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 ...

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2answers
643 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 ...
28
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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 ...
10
votes
5answers
327 views

$\varphi$ 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 ...
10
votes
2answers
844 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!
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6answers
2k views

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. ...
10
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3answers
308 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 ...
23
votes
1answer
1k views

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 ...
7
votes
2answers
359 views

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 ...
5
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0answers
84 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
8
votes
3answers
2k views

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?
2
votes
2answers
584 views

Proof: Tangent space of the general linear group is the set of all squared matrices

Let us assume we have the following definition of a tangent space: Definition of smooth path Let $X\subset\mathbb{R}^n$. Let $I$ be a real interval. \begin{equation} P \text{ is a smooth path in } ...
4
votes
0answers
65 views

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb ...
3
votes
1answer
413 views

First and second homotopy groups of a connected Lie group

I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$? Thanks for anyone who ...
10
votes
5answers
214 views

Does non-commuting $\mathfrak{g}$ imply non-abelian $G$?

Question 8.1 in Kristopher Tapp's introductory text on matrix groups asks to show that $SO(n)$ is non-abelian ($n>2$) by finding two elements of $so(n)$ that do not commute. Why is this method ...
5
votes
1answer
597 views

Non surjectivity of the exponential map to GL(2,R)

I was asked to show that the exponential map $\exp: \mathfrak{g} \mapsto G$ is not surjective by proving that the matrix $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
5
votes
2answers
507 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
4
votes
1answer
246 views

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 ...
3
votes
2answers
490 views

Image of Matrix Exponential Map

It is known that every $A$ belongs to $GL(n,\mathbb C)$ equals to $\exp(B)$ for some $n \times n$ matrix $B$. How to show the following is true? Show that a matrix $M$ belonging to $GL_n(\mathbb R)$ ...
1
vote
1answer
185 views

Lie group structure on some topological spaces

I have some basic background in Lie theory and I have some difficulties to show that some topological spaces admits a Lie group structure. More precisely, for a given Lie group $G$: 1) Why its ...
5
votes
0answers
377 views

Alternating and special orthogonal groups which are simple

I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if ...
3
votes
0answers
27 views

Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question] If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? ...
3
votes
1answer
290 views

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 ...
3
votes
1answer
354 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
18
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10answers
2k views

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 ...
15
votes
1answer
451 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 ...
15
votes
2answers
2k views

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 ...
13
votes
3answers
544 views

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

As we know, Lie group is a special smooth manifold. I want to find some geometric property, which is only satisfied by the Lie group. I only found one property: parallelizability. Can you show me ...
10
votes
3answers
2k views

Recovering the two SU(2) matrices from SO(4) matrix

Since there is a 2-1 homomorphism from SU(2)xSU(2) to SO(4) there should be a way to recover the two SU(2) matrices given an SO(4) matrix. I believe I could set this up as a system of equations ...
5
votes
2answers
174 views

Homotopy groups of some magnetic monopoles

This is a list of homotopy groups which I (as a physics researcher) encounter when studying magnetic monopole under certain configuration of gauge field profiles. \begin{gather} \pi_2(SU(2)/U(1)) ...
10
votes
2answers
137 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 ...
9
votes
2answers
775 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 ...
8
votes
1answer
2k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
6
votes
1answer
155 views

Possibilities of an action of $S^1$ on a disk.

I'm dealing with actions of the circle over differentiable manifolds. In the book I'm reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an ...
11
votes
1answer
219 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
11
votes
3answers
741 views

Lie algebra of a quotient of Lie groups

Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding ...
8
votes
2answers
181 views

Relation between $SU(4)$ and $SO(6)$

This is more of a particle physics question than maths. Since $SO(6)$ and $SU(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) ...
8
votes
2answers
652 views

Homology and Euler characteristics of the classical Lie groups

I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques ...
3
votes
1answer
276 views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
3
votes
2answers
294 views

Fundamental group of $SO(3)$

How can I show that the universal cover of $SO(n)$, for $n\ge 3$, is a double cover? And how does that reflect the fact that the fundamental group of $SO(n)$ has two elements? What is the relation ...
10
votes
2answers
134 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
6
votes
3answers
190 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
6
votes
1answer
164 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
5
votes
1answer
113 views

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 ...
4
votes
1answer
293 views

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 ...
3
votes
2answers
311 views

deformation retract of $GL_n^{+}(\mathbb{R})$

Well, I need a deformation retract from $GL_n^{+}(\mathbb{R})$ to $SO(n)$ Here is what I tried, let $A\in GL_n^{+}(\mathbb{R})$ $A=(A_1,\dots,A_n)$ where $A_i$'s are collumn vectors, Recall that the ...
3
votes
1answer
244 views

Matrix Exponential does not map open balls to open balls?

Consider the following theorem from Hall's Lie Groups, Lie Algebras and Representations: Theorem 2.27: For $0 < \varepsilon < \textrm{ln} 2$, let $U_\varepsilon = \{X \in M_n(\Bbb{C}) | ...
2
votes
0answers
62 views

The tangent bundle of a Lie group is trivial

I'm trying to recreate the proof given by Alex Youcis at http://math.stackexchange.com/a/308798/86801 and got everything except $\displaystyle (L_h)_\ast \left.\frac{\partial}{\partial x_i}\right|_e = ...
2
votes
1answer
280 views

fundamental group of $U(n)$

Is my logic is correct? $f:U(n)\rightarrow U(1)$ defined by $f(A)=detA$ is a group homomorphism so that induces $f^{*}: \pi_1(U(n))\rightarrow \pi_1(U(1))$ will be an Isomorphism right(I am not ...
2
votes
1answer
230 views

Complete reducibility of sl(3,F) as an sl(2,F)-module

I was reading the Weyl's theorem on the complete reducibility of a finite dimensional representation of semi-simple Lie algerba and wanted to apply the theorem to the following problem which was ...
1
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1answer
66 views

Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra #2

I read the book of Karin Erdmann and Mark Wildon: "An introduction to Lie algebras". In page 22 they say that: If $\dim (L) = 3$, $\dim (L') = 2$ then (a) $L'$ abelian and (b) $\operatorname{ad} x ...