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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Moduli spaces of stable sheaves on noncommutative projective schemes

In noncommutative algebraic geometry in the sense of Artin and Zhang, can we construct moduli spaces of stable sheaves on noncommutative projective schemes as (commutative)schemes ? I would appreciate ...
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D-module theoretic Chern characters and an index-type theorem

Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an ...
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A foliation with prescribed graph of foliation

**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation ** Definition of the graph of a ...
Ali Taghavi's user avatar
7 votes
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Strange formula for the dimension of a certain space of noncommutative polynomials

Consider a vector space $V_r(n)$ spanned by (noncommutative) monomials in variables $x_1,\ldots,x_r$ $$ x_{1}^{n_1}x_{2}^{n_2}\ldots x_{r}^{n_r} $$ of total degree $n.$ Inside this space consider a ...
Daniil Rudenko's user avatar
4 votes
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A Fréchet space characterization of smooth structures on topological spaces?

For a compact manifold $M$ the space of smooth functions $C^{\infty}(M)$ is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to ...
Sven Mortenson's user avatar
2 votes
1 answer
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Gelfand-Naimark and Peter-Weyl for the unitary group

Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
Jake Wetlock's user avatar
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Hermitian vector bundles and Hilbert $C^*$-modules

Let $X$ be a compact Hausdorff space and $C(X)$ its algebra of continuous complex valued functions. The Gelfand-Naimark theorem tells us that we have a duality between commutative $C^*$-algebras and ...
Jake Wetlock's user avatar
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Integral lattice in noncommutative Hodge theory

Associated to a $DG_{\mathbb{C}}$-category, $\mathcal{C}$, we have some Hodge theoretic data - $HH_{*}(\mathcal{C})$ plays the role of Hodge cohomology and $HP$ plays the role of de Rham cohomology. ...
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What is the significance of the Jiang Su algebra in classification of C$^*$ -algebras?

Something I've been thinking about for a while that I'm not sure I understand is why $\mathcal{Z}$ stability, as opposed to say $\mathcal{O}_\infty$-stability or even $\mathcal{K}$-stability is so ...
Owen Tanner's user avatar
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About the algebraic structure of the $G$-equivariant $KK$-theory

Let $ G $ be a second countable locally compact group. Let $ A $ and $ B $ be two $G$-$C^*$-algebras. Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $. Could you tell me ...
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Kernels of completely positive maps

In the excellent book “$C^*$-algebras and their automorphism groups” by Pedersen there are results on the left ideals $L_\phi$ associated to states $\phi$ on a $C^*$-algebra $A$ and more results in ...
Edwin Beggs's user avatar
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Non-commutative complex geometry

I was reading a physics paper where it was mentioned that the basic framework of Connes' differential non-commutative geometry (or actually, a slight modification of Connes in that paper) would need ...
Hollis Williams's user avatar
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Nonstandard Podles spheres as $U_c(\frak{h})$ invariants

In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
Jake Wetlock's user avatar
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What are the topics in noncommutative algebraic geometry?

Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
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Non commutative Teichmuller theory

Perhaps the first example in Teichmuller theory is the following proposition: Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \...
Ali Taghavi's user avatar
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Associating noncommutative geometries to 2D conformal field theories

I have recently been reading a bit about noncommutative geometry and string theory and it looked to be an open question (or at least this was open two decades ago) whether there are constructions ...
Hollis Williams's user avatar
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"Non-critical" zeros of $\zeta$ and the $\zeta$-cycles of Connes and Consani

In the recent preprint of Connes and Consani https://arxiv.org/abs/2106.01715 a new spectral realization of the critical zeros of $\zeta$ (edit: defined as being those on the critical line only, see ...
Archie's user avatar
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Fermions, their path integrals and effective actions

I just read the nice exposition Fermionic Path Integral on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the ...
iolo's user avatar
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A Poisson structure induced by double Poisson bracket

$\DeclareMathOperator\Sym{Sym}$Let $k$ be a field of characteristic zero. In Van den Bergh's paper Double Poisson algebras, it is shown that a double Poisson bracket on an unital associative algebra $...
Yining Zhang's user avatar
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Quivers as noncommutative curves

I've heard that an idea behind noncommutative geometry (in dim 1) is to study "noncommutative" analogues of $\text{Coh}(\text{curve})$, rather than the curve directly. Apparently the ...
Pulcinella's user avatar
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Research in spin geometry

I am currently learning differential geometry, but I have heard about the field of spin geometry and have skimmed through the book Dirac Operators in Riemannian Geometry by Thomas Friedrich. I have ...
Daniel Waters's user avatar
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Projective planes over non-division rings

Is there a "right" notion of a projective plane over a general (unital, non-division) ring? Let me explain what type of object I am looking for. Let $R$ be an arbitrary (not necessarily ...
Anton Izosimov's user avatar
15 votes
1 answer
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Reference for the Swan-Serre theorem as a monoidal equivalence

Let $X$ be a compact Hausdorff The well-known Swan--Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$...
Boris Henriques's user avatar
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Hopf "algebroid" structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer. To make ...
Bumblebee's user avatar
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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 ...
dohmatob's user avatar
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Antisymmetric tensor coordinates and tensorial spaces

I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates $X^{\mu\nu}=-X^{\nu\mu}$, with $\mu,\nu=1,...,N$, which have been ...
Gian's user avatar
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1 answer
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Invertible elements of the Hopf algebra quantum $SU(2)$

Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see https://en.wikipedia.org/wiki/Compact_quantum_group (Note that on the ...
Jake Wetlock's user avatar
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Construction of non-commutative torus using ergodic action of $\mathbb{T}^{n}$

It is well known that non-commutative torus can be constructed using universal C* algebra, by n unitary elements and twisted relations. It can also be constructed using ergodic action of torus group $\...
Ken.Wong's user avatar
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Dense subalgebra that is closed under unbounded derivation on noncommutative torus

Let $A_{\theta}$ be the noncommutative torus, we can define: $$ A^{\infty}_{\theta}\mathrel{:=}\left\{\sum_{n,m\in\mathbb{Z}}a_{n,m}U^{n}V^{m} \,\middle\vert\, a_{n,m}\in S(\mathbb{Z}^{2})\right\} $$ ...
Ken.Wong's user avatar
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1 answer
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On the stabilizer in $\mathrm{GL}(2,\mathbb{Z})$ of a real quadratic irrationality

$\DeclareMathOperator\GL{GL}$Let $\theta$ be a real quadratic irrationality with discriminant $\Delta$; let $\mathcal{O}_\Delta$ denote the resulting quadratic order of discriminant $\Delta$ in $\...
Branimir Ćaćić's user avatar
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 ...
Andrew NC's user avatar
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Why Der($A_{\theta}$) is spanned by two elements only?

In the work of Connes and Marcolli, on page 20, it state that: Just as in the classical case of a (commutative) manifold, what ensures that the derivations considered are enough to span the whole ...
Ken.Wong's user avatar
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Symmetry group for (noncommutative) manifold from spectral triple

(This post is cross-post in Mathematics Stack Exchanges https://math.stackexchange.com/questions/3992766/symmetry-group-for-noncommutative-manifold) Is there any notion of symmetry group arise for ...
Ken.Wong's user avatar
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8 votes
2 answers
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tangent bundle on noncommutative manifold

Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold $(A,H,D)$, by replacing vector bundle by finitely generated projectve module $M$. For the construction of ...
Ken.Wong's user avatar
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What is the precise relationship between real Poisson algebras and commutative $C^*$ algebras?

I've been teaching myself quantum mechanics, and I realized that I'm missing something fundamental. Namely, there are two pictures that I don't know how to reconcile: Quantum Mechanics generalizes ...
Andrew NC's user avatar
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7 votes
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$C^*$ algebras whose nontrivial projections form a non empty compact connected set

Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that ...
Ali Taghavi's user avatar
3 votes
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The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras

In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem $$H_{\text{CE}}...
Lukas Miaskiwskyi's user avatar
2 votes
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A quantity associated to a foliated manifold and its non-commutative interpretation

Let $M$ be a compact $n$-dimensional manifold. Assume that $F$ is a $k$-dimensional foliation of $M$. The graph $G(M,F)$ of this foliation is a $(n+k)$-dimensional manifold. We recall its definition: ...
Ali Taghavi's user avatar
1 vote
0 answers
99 views

Classification of all groupoids $G$ whose automorphism group is in bijective correspondence the automorphism group of $C^*_\text{red}(G)$

Is there a terminology (and a classification) for all groupoids $G$ for which all automorphisms of $C^*_\text{red}G$ are induced from a groupoid automorphism of $G$. (A groupoid automorphism has ...
Ali Taghavi's user avatar
3 votes
0 answers
165 views

"Somewhat connected" spaces or algebras

Before we state our question, we give a motivational simple example: Put $X$ for disjoint union of two circles. However $X$ is not a connected space but it has an open dense subset $U$ such that $U$ ...
Ali Taghavi's user avatar
8 votes
0 answers
521 views

Why is Hochschild homology interesting if its cohomology groups are infinite-dimensional?

I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) ...
Jake Wetlock's user avatar
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1 vote
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421 views

Fredholm $C^*$-algebras

Let $H$ be a Hilbert space. A vector subspace $W\subset B(H)$ is called a Fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operators $T$ in $W$. ...
Ali Taghavi's user avatar
2 votes
1 answer
141 views

Hilbert module over $C_0(\Lambda)$ as space of continuous sections of HIlbert bundle

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 ...
Tan1278's user avatar
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3 votes
1 answer
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Reference for "the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner"

I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded ...
dohmatob's user avatar
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5 votes
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Riemannian version of topological $K$-theory

Let $X$ be a compact Hausdorff space.Put $Vec(X)$, the space of all real (or complex) vector bundles over $X$.We put also $Vec_g(X)$, the space of all Riemannian vector bundles over $X$, that is the ...
Ali Taghavi's user avatar
9 votes
0 answers
358 views

Bernoulli-like polynomials

Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then $$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$ $$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$ where $B_n$ is a monic polynomial of degree $n.$ Now ...
Boris Tsygan's user avatar
11 votes
2 answers
1k views

Good reference for topological Hochschild homology

I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC). I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
Sunny's user avatar
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7 votes
3 answers
615 views

Noncommutative torus as a von Neumann algebra

Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
truebaran's user avatar
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6 votes
2 answers
383 views

Survey of recent developments of the Gelfand-Kirillov dimension

It is almost two decades since the now classical books by McConnell and Robinson's [ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
jg1896's user avatar
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3 votes
1 answer
361 views

A set of objects classically generates the full subcategory of compact objects iff it generates the whole category

Sorry in advance if my question doesn't have the level of this community. I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
T. Wildwolf's user avatar

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