2
votes
1answer
22 views

Can topological groups be smoothed into lie groups?

I've been thinking about this for the past couple of days and I'm really not sure of the answer.. By "smoothed" I mean that for any arbitrary precision we can find a Lie group which approximates the ...
2
votes
1answer
38 views

Are bounded subsets of Lie groups totally bounded

Let $ G $ be a finite dimensional real Lie group, and take a bounded ball $ B_R(e) \subset G $ in it, coming from the Riemannian metric, which itself is induced from an inner product on $ \mathfrak{g} ...
1
vote
2answers
55 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
0
votes
1answer
32 views

Elements Outside the Identity Component $SO^+(1,\,3)$ of the Lorentz Group $O(1,\,3)$

I have been answering a question on Physics Stack exchange to do with the difference between the "proper orthochronous" (i.e. identity component of) the Lorentz group $SO^+(1,\,3)$ and the Lorentz ...
2
votes
0answers
36 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
1
vote
1answer
39 views

On homomorphisms of the compact symplectic group

Each $q$ in the symplectic group $Sp(1)$, defines an operator on the imaginary quaternions: $\rho(q):Im\mathbb{H} \rightarrow Im\mathbb{H}$ by $\rho(q)(x)=qx\bar{q}$ We can see here that $\rho:Sp(1) ...
11
votes
2answers
155 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
3
votes
1answer
46 views

Unique Square Root Neighbourhood in Topological Group

For a Lie Group $\mathfrak{G}$ and any neighbourhood $\mathcal{V}\subset\mathfrak{G}$ of the identity $\mathrm{id}\in\mathfrak{G}$, $\exists$ neighbourhood $\mathcal{U}\subset\mathcal{V}$ of ...
1
vote
0answers
29 views

Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you ...
4
votes
0answers
42 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
5
votes
2answers
176 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
2
votes
1answer
63 views

Is there a local group that is not locally isomorphic to a topological group?

Let $\:\langle \hspace{.02 in}\mathbf{U},\hspace{-0.03 in}\mathcal{T}_u\rangle\:$ be a Hausdorff space. $\;\;\;$ Let $\hspace{.02 in}\Gamma\hspace{.02 in}$ be a closed subset of $\hspace{.04 ...
5
votes
0answers
145 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
1
vote
0answers
43 views

right multiplication by elements of a discrete subgroup preserve left haar measure?

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar ...
1
vote
0answers
52 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
2
votes
1answer
101 views

Mal'cev completion of nilpotent groups

Is the $\mathbb{R}$-Mal'cev completion of a finitely generated torsion free nilpotent group connected and simply connected?. Thanks!
10
votes
1answer
115 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
1
vote
0answers
101 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
3
votes
1answer
101 views

Quaternionic general linear group is open

Is there an elegant proof of the following fact: "The quaternionic general linear group $GL(n, \mathbb{H})$ is open in $M_n(\mathbb{H})$", where $M_n(\mathbb{H})$ is the set of all $n \times n$ square ...
11
votes
1answer
159 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 ...
2
votes
0answers
43 views

Transitive topological action

Let $G$ be a topological group, $A$ be set and $\mu\colon G\times A\to A$ a transitive action. I'm trying to prove the statement below is false. There exists only one topology in $A$ such ...
3
votes
0answers
37 views

The simply-connectedness of quotient space

If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
4
votes
1answer
112 views

orthogonal group of a quadratic vector space

I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
1
vote
0answers
60 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
1
vote
2answers
68 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
7
votes
0answers
88 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
3
votes
2answers
136 views

why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
-1
votes
0answers
92 views

Covering space (Lie groups and their maximal tori)

Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
15
votes
1answer
497 views

Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
1
vote
1answer
54 views

Conjugacy classes of a compact matrix group

Let $G$ be a compact matrix group. May I know why the conjugacy classes of $G$ is necessarily closed? I tried to argue by taking limits but to no avail so is there a hint on how to tackle this ...
3
votes
2answers
317 views

Local Isomorphism on Topological Groups

I'm currently studying Lie Groups by "Theory of Lie Groups I", C. Chevalley. He talks about Topological Groups on chapter two. To be more precise, on page 38 he presents two examples in order to show ...
4
votes
1answer
143 views

Two Lie groups which are isomorphic but not homeomorphic

I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does ...
1
vote
0answers
216 views

Identity component of a Lie group

Could any one help me to solve this problem? Let the identity component $G_0$ of a Lie Group $G$ be the connected component of the identity element $e\in G$. Let $\mu$ and $i$ be the multiplication ...
1
vote
0answers
72 views

$1$-parameter subgroups in $GL_n(\mathbb{C})$

I came across this link on planetmath and a few facts on that link are confusing me. According to planetmath, any $1$-parameter subgroup in $GL_n(\mathbb{C})$ arises from the exponential map. That ...
11
votes
4answers
358 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
6
votes
1answer
230 views

A compact Lie group has descending chain condition on closed subgroups.

Proposition: Let $G$ be a compact Lie group and let $$G\supset G_1\supset G_2\supset\ldots$$ be a chain of closed subgroups $G_i$ of $G$. Then this chain must eventually stabilize. Question: The hint ...
4
votes
2answers
812 views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
6
votes
1answer
1k views

The special orthogonal group is a manifold

How can we show that $SO(n)$ is an $n^2$-manifold. It would be tempting to say that $SO(n)$ is an open set of $\mathbb R^{n^2}$ but this is not the case since $SO(n)$ is given as the intersection of ...
1
vote
1answer
83 views

Centralizers in reductive Liegroups = unimodular?

Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
9
votes
1answer
201 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
5
votes
0answers
237 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
10
votes
2answers
938 views

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
21
votes
1answer
460 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...