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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36
votes
7answers
4k 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 ...
30
votes
1answer
2k 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 ...
28
votes
10answers
6k 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 ...
25
votes
0answers
264 views

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 ...
24
votes
1answer
817 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 ...
21
votes
2answers
355 views

How can I lift a path to $\mathrm{Spin}(n)$?

Suppose I am given an explicit differentiable path $\gamma\colon[a,b]\to SO(n)$, with $\gamma(a)=\gamma(b)=I$. Then $\gamma$ either does or does not lift to a closed loop in $\mathrm{Spin}(n)$. How ...
21
votes
1answer
496 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 $\rho : G\times X\rightarrow X$ ...
20
votes
4answers
1k views

Irreducible representations of Poincaré group

I am looking for any reference on Wigner's classification of irreducible representations of the Poincaré group. I know the classification, but is there any reference where the representations ...
19
votes
4answers
562 views

How is $\operatorname{GL}(1,\mathbb{C})$ related to $\operatorname{GL}(2,\mathbb{R})$?

I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. ...
18
votes
1answer
388 views

Can we prove that there are countably many isomorphism classes of compact Lie groups without appealing to the classification of simple Lie algebras?

It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply ...
17
votes
11answers
437 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
17
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 ...
15
votes
6answers
568 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 ...
15
votes
3answers
1k 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 ...
15
votes
1answer
147 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
15
votes
1answer
306 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
14
votes
1answer
245 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
14
votes
1answer
171 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
14
votes
0answers
201 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
13
votes
2answers
270 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 ...
13
votes
3answers
3k views

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

Since there is a $2$-$1$ homomorphism from $SU(2)\times SU(2)$ 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 ...
12
votes
4answers
3k views

What is a Lie Group in layman's terms?

I'm having trouble getting my head arround the concept. The folks at mathoverflow were too clever to answer me. Can someone explain it to me?
12
votes
3answers
664 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 ...
12
votes
3answers
423 views

What is the main use of Lie brackets in the Lie algebra of a Lie group?

I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at ...
12
votes
6answers
4k 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. ...
12
votes
1answer
932 views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
12
votes
2answers
204 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 ...
12
votes
2answers
191 views

Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group ...
12
votes
1answer
439 views

Diffeomorphic, group-isomorphic Lie groups that are not isomorphic as Lie groups

Do there exist two Lie groups which are diffeomorphic as smooth manifolds, have isomorphic group structures, yet are not isomorphic as Lie groups? Of course, for this to happen, any diffeomorphism ...
12
votes
0answers
198 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
11
votes
5answers
373 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 ...
11
votes
2answers
945 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
11
votes
2answers
658 views

The action of SU(2) on the Riemann sphere

One way to get the famous double cover $\text{SU}(2) \to \text{SO}(3)$ is to note that $\text{SU}(2)$ is isomorphic to the group of unit quaternions and to let unit quaternions $q$ act on the subspace ...
11
votes
1answer
432 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
1answer
222 views

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
11
votes
1answer
283 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact ...
10
votes
2answers
1k 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!
10
votes
5answers
505 views

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 ...
10
votes
5answers
312 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 ...
10
votes
3answers
3k 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?
10
votes
2answers
861 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 ...
10
votes
2answers
2k 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 ...
10
votes
1answer
3k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
10
votes
2answers
232 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 $4d$) ...
10
votes
2answers
237 views

Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?)

This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering ...
10
votes
1answer
92 views

Understanding Dynkin Diagrams - any organising ideas - are they now adequately understood?

Some 30 or so years ago JH Conway posed a question about the ubiquity of the Dynkin Diagram - not necessarily in public, but I heard him ask it. I think it was in the context of "what would be ...
10
votes
1answer
134 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
9
votes
2answers
1k 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 ...
9
votes
3answers
3k views

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 ...
9
votes
2answers
731 views

Is the universal cover of an algebraic group an algebraic group?

Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, ...