119 views
Noncommutative algebraic geometry in the case of skew fields
Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry.
But one might wonder whether things are any better in the case of ...
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2answers
435 views
Why are Hopf algebras called quantum groups?
Why are noncommutative nonassociative Hopf algebras called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
