All Questions
14
questions
7
votes
0
answers
184
views
Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44.3k
12
votes
3
answers
1k
views
Extending group actions to vector bundles
Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$.
Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
- 129
5
votes
0
answers
294
views
What is the image of the diagonal map on the cohomology of Lie groups
Consider a simple Lie group $G$ and its mod $p$ cohomology $H^*(G, \mathbb{Z}_p)$.
A good reference is the book
Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by ...
- 1,429
10
votes
1
answer
638
views
how to view homology of affine Grassmannian as a subring of symmetric function
Let $G=SL_n$, it is proven that $R:=H_*(Gr_G)\cong \mathbb{C}[\sigma_1,...,\sigma_{n-1}]$ where $\sigma_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure ...
- 849
3
votes
0
answers
325
views
Quotient space of Grassmannian
The Grassmannian $G(k,2k)$ is equipped with a nice $\mathbb Z_2$ action with respect to a non-degenerate symplectic bilinear form: $1.V=V^{perp}$. Is there a reference where the ring of polynomial ...
- 611
14
votes
1
answer
2k
views
what is the universal cover of GL(2,R)?
In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs (M,f) ...
- 303
2
votes
1
answer
244
views
How to extend an equivariant map from a compact Lie group
Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$...
- 1,030
3
votes
2
answers
677
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,689
6
votes
0
answers
439
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,689
5
votes
2
answers
740
views
Equivariant Cohomology of a Complex Projective Variety
Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
- 4,890
2
votes
1
answer
1k
views
Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 4,890
18
votes
5
answers
6k
views
Does a finite-dimensional Lie algebra always exponentiate into a universal covering group
Hi,
I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.
Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,.....
- 649
8
votes
3
answers
2k
views
Cohomology rings of $ GL_n(C)$, $SL_n(C)$
Can anyone provide me with the reference for the following fact
(idea of the proof will be appreciated too):
Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
- 1,773
26
votes
2
answers
5k
views
Cohomology of Lie groups and Lie algebras
The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
- 23.4k