# 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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### 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 ...
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### Is there an easy way to show which spheres can be Lie groups?

I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it ...
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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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### 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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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 neighborhood....
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### Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. ...
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### how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$

Could any one give me hint for this one? how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$, well, Is it the same: there is a 2-fold covering map from $SU(2)$ to $SO(3)$? what is that map will be?
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### $\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$

I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$. At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
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### Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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### Under what conditions is the exponential map on a Lie algebra injective?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map. In his blog, Terrence Tao notes that if a Lie group is not simply-connected, ...
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### What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to ...
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### Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
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### How do you find the Lie algebra of a Lie group (in practice)?

Given a Lie group, how are you meant to find its Lie algebra? The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them? My group is the set ...
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### Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.

Since $T_p G$ is isomorphic to $T_e G$ for all $p\in G$, it makes sense that each vector in $T_p G$ can be identified with a vector in $T_e G$. Hence, to make the map from $TG$ one to one, we must ...
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### Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
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### Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

I am trying to answer the following question: Which Lie groups have a Lie algebra admitting an $\text{Ad}$-invariant inner product? First of all, all compact Lie groups satisfy this condition ...
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### fundamental group of $GL^{+}_n(\mathbb{R})$

I would like to know whether the $GL^{+}_n(\mathbb{R})$ the set of all invertible matrices with positive determinant is simply connected or not? I guess it is not simply connected but that is just a ...
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### How to evaluate the derivatives of matrix inverse?

Cliff Taubes wrote in his differential geometry book that: We now calculate the directional derivatives of the map $$M\rightarrow M^{-1}$$ Let $\alpha\in M(n,\mathbb{R})$ denote any given matrix. ...
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### Pushforward of Inverse Map around the identity?

Let $G$ be a Lie group and $i:G \rightarrow G$ denote the inversion map. (Notation: $f_*$ is the pushforward map $F_*:T_pG \rightarrow T_{i(p)}G$ which takes $(F_{*}X)(f)=X(f\circ F)$ and $X$ is a ...
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### Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure

$\newcommand{\diff}{\mathrm{d}}$ TL;DR Having read this I know something about Haar measures, in particular that a left-invariant one exists and is unique on any Lie group $G$. I know that defining: ...
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### Visualizing the fundamental group of SO(3)

Recently I became interested in trying to visualize the fact that $\pi_1(\text{SO}(3)) = \mathbb{Z}/2\mathbb{Z}$. For whatever reason, the plate trick doesn't do it for me, so I've been looking for ...
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### On lifting an action of $G$ on $X$ to an action of $G'$ on $\tilde{X}$.

I am reading the section on covering actions from Glen Bredon's Tranformation groups. Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path ...
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### Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.

I'm trying to prove that $Sp(2m)$ is a Lie group using this: defining a function $f(A)=A^tJA-J$ and trying to see that this is a submersion. But I've not realized yet what is the domains and the range ...
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### An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0$$ be a short exact sequence of $\mathfrak{g}$-...
This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
### Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis
For a lie algebra $\mathbb{g}$ we can define the adjoint representation as: $ad: \mathbb{g} \rightarrow End(\mathbb{g})$ as the map such that $ad_x(y)=[x, y]$ for all $\in \mathbb{g}$ I am ...