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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Finite groups and noncommutative algebraic geometry

DISCLAIMER: My relationship with noncommutative algebraic geometry is that of a curious, ignorant bystander. I confess that I know very little about noncommutative algebraic geometry, but I am ...
semisimpleton's user avatar
11 votes
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Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry. For graphs this had been an open ...
David Roberson's user avatar
3 votes
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Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
Qwert Otto's user avatar
4 votes
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Universal property of the category of quasicoherent sheaves of a blowup

We know that if $Z \rightarrow X$ is a closed subscheme of X of ideal $\mathcal{I}$, then if $\pi : Bl_Z X \rightarrow X$ is the projection, $\pi^* \mathcal{I}$ is invertible. Does the category of ...
Nikola Tomić's user avatar
6 votes
1 answer
297 views

How are coordinate charts constructed in noncommutative geometry?

In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
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Derivations in Connes' noncommutative geometry

In Connes' noncommutative geometry, the starting point is a spectral triple $(A,D,H)$ where $A$ is a commutative C* algebra, e.g. as in Connes "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS&...
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Dependence of functional integral on the function space

In physics, the following functional integral is considered \begin{gather} Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf )) \end{gather} It is usually said that the integration is performed ...
0x11111's user avatar
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Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
Qwert Otto's user avatar
6 votes
2 answers
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Is there a notion of point in noncommutative geometry?

It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
Esmond's user avatar
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Lie algebra cohomology of the space of vector fields

For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
Qwert Otto's user avatar
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There exists noncommutative geometric invariant theory?

In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...
jg1896's user avatar
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Proof of a result by Zhang in Artin's seminal paper

In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function ...
jg1896's user avatar
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Automorphism group of the first Weyl field

A related question is this one (Automorphism group of the quantum Weyl field). Let $A_1$ denote the rank 1 Weyl algebra (over the complex numbers), and $D_1$ its skew field of fractions, called the ...
jg1896's user avatar
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Literature for noncommutative birational invariants

Let $k$ be an algebraically closed field of zero characteristic. All fields under discussion are fields over $k$, and all division rings are division algebras over $k$. There is rich theory of ...
jg1896's user avatar
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Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
Qwert Otto's user avatar
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Arithmetic derivatives and non-commutative generalizations

In the theory of arithmetic derivatives, in the simplest case an arithmetic derivative on $\mathbb{N}$ is defined via the rule $(a \times b)'= a \times b' + a' \times b$, mirroring the product rule ...
Ilk's user avatar
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Lattices and noncommutative algebras in noncommutative geometry

This a question that I've asked in mathematics stack exchange without having received any response : I am interested in the relation between lattices and noncommutative algebras in the context of ...
Esmond's user avatar
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Confusion around a (necklace) cobracket in Ginzburg's article Calabi-Yau Algebras

Something has been puzzling me for quite a while in Ginzburg's article Calabi-Yau Algebras. At some point he considers the free graded algebra $\mathbb{C}\langle x_1, \dots, x_n, \theta_1, \dots \...
Vik S.'s user avatar
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1 answer
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Hochschild cohomology of a group algebra

Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
Qwert Otto's user avatar
3 votes
1 answer
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Is a compact set of extreme points contained in a compact face?

I have run into the following question in convex analysis, which I haven't found answered in the literature: Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
Sean's user avatar
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1 answer
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Gluing data for modules over a ring with idempotents

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
Sergey Guminov's user avatar
3 votes
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Braided monoidal categories as generalized "braided" schemes

It's well know by the Gabriel-Rosenberg reconstruction theorem that a (quasi-separated) scheme $X$ is completely determined by its category of quasicoherent sheaves $\mathbf{QCoh}(X)$. The latter is ...
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Poisson closure of singular support of a $\mathfrak{g}$ module

This is a question about when two Poisson subvarieties of the dual of a finite dimensional complex Lie algebra agree. I begin with some definitions, which are essentially standard. Singular Support. ...
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Is the wrapped Fukaya category a symplectomorphism invariant?

