All Questions
Tagged with lie-groups topological-groups
79 questions
9
votes
1
answer
657
views
Must an inverse limit of simply connected groups be simply connected?
While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
- 7,193
12
votes
2
answers
883
views
Does almost every pair of elements in a compact Lie group generates the connected component?
It is known that almost every pair of elements in a connected compact Lie group (topologically) generates the group.
Obviously this isn't true for non-connected groups but
Given a compact Lie ...
- 262
4
votes
2
answers
2k
views
A question about unitary and anti-unitary matrices
The question is the following: Let $U:\mathbf{C}^n\to \mathbf{C}^n$ be a unitary operator; let $\tilde{U}:\mathbf{C}^n\to\mathbf{C}^n$ be an antiunitary operator.
Can one deform $U$ to $\tilde{U}$ ...
- 327
6
votes
1
answer
249
views
Growth function of locally compact groups
Every locally compact second countable group $G$ has a regular left-invariant measure $h$, the Haar measure. On the other hand the Birkhoff–Kakutani Theorem asserts that such groups also admit a ...
0
votes
1
answer
434
views
Reference request: Any connected Lie group has a countable base for its topology
I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?
- 14.2k
4
votes
0
answers
640
views
Closed subgroups of a connected Lie group
Is it true that for any closed subgroup $H$ of a connected Lie group $G$, the group of connected components $\pi_0(H)$ is finite or countable? (inspired by the comment of nfdc23 to this question ).
- 14.2k
2
votes
0
answers
321
views
Surjective homomorphisms of non-connected Lie groups
Let $\psi\colon B\to C$
be a homomorphism of real Lie groups, where the group $C$ is connected.
Let $B^0$ denote the identity component of $B$, and we set $\pi_0(B)=B/B^0$, then $\pi_0(B)$ is a ...
- 14.2k
3
votes
1
answer
141
views
Measure on orbits of $N$ under conjugation by $H$
Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, ...
- 6,180
2
votes
0
answers
82
views
Uniquely divisible neighborhoods of identity in topological groups
Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a ...
- 1,959
4
votes
3
answers
2k
views
Why is the unipotent radical of a parabolic subgroup unimodular?
Let $G$ be a connected, reductive group defined over a local field $F$. Let $P$ be a parabolic subgroup of $G$ which is defined over $F$, and let $N = \mathscr R_u(P)$ be the unipotent radical of $P$....
- 6,180
2
votes
0
answers
62
views
Codistal subgroups of locally compact groups
Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is codistal in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal ...
- 4,728
4
votes
0
answers
264
views
When can a locally compact group be approximated by discrete subgroups?
This question is about partitioning a (locally) compact group into cells by using discrete subgroups.
Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...
- 6,287
7
votes
4
answers
2k
views
Topological structure of SO(n) as a product
I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] ...
3
votes
1
answer
142
views
What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,040
3
votes
1
answer
380
views
A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...
- 356
6
votes
2
answers
392
views
Union of conjugates of a closed subgroup of a compact group
Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$.
Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of $G$...
- 11.3k
7
votes
2
answers
2k
views
Conditions for a topological group to be a Lie group
In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157):
Let $G$ be a locally compact group....
- 1,105
1
vote
2
answers
481
views
Cross section for closed Lie subgroup in a Lie group
Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?
- 23
20
votes
2
answers
2k
views
Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?
The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...
- 2,974
6
votes
3
answers
1k
views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
- 17.5k
9
votes
2
answers
717
views
Topology on extensions of topological groups
Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups.
Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...
- 125
2
votes
2
answers
423
views
Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$
What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups ...
- 125
2
votes
1
answer
571
views
On homeomorphic compact connected topological groups
I wish to thank Professor Claudio Gorodski for his very helpful
answers to my question on the webcite:
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...
- 491
17
votes
0
answers
1k
views
What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
- 47.3k
5
votes
3
answers
1k
views
On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
- 7,609
8
votes
1
answer
730
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 1,414
-3
votes
4
answers
5k
views
What is the situation with Hilbert's Fifth Problem?
The common knowledge in this regard seems to be that Hilbert's Fifth Problem was completely solved by Gleason, Montgomery, and Zippin. However, such wisdom was contested by Peter Olver:
Olver, Peter ...
12
votes
4
answers
2k
views
Which compact groups have finitely many irreducible representations of each dimension?
If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite ...
- 118k
6
votes
0
answers
2k
views
Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups
Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?
- 4,351