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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1answer
811 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
1answer
310 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
129 views

What does $GL_n(R)$ look like?

Exactly as in the title - what does the general linear group "look like" (you are free to interpret this however you like) as submanifold of $R^{n^2}$? What should I imagine when I think of it? (I ...
3
votes
1answer
277 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
329 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
142 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
384 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, ...
2
votes
1answer
262 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
vote
1answer
184 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 ...
1
vote
1answer
77 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
168 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 ...
1
vote
2answers
332 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 ...
0
votes
0answers
26 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 ...
8
votes
1answer
188 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 ...
8
votes
1answer
313 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, ...
6
votes
2answers
297 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 ...
6
votes
1answer
449 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
0answers
63 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
310 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. ...
5
votes
0answers
259 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
4
votes
1answer
49 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 ...
4
votes
2answers
172 views

List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find the ...
4
votes
2answers
176 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
143 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
4
votes
1answer
172 views

How to prove that every Lie group is the semidirect product of a connected Lie group and a discrete group?

Every Lie group is the direct product of a connected Lie group and a discrete group. I think the component of the identity could be useful.
4
votes
0answers
420 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
3
votes
2answers
68 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
3
votes
1answer
47 views

More elegant proof of that this diagram commutes

Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by $(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$. Let $A \in M^n (\mathbb C)$ and define ...
3
votes
1answer
66 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...
3
votes
1answer
100 views

Subgroups of $SO(4)$ with free transitive action on $S^3$

By considering $S^3$ as the group manifold of $SU(2)$, the ordinary action of $SO(4)$ on the three sphere can be written as the $SU(2)\times SU(2)/\mathbb{Z}_2$ given by the group action of ...
3
votes
1answer
43 views

Commutativity and Maximal Tori in Connected, Compact Lie Groups

Let $G$ be a path-connected, compact Lie Group. Let $x \in G$ and let $T_x \subset G$ denote the union of all the maximal tori in $G$ that contain $x$. Question: Is it true that if $y \notin T_x$, ...
3
votes
1answer
362 views

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 ...
3
votes
2answers
341 views

Local Isomorphism on Topological Groups

I'm currently studying Lie Groups by "Theory of Lie Groups I", C. Chevalley. He talks about Topological Groups on chapter two. To be more precise, on page 38 he presents two examples in order to show ...
3
votes
1answer
162 views

How to write down the pull back of a differential form by exponential map?

The exponential map $e_{m}: M(n,\mathbb{R})\rightarrow M(n,\mathbb{R})$ is defined by $$e_m(\alpha)=me^\alpha,\quad e^\alpha=1+\alpha+\frac{\alpha^2}{2}+\frac{\alpha^3}{3!}+\cdots$$ Now fix $q\in ...
3
votes
1answer
72 views

Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$

Let $G=GL_n(\mathbb{C})$. Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ g.(x,y)=(gx,y). $$ Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ ...
3
votes
1answer
331 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
3
votes
1answer
450 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
2
votes
0answers
46 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
2
votes
2answers
109 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
2
votes
1answer
84 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
2
votes
1answer
127 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 $S_{n}$?
2
votes
2answers
90 views

Computing the differential of the map $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$

Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map $$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. Then why does ...
2
votes
1answer
95 views

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I'm given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in ...
2
votes
0answers
214 views

What is the constant $e$, fundamentally? [duplicate]

Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask ...
1
vote
1answer
60 views

Is it true for solving differential equations by getting constant coefficient matrix with magnus expansion

The magnus expansion is given in detail http://en.wikipedia.org/wiki/Magnus_expansion. While implementing magnus expansion to differential equations we have an iteration formula as follows $$Y'(t) = ...
1
vote
1answer
41 views

What 1D $\mathbb{C}$-Subspaces are Stabilized by Elements of a Specific 2-Torus in $SO(7)$?

Consider the 2-torus $T \subset SO(7)$ defined by $T = \left\{ \mathrm{diag}(R_{\theta_1}, R_{\theta_2}, R_{-(\theta_1 + \theta_2)}, 1) \mid \theta_1, \theta_2 \in \mathbb{R} \right\}$, where ...
1
vote
1answer
134 views

Sp(1) and SO(3) Locally Isomorphic - Trouble with part of proof

My apologies if this is fairly basic. I'm trying to understand the proof that $Sp(1)$ and $SO(3)$ are locally isomorphic but are not isomorphic but have run into some trouble. Here's what my ...
1
vote
1answer
119 views

Weyl character formula and finding the trace.

Let $v$ be a positive integer. I have a representation $\rho_v$ of $USp(4) = \{g\in M_2(\mathbb{H})\,|\,g^{T}\bar{g}\}$, where $\mathbb{H}$ is Hamiltons quaternions. The representation $\rho_v$ has ...
1
vote
0answers
180 views

How to prove the lie algebra of $n\times n$ traceless matrices is semi-simple?

The Lie algebra of all the $n \times n$ matrices is not semi-simple. However, if we restrict ourselves to traceless $n\times n$ matrices, we do obtain a semi-simple (in fact, simple) Lie algebra which ...
1
vote
1answer
425 views

Representations of U(n) using bosons and fermions

I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper. Especially I don't understand their first sentence, "Using N bosons (fermions) ...