All Questions
Tagged with loop-spaces lie-groups
8 questions
8
votes
0
answers
176
views
Comparison of two well-known bases of the integral homology group of based loop group
Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways:
(1) Via Bott-Samelson'...
- 461
8
votes
1
answer
340
views
Delooping the quotient space $SU/SU(n)$
Let $SU$ denote the infinite unitary group. Does the quotient space $SU/SU(n)$ admit a delooping $X$? One could also ask that this space $X$ sit in a fiber sequence $BSU(n)\to BSU\to X$, but this is ...
- 5,770
8
votes
1
answer
266
views
When does $BG \to BA$ loop to a homomorphism?
If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
- 35.5k
1
vote
0
answers
150
views
Symplectic structures on the grassmannian model of the based loop group
$\newcommand{\Ad}{\operatorname{Ad}}$
In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...
- 627
13
votes
1
answer
731
views
free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...
- 2,035
7
votes
1
answer
1k
views
Haar measure on infinite dimensional Lie groups?
Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like ...
- 548
2
votes
2
answers
1k
views
A question about the affine Grassmanian
For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as:
$$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$
Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
- 751
17
votes
2
answers
1k
views
What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?
Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
- 8,167