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} ...
0
votes
0answers
49 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
2
votes
1answer
47 views

Homotopy Type of Diffeomorphism Group of Lie Group

Let $G$ be a finite dimensional connected Lie group and $Diffeo(G)$ be the diffeomorphism group of the underlying manifold. Is it true that $Diffeo(G)$ has the homotopy type of a finite dimensional ...
2
votes
1answer
47 views

Lie Automorphisms

Take $X$ to be a Lie group. Define a Lie automorphism of $X$ to be a group isomorphism from $X$ to itself which is also a homeomorphism. Define $Aut(X)$ to be the group of Lie automorphisms of $X$ ...
0
votes
0answers
24 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
1
vote
1answer
26 views

Showing a simple Lie group is connected and compact.

I'm working on a presentation on simple Lie groups and would like to show by example that the simple Lie groups are connected, but I'm not really sure how to do this. I'd also like to show that one of ...
2
votes
1answer
48 views

Lie Group and Universal Covering Groups

Is every Lie group realized as the quotient of its universal covering group by a discrete group of isometries? Basically, a Lie group analog for the uniformization theorem. It seems reasonable but I'm ...
0
votes
1answer
28 views

Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point. I know that the exponential map lets us move between the Lie algebra and the ...
11
votes
2answers
154 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 ...
2
votes
2answers
70 views

Understanding that $GL_n(\mathbb{R})$ has two connected components

I am trying to understand the proof of the theorem: $GL_n(\mathbb{R})$ has two components. The proof says that The group of matrices with positive and negative determinant, ...
0
votes
1answer
38 views

What are the classical compact connected lie groups?

I think they are $SO(n),SU(n),Sp(n)$, and $Spin(n)$ but I'm not sure. Any help would be appreciated.
4
votes
1answer
54 views

smooth lie group action

Let $\theta:G\times M\to M$ be a smooth left action of a Lie group $G$ on the manifold $M$. Suppose $G$ is compact and $M$ is Hausdorff. Let $K$ be a compact set in $M$. Is it true that $G_K:=\{g\in ...
0
votes
1answer
79 views

Is $GL(n;R)$ closed as a subset of $M_n(R)$?

Let $M_n(R)$ denote the space of all $n×n$ matrices with real entries. The general linear group over real numbers,denoted $GL(n,R)$, is given by $GL(n,R)=${$A∈M_n(R)|det(A)\neq0$}. Is ...
2
votes
1answer
128 views

How to prove the space of orbits is a Hausdorff space

Let $M$ be a connected smooth n-dimensional manifold and $G$ a lie group acting smoothly on $M$. for $x\in M$, the orbit $G\cdot x=\{g(x)\mid g\in G\} $ is a sub-manifold of $M$ and if the action is ...
3
votes
1answer
113 views

$\mathbb{RP}^3$ is homeomorphic to the solid ball with antipodal points identified

I am reading the book Application of Path integrals by Schulman, which has a chapter on applications of homotopy theory to path integrals. In that he says we can geometrically describe $SO(3)$ by a ...
5
votes
1answer
139 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
3
votes
1answer
100 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 ...
4
votes
1answer
70 views

Is Bruhat cell dense in p-adic topology?

I've seen in literature a statement like 'there exists an open and dense Bruhat cell'. In $GL(2,F)$ for example, where $F$ is a p-adic field, let $\omega=\begin{pmatrix} & 1 \\ 1 & ...
0
votes
0answers
37 views

Topology of Lie groups. [duplicate]

I do not understand the topology of a Lie group clearly. Let $G$ be a Lie group and $T_eG$ be its tangent space at the identity $e \in G$. Why $Aut(T_eG)$ is an open subset of the vector space of ...
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 ...
8
votes
1answer
79 views

Are there more embeddings $U(2) \hookrightarrow SO(4)$?

It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$. My ...
4
votes
1answer
32 views

Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.

I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
3
votes
1answer
177 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ ...
9
votes
2answers
978 views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
-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 ...
0
votes
1answer
165 views

$3\times 3$ symmetric matrix with signature $(2,1)$

I need to show the set of $3\times 3$ real symmetric matrices with signature $(2,1)$ is an open connected subset in the usual topology of $\mathbb{R}^6$. To show connectedness I did like the ...
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 ...
0
votes
1answer
36 views

an equation in a component of identity in a lie group

could any one help me how to solve : prove that there exist solution for the equation $x^2=y$ in identity component of a lie group. I dont know how to start this one, what is the specia; about ...
0
votes
1answer
43 views

Local Isomorphism on Lie Groups II

Yesterday I asked about an example in Chevalley's book "Theory of Lie Groups I". Well, the second example on pages 38 and 39 made me work a lot to understand. ...
2
votes
1answer
53 views

complex projective space with some identification with groups

Could any one tell me how to show : $\mathbb{C}P^n ≅ SU ( n + 1)/ U ( n)$? $\mathbb{C}P^n ≅ S^{2 n +1}/ U (1)$ ? what is transitive group of $\mathbb{C}P^n$?
1
vote
0answers
215 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 ...
5
votes
2answers
214 views

Nilpotent Lie group homeomorphic to $\mathbb R^n$?

Is it true that a connected, simply connected, nilpotent $n$-dimensional Lie group $G$ is homeomorphic to $\mathbb R^n$? EDIT: Maybe a possible argument is the following: Since $G$ is simply ...
11
votes
4answers
357 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
229 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
808 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 ...
4
votes
0answers
127 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...
31
votes
6answers
2k views

How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
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 $\rho : G\times X\rightarrow X$ ...
1
vote
2answers
186 views

lie groups and topology

Is there a relationship between Lie groups and topology and is there a succinct explanation that can be provided? Is there a good online reference that discusses this.
2
votes
2answers
188 views

Concerning the definition of effective quotient orbifold

I've been trying to figure out orbifolds, and in all of the sources I seem to be confused with the orbifold structure on quotient orbifolds. A quotient orbifold is defined as follows. Let $M$ be a ...