Questions tagged [noncommutative-geometry]
Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
459
questions
6
votes
1
answer
382
views
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 ...
11
votes
0
answers
432
views
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 ...
3
votes
0
answers
100
views
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 ...
4
votes
1
answer
138
views
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 ...
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 ...
3
votes
1
answer
200
views
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&...
0
votes
0
answers
130
views
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 ...
9
votes
1
answer
216
views
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}(\...
6
votes
2
answers
1k
views
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 ...
6
votes
0
answers
248
views
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(...
1
vote
0
answers
133
views
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}$...
6
votes
0
answers
490
views
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 ...
2
votes
0
answers
93
views
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 ...
1
vote
0
answers
95
views
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 ...
5
votes
0
answers
276
views
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. ...
3
votes
0
answers
201
views
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 ...
1
vote
1
answer
323
views
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 ...
5
votes
0
answers
128
views
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 \...
9
votes
1
answer
354
views
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,...
3
votes
1
answer
144
views
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 ...
2
votes
1
answer
249
views
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 ...
3
votes
0
answers
111
views
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 ...
1
vote
0
answers
100
views
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. ...
5
votes
0
answers
155
views
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 ...
-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?
...
1
vote
0
answers
98
views
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))$, ...
3
votes
0
answers
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 $\...
3
votes
0
answers
123
views
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, ...
5
votes
0
answers
145
views
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}^\...
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 ...
1
vote
0
answers
88
views
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 ...
10
votes
0
answers
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{...
3
votes
0
answers
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{...
2
votes
0
answers
191
views
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]=...
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 ...
2
votes
0
answers
90
views
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}\}$$
What is an ...
0
votes
0
answers
219
views
Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)
I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows:
$$\hat{X}^{r}=\hat{x}-i(r-...
2
votes
1
answer
192
views
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 ...
2
votes
0
answers
86
views
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 $...
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 $...
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^* ...
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 ...
2
votes
0
answers
83
views
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, \...
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 ...
1
vote
0
answers
102
views
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 ...
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 ...
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 $...
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 ...
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 ...
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 ...