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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34
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7answers
3k 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 ...
27
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 ...
25
votes
10answers
4k 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 ...
21
votes
2answers
336 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
471 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
989 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
1answer
665 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 ...
18
votes
4answers
541 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
379 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
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
1answer
296 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
6answers
474 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 ...
13
votes
2answers
231 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
609 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 ...
13
votes
3answers
916 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 ...
13
votes
1answer
225 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 ...
12
votes
2answers
171 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
175 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
409 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
184 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 ...
11
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 ...
11
votes
2answers
854 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
1answer
328 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
186 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 ...
11
votes
0answers
161 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 ...
10
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?
10
votes
5answers
338 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
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
3answers
331 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 ...
10
votes
6answers
3k 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
votes
2answers
762 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
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?
10
votes
5answers
250 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
2answers
614 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 ...
10
votes
2answers
231 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
86 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
123 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
637 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 $G_{an}$, ...
9
votes
1answer
214 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
9
votes
2answers
1k 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 ...
9
votes
3answers
541 views

Proper, smooth action with singular orbit space

This is a problem from Lee, Smooth Manifolds (Problem 9.4). It's not an homework problem, I'm just systematically solving every problem of that book, and I got stuck on this one. Usually I try not to ...
9
votes
1answer
2k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
9
votes
1answer
638 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 ...
9
votes
1answer
379 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
9
votes
2answers
451 views

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 ...
9
votes
1answer
136 views

Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
9
votes
1answer
365 views

The center of a simply connected semisimple Lie group

I am learning about Lie groups, and I have the following basic question: Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
9
votes
3answers
521 views

Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?

Let $G$ be a compact (Hausdorff) group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. (Hence $G$ is a Lie group.) Is it true that every irreducible representation ...
9
votes
1answer
142 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
9
votes
0answers
108 views

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

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...