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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11
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1answer
393 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 ...
9
votes
2answers
214 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 $4d$) ...
9
votes
1answer
3k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
6
votes
1answer
157 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 ...
1
vote
3answers
407 views

Minimization on the Lie Group SO(3)

Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian ...
10
votes
2answers
805 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 ...
7
votes
1answer
277 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
2answers
199 views

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 ...
5
votes
2answers
190 views

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
5
votes
2answers
181 views

Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?

This might be a dumb question; I know only enough group theory to be able to ask dumb questions. Ken W. Smith has pointed out that one way to get intuition about the determinant is to observe that it ...
5
votes
2answers
199 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
5
votes
1answer
665 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?
5
votes
2answers
428 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 ...
4
votes
1answer
109 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
4
votes
1answer
276 views

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 ...
3
votes
1answer
58 views

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, ...
3
votes
0answers
530 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 = ...
3
votes
1answer
474 views

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, ...
3
votes
2answers
503 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 ...
1
vote
0answers
48 views

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 ...
6
votes
3answers
236 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
489 views

Representations of a non-compact group are labeled by its maximal compact subgroup?

I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work. Is there some idea that the representations of a non-compact group are ...
5
votes
1answer
263 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 ...
5
votes
1answer
462 views

Expression for the Maurer-Cartan form of a matrix group

I understand the definition of the Maurer-Cartan form on a general Lie group $G$, defined as $\theta_g = (L_{g^{-1}})_*:T_gG \rightarrow T_eG=\mathfrak{g}$. What I don't understand is the expression ...
5
votes
2answers
158 views

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, ...
5
votes
1answer
919 views

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 ...
4
votes
2answers
214 views

free subgroups of $SL(2,\mathbb{R})$

In the example section of the wikipedia article on the the Ping Pong lemma, you can see how to construct a free subgroup of $SL(2,\mathbb{R})$ with two generators $$ a_1 = \begin{pmatrix} 1 & 2 ...
4
votes
1answer
316 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}) | ...
4
votes
1answer
315 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
0answers
119 views

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I'll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential ...
3
votes
1answer
320 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 ...
2
votes
1answer
403 views

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 ...
2
votes
1answer
171 views

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 ...
2
votes
1answer
145 views

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
1
vote
2answers
162 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...
1
vote
1answer
41 views

Group of isometries is closed in $GL_{n+1}$

I solved the following exercise. Could someone please check my work? Prove that $\operatorname{Isom_n}{\mathbb R^n}$ is a matrix group. Is it compact? My work: We recall that ...
1
vote
2answers
120 views

Help with a proof that SO(n) is path-connected.

I've found lots of different proofs that SO(n) is path connected, but I'm trying to understand one I found on Stillwell's book "Naive Lie Theory". It's fairly informal and talks about paths in a very ...
1
vote
1answer
205 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
1
vote
1answer
82 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 ...
1
vote
1answer
183 views

Are there any good ways to see the universal cover of $GL^{+}(2,\mathbb{R})$?

Are there any good ways to see the universal cover of $GL^{+}(2,\mathbb{R})$? Here $GL^{+}(2,\mathbb{R})$ stands for the identity component of $GL(2,\mathbb{R})$, i.e. positive determinant matrices. I ...
0
votes
1answer
43 views

Proof that these two definitions are equivalent

Where can I find a proof of or how can I prove that these two definitions are equivalent? Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$. Definition 2: ...
0
votes
0answers
29 views

Question about a notation.

I have a question about the notation in the paper. On page 8, the 3-rd line from bottom, it is said that $h_J(t)$ is the corresponding diagonal matrix in $GL_4$. I think that $J$ is some set such that ...
9
votes
1answer
362 views

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, ...
8
votes
1answer
208 views

Can $(\Bbb{R}^2,+)$ be given the structure of a matrix Lie group?

I have an assignment problem that is coming from Brian Hall's book Lie Groups, Lie Algebras and Representations: An Elementary Introduction. Suppose $G \subseteq GL(n_1;\Bbb{C})$ and $H ...
6
votes
2answers
350 views

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 ...
5
votes
0answers
61 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
5
votes
1answer
71 views

If a connected Lie group is divisible, is its exponential map surjective?

A group $G$ is divisible if for all $g \in G$ and $k \in \mathbb{Z}_+$ there is an element $h \in G$ such that $h^k = g$; we call such an $h$ a $k$th root of $g$. In an answer to a recent question ...
5
votes
0answers
71 views

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 ...
5
votes
1answer
276 views

Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$

I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the ...
5
votes
1answer
425 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...