Say, let $\phi\colon W_1\to W_2$ be a symplectomorphism of Weinstein manifolds(or with stronger assumption that $W_1$ is Liouville homotopic equivalent to $W_2$, but with non-compact support), do they ...
TheWildCat's user avatar
-3 votes
1 answer
319 views

Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor. They form a category with usual structures. Question. Is this category equivalent to the category of $C^*$ algebras? ...
Ali Taghavi's user avatar
1 vote
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Formula for the KK-theory groups $KK(A, C(S))$

I am studying $C^*$-algebras and their KK-theory. Let $A$ be a (unital if you wish) $C^*$-algebra and $S$ be a compact Hausdorff space. I am interested in computing the KK-theory groups $KK(A, C(S))$, ...
Luiz Felipe Garcia's user avatar
3 votes
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154 views

What's the purpose of the operator $(\Delta^{-1}+\lambda)^{-1}$ in Tomita-Takesaki modular theory?

I was reading Tomita-Takesaki modular theory (from all the books, and articles), the goal is to relate a von Neumann algebra $\mathcal{A}$ with its commutant $\mathcal{A}'$ on a Hilbert space $\...
MrPajeet's user avatar
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Relationship between the Penrose groupoid algebra and the group algebra of the symmetry group of a Penrose tiling

Given a Penrose tiling, there are two C*-algebras associated with it: the $\mathcal{O}_\text{Penrose}$ algebra (Penrose groupoid algebra) and the group algebra of the symmetry group of the tiling, ...
Mirco A. Mannucci's user avatar
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Questions about the $K$-theory of the algebraic standard Podleś sphere

Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
Branimir Ćaćić's user avatar
4 votes
0 answers
96 views

Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
Josh Lackman's user avatar
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A question on Stable rank 1

My apology in advance if my question is elementary According to the initial definition of topological stable rank introduced by Marc Rieffel we have the following: An algebra has tsr 1 if the space ...
Ali Taghavi's user avatar
10 votes
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786 views

Noncommutative condensed sets

Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories \begin{align*} \mathrm{CHaus}^{\mathrm{...
Luiz Felipe Garcia's user avatar
3 votes
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162 views

Hochschild homology of stable categories as topological chiral homology

Sorry for repost from Math Stack Exchange: Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$. Its Ind-completion $\mathscr{...
Chris Kuo's user avatar
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The trigonometric $C^*$-algebra

The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
Ali Taghavi's user avatar
6 votes
1 answer
151 views

Every element of a $W^*$-algebra is a linear combination of exponential unitaries?

I am trying to understand a proof in this paper (specifically theorem 5.4). In it, a fact is used that every element of the $W^*$-algebra $A$ is a linear combination of exponential unitaries. I've ...
Ashley Shade's user avatar
2 votes
0 answers
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The group of quasi unitary elements of a (simple) Banach algebra

For a Banach algebra $A$ with invertible group $G(A)$ we define the following group: $$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$ What is an ...
Ali Taghavi's user avatar
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0 answers
219 views

Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)

I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows: $$\hat{X}^{r}=\hat{x}-i(r-...
Hasib's user avatar
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1 answer
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Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
truebaran's user avatar
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2 votes
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Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents

The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $...
Ali Taghavi's user avatar
4 votes
1 answer
131 views

A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
Ali Taghavi's user avatar
2 votes
0 answers
159 views

Simple modules of quantum planes

Let $k$ be an algebraically closed field. Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$. We often call $R$ a quantum plane. If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...
Walterfield's user avatar
8 votes
2 answers
182 views

Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras

There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
Motmot's user avatar
  • 293
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The generators of twisted homogeneous coordinate rings

Let $X$ be a projective scheme over an algebraically closed field $k$ of characteristic $0$. Let $\sigma$ is an automorphism of $X$ and $\mathcal{L}$ be an invertible sheaf on $X$. Let $B := B(X, \...
Walterfield's user avatar
8 votes
0 answers
454 views

Analytification of DG-categories over $\mathbb C$?

In recent notes of complex geometry by Clausen–Scholze, they gave a theory of analytification of finite type $\mathbb C$-schemes. It seems to me that there is a non-commutative analogue which works ...
Z. M's user avatar
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A locally convex $C^*$ algebraic structure on the disk algebra

A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
Ali Taghavi's user avatar
4 votes
0 answers
105 views

KK-theory for commutative $C^*$-algebras

The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative ...
Jake Wetlock's user avatar
  • 1,114
2 votes
1 answer
278 views

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 $...
Fernando Peña Vázquez's user avatar
2 votes
0 answers
204 views

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 ...
Sean's user avatar
  • 135
5 votes
1 answer
279 views

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 ...
curious math guy's user avatar
4 votes
0 answers
88 views

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 ...
user498029's user avatar

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