Questions tagged [noncommutative-geometry]
Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
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"The" kronecker foliation or "a" kronecker foliation?
Consider the following two foliations of torus:
1)The Kronecker foliation with slope $\sqrt{2}$
2)The Kronecker foliation with slope $\pi$
As I learn from the literature, these two foliations are ...
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votes
1
answer
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Commutative spectral triples
The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these categories....
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9
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What is the significance of non-commutative geometry in mathematics?
This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more ...
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2
answers
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How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)
What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
...
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33
votes
3
answers
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Reference request for translating from Top to C*-alg
Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
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3
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Noncommutative smooth manifolds
Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples.
Using his definition it is unclear how to separate the smooth structure from the metric.
...
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4
answers
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What's a noncommutative set?
This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...
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If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...
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4
votes
0
answers
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Quantization of $S^2$ as $C^*$-algebra?
The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...
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Non-commutative algebraic geometry
Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which ...
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Theories of Noncommutative Geometry
[I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community. I have also voted to reopen it. -- PLC]
There are many ...
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Is there a quaternionic algebraic geometry ?
Let $\mathbb{H}$ be the skew-field of quaternions. I'm aware of the
Theorem 1. A function $f:\mathbb{H}\to\mathbb{H}$ which is $\mathbb{H}$-differentiable on the left (i.e. the usual limit $h^{-1}\...
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6
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Why Drinfel'd-Jimbo-type quantum groups?
Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
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Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc]
About 20 years ago I read in textbook that
"all irreducible representations of compact groups are finite-dimensional", but
me and the proof of this fact never met each other :)
May I ask dear MO ...
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31
votes
1
answer
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What is about nonassociative geometry?
At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed ...
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3
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Is the group von Neumann algebra construction functorial?
Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
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5
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Non-commutative geometry from von Neumann algebras?
The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^*$-algebras with unital $*$-homomorphisms to the category of compact Hausdorff spaces with ...
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2
answers
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Why is "naive" definition of non-commutative spectrum bad?
It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative setting....
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1
answer
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Convolution algebras for double groupoids?
There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
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14
votes
2
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Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
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14
votes
2
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Relationship between Hochschild cohomology and Drinfeld centers
Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$.
I was reading nlab's entry on Hochschild cohomology ...
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9
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Examples of noncommutative analogs outside operator algebras?
Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:
A $C^\ast$-algebra is a ...
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13
votes
6
answers
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Hopf algebras arising as Group Algebras
Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
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1
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Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
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3
answers
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Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
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12
votes
2
answers
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Mysterious quotes (at least for me)
I heard two quotes, one from Alain Connes and an other one from Orlov.
Alain Connes was talking about noncommutative geometry and he said the following:
" a noncommutative algebra creates its own ...
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11
votes
1
answer
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Current status of the linearity of mapping class group
In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
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11
votes
0
answers
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Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
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3
answers
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The "right" $C^*$ algebraic proof of Bott Periodicity
In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity:
$\bullet$ An argument based on Moyal quantization found in "Elements of Noncommutative ...
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10
votes
2
answers
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Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
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9
votes
1
answer
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Point modules of quantum projective space $\mathbb{P}^n$
Let $A$ be a quantum $\mathbb{P}^n$ defined by
$$
A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}.
$$
I would like to know the set $X$ of isomorphism ...
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0
answers
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Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
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4
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Kontsevich, and Geometric, Quantization and the Podles sphere
There exist a large family of noncommutative spaces that arise from the quantum matrices. These algebraic objects $q$-deform the coordinate rings of certain varieties. For example, take quantum $SU(2)$...
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votes
0
answers
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Noncommutative geometry and line length
I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...
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votes
1
answer
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$*$-algebras, completions, and $K$-theory
What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
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6
votes
1
answer
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Are all transversely oriented, transversely measured foliations given by closed forms?
This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other ...
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5
votes
1
answer
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Dirac operator on a Morita equivalent algebra
Let $(A,H,D)$ be a spectral triple and let $B$ be an algebra which is Morita equivalent to $A$. Then there exists a finitely generated, projective $A$ module $E$ such that $B=End_A(E)$. Endow $E$ with ...
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votes
1
answer
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Homotopic morphisms between curved A-infinity algebras
I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...
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votes
0
answers
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Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra
Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...
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0
answers
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
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votes
1
answer
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On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra
Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.
Obviously the singularities of this systems are just the idempotents of the ...
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4
votes
1
answer
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Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma
I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
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4
votes
0
answers
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Spectrum of the hypoelliptic transverse signature operator
Let $D$ be the transverse signature operator constructed by Connes and Moscovici in the paper "Local index formula in Noncommutative Geometry":this is first order hypoelliptic pseudodifferential ...
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4
votes
1
answer
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A Question on Koszul duality and $B(\infty)$ structures on $HH^*$
The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type.
There is an equivalence of Gerstenhaber ...
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votes
1
answer
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A question on $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions:
All elements of $A$ are left zero divisor.
All elements are right zero divisor.
All elements are ...
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4
votes
1
answer
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Reference request for instantons
I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
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votes
0
answers
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A topological space extracting from a group action
Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to \mathbb{C}$...
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votes
2
answers
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Reference for Connes Bourbaki membership or otherwise
Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?
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votes
0
answers
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Relative version of Hopf cyclic cohomology
In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...
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votes
1
answer
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Normalization of cyclic cocycles
This question is a continuation of the discussion
Normalization of Hochschild cocycles
but this time in the cyclic context. I would like to ask whether the following is true:
The inclusion of ...
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