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0
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0answers
18 views

Quick question about conjugate equivalence bimodules and inner products

let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...
1
vote
1answer
77 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
5
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0answers
120 views
+50

Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...
2
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0answers
53 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
0
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0answers
65 views

Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
3
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0answers
214 views

Van den Bergh Duality, Serre Daulity and Poincaré duality [closed]

All three duality theorems: Van den Bergh Duality, Serre Duality and Poincaré duality seem to be very similar, is there an explicit relationship between the three? For example can van den Bergh ...
0
votes
1answer
85 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...
3
votes
1answer
194 views

Why should noncommutative CYs be dgas?

For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms $$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$ in the ...
3
votes
0answers
78 views

Noncommutative Poincare inequalities

This is a question on how (or if) people in the community think about the Poincare inequality in noncommutative geometry. In geometry, the Poincare inequality (when it exists) gives a bound on a ...
13
votes
2answers
490 views

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 ...
7
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0answers
185 views

Non Commutative Hyperspaces

Let $X$ be a compact metric space. Recall that the hyperspace $2^{X}$ is the set of all non empty compact subsets of $X$ with the Hausdorff metric. Assume that $\mathcal{C}$ is the category of all ...
5
votes
1answer
196 views

Morita equivalence for operator algebras and tensor products, question about proof

This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...
1
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0answers
75 views

Non Commutative analogues of a commutative fact

What is a relevant non commutative analogues for the following fact, in term of spectral triples and cyclic cohomology?: "If $M$ is a compact oriantable manifold without boundary and $X\subset M$ is ...
3
votes
0answers
162 views

2 - Calabi Yau algebras and bimodule coherence

Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero. In H. Minamoto "Ampleness of two-sided tilting complexes", ...
1
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1answer
210 views

Computing noncommutative geometries

I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...
5
votes
1answer
218 views

A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version. In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...
0
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0answers
60 views

Problem with Smoothness and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
1
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0answers
103 views

Non commutative analogy of compact-open topology

Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase: For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$. We can ...
0
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0answers
82 views

On non-unital ring and algebraic geometry

When I learned abstract algebra many years ago,I noticed the author deals with commutative ring say,$A$ has the proposition:$A^2=A$(without assuming it has identity).It seems that many proposition of ...
1
vote
0answers
40 views

When is the dual module isomorphic to conjugate module of a *-algebra

Let $A$ be a $\ast$-algebra, and let ${\cal M}$ be a module over $A$. As usual, we denote the dual module of $M$ by $M^*$. Moreover, we denote the conjugate module of $M$ by $\overline{M}$, where as ...
6
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0answers
219 views

Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves. Recall that $\mathbf{D}(X)$ determines the Grothendieck ...
1
vote
1answer
166 views

algebra-geometry duality

For topological spaces $S$ and $T$, denote by $C(S)$ and $C(T)$ the corresponding algebras of continuous real-valued functions. What are the necessary conditions that we need to impose on $S$ and $T$ ...
4
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0answers
121 views

“Definitive” Noncommutative Space

Let $Y$ be a (locally compact) non-Hausdorff topological space. I want to know if there is a necessary and/or sufficient condition for $Y=X/G$, that is, $Y$ is the orbit space of a locally compact ...
6
votes
2answers
211 views

Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one

A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and ...
2
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0answers
50 views

Construct the Haar Functional using $R$-Matrices

Let $H$ a cosemi-simple coquasi-triangular Hopf algebra, arising from an $R$-matrix using the standard FRT construction. Semi-simplicity implies the existence of a Haar function. Is there any way in ...
3
votes
1answer
147 views

example of a compact quantum group at a root of unity?

In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some ...
14
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3answers
966 views

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 $H$ ...
4
votes
1answer
197 views

Noncommutative Erlangen Program

Has Klein's "Erlangen Program" been generalized/extended to the noncommutative setting (say, à la Connes)? Is there a classification of "noncommutative klein geometries" at least in very low ...
18
votes
1answer
1k views

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 ...
4
votes
1answer
193 views

Do we have a “topological assembly map” in the Baum-Connes conjecture?

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \text{a-ind}: ...
2
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1answer
351 views

C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...
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3answers
407 views

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 ...
4
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0answers
178 views

Non-commutative Formal Group Laws

Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of ...
2
votes
2answers
415 views

Possible directions in noncommutative geometry

I recently came across noncommutative geometry and found it rather interesting. I should mention that I'm a graduate student considering options for my research and if I were to name an area which I'm ...
1
vote
1answer
799 views

PhD in operator algebras and non-commutative geometry [closed]

I do not know whether it is a good place to ask this question or not. I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
7
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0answers
209 views

Derived (non-commutative) geometry, geometric constructions in explicit form

I'm interested in the following construction. Start with derived category of coherent sheaves witch equivalent to derived category of representations of some dg-algebra. Quasi-isomorphic dg-algebras ...
4
votes
2answers
384 views

$q$-Deforming Woronowicz's Leibniz Rule

The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and $$ \text{d}:A \to \Omega, $$ is a bimodule map, ...
8
votes
1answer
547 views

What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
4
votes
1answer
79 views

Example of computation of moduli space of $n$-pointe modules?

I am looking for an example of computation of the isomorphism classes of $n$-point modules over a non-commutative generated graded algebra (assuming all good properties such as Noetherian property). ...
3
votes
1answer
335 views

Geometry and quantization

I know that lots of effort is being put into quantization of geometry(NCG).This effort of course comes with the idea of operator algebra being a powerful machinery. Has any effort been given in the ...
18
votes
4answers
2k views

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 ...
8
votes
1answer
396 views

Eigenvalues of the free sphere

Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x_1,\ldots,x_n:\mathbb R^n\to\mathbb R$, and in fact we have the ...
3
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0answers
132 views

AS Cohen Macaulay algebras and dualizing complexes

Let $A$ be an $\mathbb N$-graded algebra such that $A_0 = k$ is a field. This are usually called graded connected algebras. One can define a torsion functor with respect to the ideal $\mathfrak m = ...
4
votes
0answers
414 views

Open question: non-commutative site following Grothendieck, Quillen, Connes and Crane for quantum gravity.

This is an open question and it's to find out who is interested in this kind of thing, who can benefit from thinking about this. It is very brief but hopefully will only be unclear to people who are ...
2
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0answers
76 views

Fat modules on some algebras.

Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely ...
2
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0answers
52 views

Boundedness of modules on AS regular algebras

Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules ...
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0answers
51 views

Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$?

Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity ...
3
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0answers
93 views

Balanced dualizing complex vs rigid dualizing complex?

In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing ...
2
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1answer
253 views

Kasparov's Dirac element and the index map

In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology ...
8
votes
1answer
224 views

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 ...