# Tagged Questions

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### 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 ...
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### 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 ...
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### $\mathbb{RP}^3$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $3\times 3$ 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 $\mathbb{C}P^n$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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 ...