# Tagged Questions

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the (group-...

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### One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear real matrices. It is easy to see that a real matrix is ...
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### The description of abelian Lie groups

There is a problem in my problem sheet which asks me to describe all abelian connected Lie groups (moreover this is the first problem so it should be rather easy). I don't understand how this ...
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### Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
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### Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
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### Lie group homomorphism from $\mathbb{R}\rightarrow S^1$

I need to prove that every Lie group homomorphism from $\mathbb{R}\rightarrow S^1$ is of the form $x\mapsto e^{iax}$ for some $a\in\mathbb{R}$. Here is my attempt: As it is group homomorphism so it ...
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### The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
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### fundamental group of $U(n)$

Is my logic correct? $f:U(n)\rightarrow U(1)$ defined by $f(A)=\det A$ is a group homomorphism so that the induced homomorphism $f^{*}: \pi_1(U(n))\rightarrow \pi_1(U(1))$ will be an isomorphism, ...
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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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### Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
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### Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
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### Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite

I am gathering material for an exposition on Lie theory and I am after proofs that a Lie group with finite centre is compact if and only if its Killing form is negative definite. I know of one, ...
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### Can somebody explain the plate trick to me?

I learned of the plate trick via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a link to an animation of the "belt trick" which is ...
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### Generators of compact Lie groups

Suppose $G$ is a compact connected Lie group and let $\{X_i\}$ be a basis for its Lie algebra $\mathfrak g$. We know that the exponential $\exp:\mathfrak g \to G$ is surjective but when is it the ...
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### Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
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### Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

$SO(3)$ denotes 3x3 rotation matrices. This is Lie group, with corresponding Lie algebra being $\mathrm{Skew}_3$, the space of 3x3 skew-symmetric matrices. The link between them is the matrix ...
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### Does the algebraic structure of a Lie group restrict the possible dimensions of other Lie groups isomorphic to it?

In a recent question, I initially doubted that $\mathbb{C}^\times\cong S^1$, my intuition being that $\mathbb{C}^\times$ has one more "dimension" than $S^1$ - in rigorous terms, $S^1$ is (or rather, ...
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### How to prove that every Lie group is the semidirect product of a connected Lie group and a discrete group?

Every Lie group is the direct product of a connected Lie group and a discrete group. I think the component of the identity could be useful.
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### Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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### Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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### Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.

$\DeclareMathOperator{\inv}{inv}$ I am trying to understand the proof of the following from this document: Let $M$ be a smooth manifold which admits a group structure such that the multiplication ...
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### Prove that $O(n)$ is a maximal compact subgroup of $GL(n,\mathbb R)$

The indication is: Let $P$ is a symmetric positive definite matrix such that the norm of $p^k$ is smaller than a constant $C$ for every integer $k$, then $P=I_n$.
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### How to obtain a Lie group from a Lie algebra

Please pardon me, if the question is too simple. How can we obtain a Lie group form a given Lie algebra (say in 3 dimension) in practise? Can someone illustrate it in 3 dimension? Is there a ...
307 views