All Questions
Tagged with noncommutative-geometry rt.representation-theory
25 questions
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
5
votes
0
answers
288
views
Representation functor on modules
Let $k$ be a field and $A$ a unital associative $k$-algebra.
The representation functor associates, to each object in non-commutative geometry, a genuine geometric object on the representation variety ...
- 985
2
votes
0
answers
66
views
Noncommutative transcendence degree of representation algebras
Let $G$ be a reductive group, for example $\text{GL}_n(\mathbb{C})$. Let $V$ denote its defining representation, and let $R$ denote the tensor algebra on the irreducible representations of $G$. It may ...
- 2,907
1
vote
0
answers
82
views
How to define explicitly the Kasparov product $ x \otimes_B y \in KK_{i+j}^G (A,C) $ of $x \in KK_i^G (A,B)$ and, $y \in KK_j^G (B,C)$?
Let $A,B,C$ be separable $G-C^*$ - algebras. Then there is a biadditive pairing for $i,j \in \mathbb{Z}_2$,
$$ KK_i^G (A,B) \times KK_j^G (B,C) \to KK_{i+j}^G (A,C) $$
If $x \in KK_i^G (A,B)$ and, $y \...
- 595
8
votes
2
answers
208
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 ...
- 293
2
votes
1
answer
287
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 $...
- 541
2
votes
0
answers
147
views
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 ...
- 595
4
votes
0
answers
127
views
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 $...
- 661
3
votes
1
answer
446
views
A set of objects classically generates the full subcategory of compact objects iff it generates the whole category
Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
- 224
4
votes
1
answer
243
views
number of indecomposable summands of an extension of two modules
I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
- 333
7
votes
2
answers
521
views
Kazhdan constant and finite index subgroups
I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?
Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...
- 165
3
votes
0
answers
39
views
Second order signature operator in diffeomorphism invariant geometry as an image under right regular representation
I would like to understand the following statement taken from this paper, dealing with the so called Transverse Index Theory or in other words with the index theory for diffeomorphism invariant ...
- 9,330
13
votes
1
answer
669
views
Is a "smooth" finite-dimensional algebra separable modulo its radical?
Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
- 5,407
10
votes
3
answers
508
views
Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...
- 288
10
votes
2
answers
833
views
Update on list of open problems for Cherednik/Symplectic Reflection Algebras
Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
- 1,066
3
votes
0
answers
324
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", ...
32
votes
4
answers
9k
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 ...
- 24.8k
2
votes
1
answer
534
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 $K^0_G(...
- 9,009
13
votes
4
answers
2k
views
Geometric interpretation of Universal enveloping algebras
Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How ...
- 13.2k
9
votes
4
answers
3k
views
Gluing perverse sheaves?
It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $...
- 5,505
12
votes
3
answers
1k
views
construct scheme from quivers?
I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
- 1,305
14
votes
3
answers
2k
views
What are the motivations for studying Cherednik (symplectic reflection, graded Hecke) algebras?
Several times I have come across these algebras and I wonder why any of these are interesting; I'm very sure they are, but I could not find an answer in the literature.
For example (the very general ...
- 5,233
3
votes
1
answer
547
views
Is D-module on flag variety of Lie algebra a scheme?
This question was motivated by the answers in D-module as quasi coherent sheaves on deRham stack. What I am interested in is the case of D-module on flag variety of Lie algebra. So,in this case, if we ...
- 5,505
8
votes
3
answers
1k
views
Is there an analogue Beilinson-Bernstein localization for quantized enveloping algebra
I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for &...
- 81
7
votes
2
answers
921
views
What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra
If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of ...
- 5,505