Tagged Questions

1
vote
0answers
36 views

Bicovariant Calculi on the Quantum Unitary Groups

The bicovariant differential calculi on quantum-$SU(n)$ have been classified (by Schmudgen I think) and have been shown to have non-classical dimension. My question is whether or n …
5
votes
1answer
206 views

Noncommutative smooth manifolds

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth struct …
5
votes
0answers
264 views

Functorial point of view of spectrum (Looking for reference)

I think I should elaborate a bit. What I am asking is the definition of spectrum of a category as a stack in functor view of points. In noncommutative algebraic geometry. We defin …
2
votes
0answers
69 views

Obstructions to existence of finitely summable spectral triples

Connes proved in his beautiful paper "Compact metric spaces, Fredholm modules, and hyperfiniteness" published in 1981 that if $(A,H,D)$ is a finitely summable spectral triple with …
12
votes
2answers
322 views

How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)

What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to cor …
18
votes
7answers
1k views

Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)

In this question, Harry Gindi states: The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence. Moreover, in th …
18
votes
6answers
429 views

Applications of Noncommutative Geometry

This is related to Anweshi's question about theories of noncommutative geometry. Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative …
12
votes
5answers
567 views

“Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on …
7
votes
1answer
181 views

What properties “should” spectrum of noncommutative ring have?

There are already a lot of discussion about the motivation for prime spectrum of commutative ring. In my perspective(highly non original), there are following reasons for the impor …
30
votes
9answers
1k views

Non-commutative algebraic geometry

Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutativ …
7
votes
2answers
158 views

Is the group von Neumann algebra construction functorial?

Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $ …
10
votes
3answers
433 views

The ‘real’ use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics

In this question, Orbicular made the following comment to Feb7 and my own answers; Please keep in mind that - even though it is stated very often - noncommutative geometry does …
16
votes
6answers
826 views

Theories of Noncommutative Geometry

[I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community. I have also voted to reopen it. -- PLC] …
4
votes
3answers
142 views

Existence of non-commutative desingularizations

Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as …
2
votes
1answer
94 views

How is this action of monoidal derived category induced?

I am reading a paper concerning the action of monoidal category to another category. Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=R\bigotimes _{k}R^{ …

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