All Questions
Tagged with cohomology rt.representation-theory
33
questions
0
votes
0
answers
95
views
Is there a "cohomology theory" for involutive algebras?
I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
- 4,873
5
votes
2
answers
452
views
How to define cohomology of algebraic structures?
I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain
\begin{align*}
\cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
- 775
5
votes
0
answers
217
views
Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$
Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
- 689
2
votes
1
answer
240
views
Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem
I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...
- 121
3
votes
0
answers
163
views
Obstruction to delooping
Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
- 5,038
6
votes
2
answers
321
views
Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack
Let $G$ be an affine group scheme over a field $k$ of characteristic zero.
I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) ...
- 1,635
2
votes
0
answers
260
views
Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
- 5,038
4
votes
0
answers
160
views
Reference for equivariant derived Künneth formula
I'm looking for a reference for the following statement in as much generality as possible, assuming it is correct.
Let's $X$ and $Y$ be "spaces" with a $G$-action. We can take the $G$-product defined ...
- 41
7
votes
0
answers
237
views
$X$ with $H^*(X)=$affine Verma module?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $...
- 5,565
1
vote
1
answer
209
views
About Hom and weight space of nilpotent Lie algebra cohomology
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$
the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
- 1,865
9
votes
0
answers
292
views
cohomology of flag variety
I recently ran into a 30+ years old literature by Andersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the ...
- 91
5
votes
0
answers
109
views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
- 229
4
votes
2
answers
284
views
Cohomology Ring of a small category $\mathsf{C}$
Assume that $\mathcal{C}$ is a small category and that $\mathcal{F} \in \mathsf{ob(Ab^{\mathsf{C}})}$, is a covariant functor. When our category has finitely many objects then a classical theorem from ...
- 431
27
votes
4
answers
3k
views
Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can ...
- 7,689
14
votes
1
answer
917
views
Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result?
Fix $n \ge 2$. Let $V_n$ be the $\binom{n}{2}$-dimensional vector space (over $\mathbb{C}$) generated by a set of vectors $\{w_{ij} : 1 \le i < j \le n\}$. Let $\bigwedge^* V_n$ be the exterior ...
- 193