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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On the definition of the Cherednik algebra of a variety with a finite group action

Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
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Convergence in Hausdorff distance of intersection of closed linear subspaces with a given closed convex set

I've run into the following problem when doing some work with non-commutative metric spaces, which seems like something people may have thought about before but I can't find anything on this problem ...
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Quasi-coherent cohomology in non-commutative algebraic geometry

In non-commutative algebraic geometry, the motto so to speak is to replace the study of a scheme $X$ with the study of the category $D_{qcoh}(X)$ of quasi-coherent sheaves and study the properties ...
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Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 "...
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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 \...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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15 votes
1 answer
610 views

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 $C(X)$...
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1 answer
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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 ...
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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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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 ...
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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 ...
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2 votes
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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 $\...
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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: $$ A^{\infty}_{\theta}\mathrel{:=}\left\{\sum_{n,m\in\mathbb{Z}}a_{n,m}U^{n}V^{m} \,\middle\vert\, a_{n,m}\in S(\mathbb{Z}^{2})\right\} $$ ...
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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 $\...
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8 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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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 ...
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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 ...
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8 votes
2 answers
354 views

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 ...
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1 answer
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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 ...
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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 ...
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3 votes
0 answers
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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}}...
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2 votes
0 answers
99 views

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: ...
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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 ...
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3 votes
0 answers
160 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 $U$ ...
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8 votes
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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) ...
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1 vote
2 answers
362 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 $W$. ...
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2 votes
1 answer
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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 ...
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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 ...
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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 ...
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9 votes
0 answers
322 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 ...
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