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
56 views

preserves eigen spaces?

"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and ...
2
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1answer
110 views

Weyl group of $\mathfrak{sl}(2,\mathbb{C})$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{gl}(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
4
votes
2answers
111 views

Symmetries in a nonlinear heat equation

I have to solve the following nonlinear PDE: $$\partial_t u(x,t)=ku(x,t)^2 \partial_{xx}u(x,t)$$ where $k$ is a constant with $k>0$. Is it possible to find some symmetry in this equation which ...
2
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2answers
1k views

What are elements in $SU(1, 1)$?

I am reading some papers in physics. I don't know some notations in those papers. For example, $SU(1, 1)$, $U(1)$. I think these are Lie groups which consist of matrices. But I don't know what kind of ...
1
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1answer
68 views

Conjugacy classes of a compact matrix group

Let $G$ be a compact matrix group. May I know why the conjugacy classes of $G$ is necessarily closed? I tried to argue by taking limits but to no avail so is there a hint on how to tackle this ...
8
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1answer
1k views

On 'backslash-forward slash' notation

I am curious about a notation that I have seen, but I have only seen it in contexts beyond my current level of ability and so haven't learned its meaning. Also, it's often difficult to search for the ...
-2
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1answer
100 views

Three dimensional Lie algebra L with dim L' = 1

Now suppose the derived algebra has dimension 1. Then there exits some non-zero $X_1 \in g$ such that $L' = span\{X_1\}$. Extend this to a basis $\{X_1;X_2;X_3\}$ for g. Then there exist scalars ...
4
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3answers
510 views

Normal subgroup and Lie algebra

I have an exercise of Lie group as follows: "Let $G,H$ be closed connected subgroup of $GL_n(\mathbb{R})$, and $H$ be subgoup of $G$. Suppose that $Lie(H)$ is an ideal of $Lie(G)$. Prove that $H$ is a ...
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2answers
318 views

Lie group homomorphism from $\mathbb{R}\rightarrow S^1$

I need to prove that every Lie group homomorphism from $\mathbb{R}\rightarrow S^1$ is of the form $x\mapsto e^{iax}$ for some $a\in\mathbb{R}$. Here is my attempt: As it is group homomorphism so it ...
2
votes
1answer
181 views

connected $\Rightarrow$ path connected?

Well, so far, I have noticed that whenever a matrix lie group is connected it is path connected, so is it true that in matrix lie group connected $\Rightarrow$ path connected?If yes, could anyone tell ...
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0answers
62 views

Followup question in Brian Hall's Lie Groups and Algebras.

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then ...
4
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2answers
598 views

Example of two-dimensional non-abelian Lie algebra?

can some one give me an example of two-dimensional non-abelian Lie algebra?
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2answers
759 views

Two Dimensional Lie Algebra

I read in mark wildon book "introduction to lie algebras" "Let F be any field. Up to isomorphism there is a unique two-dimensional nonabelian Lie algebra over F. This Lie algebra has a basis {x, y} ...
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0answers
53 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
0
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2answers
209 views

compactness and fundamental group of $SL(n,\mathbb{C})$

could any one help me to prove the heading? $SL(n,\mathbb{C})$ is closed I can prove only, what are the other tools I need? $SL(n,\mathbb{C})$ connected?simply connected?
3
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1answer
455 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, ...
7
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3answers
248 views

Why is every representation of $\textrm{GL}_n(\Bbb{C})$ completely determined by its character?

I know that every (Lie group) representation of $\textrm{GL}_n(\Bbb{C})$ is completely reducible; this I believe comes from the fact that every representation of the maximal compact subgroup ...
2
votes
2answers
176 views

Lie Groups question from Brian Hall's Lie Groups, Lie Algebras and their representations.

In page 60 of Hall's textbook, ex. 8 assignment (c), he asks me to prove that if $A$ is a unipotent matrix then $\exp(\log A))=A$. In the hint he gives to show that for $A(t)=I+t(A-I)$ we get ...
3
votes
2answers
801 views

Jacobian of Reprojection Error

I am writing a program to find the transformation between two sets of 3D points extracted from a moving stereo camera. I am using an 'out of the box' Levenberg-Marquardt implementation to find this ...
12
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1answer
715 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 ...
5
votes
0answers
98 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
5
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2answers
390 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
452 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 ...
0
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0answers
76 views

a question on weyl group and its action on $\mathfrak{t}$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
2
votes
2answers
100 views

a question on weyl group

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
1
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1answer
149 views

Derivative wrt. to Lie bracket.

Let $\mathbf{G}$ be a matrix Lie group, $\frak{g}$ the corresponding Lie algebra, $\widehat{\mathbf{x}} = \sum_i^m x_i G_i$ the corresponding hat-operator ($G_i$ the $i$th basis vector of the tangent ...
4
votes
3answers
232 views

Unitary representation of $SO(3)$

Definition: $\mathcal{H}$ be a Hilbert space and $U(\mathcal{H})$ denote the unitary operators on it, If Unitary representation of a matrix lie group $G$ is just a homomorphism $\Pi:G\rightarrow ...
1
vote
1answer
96 views

Some representation of $SU(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, could any one tell me how is that Linear Transformation look like explicitly?And would you please tell me how this is an ...
3
votes
1answer
456 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 ...
0
votes
1answer
139 views

confused in the term “closed” in closed subgroup

well, In Brian C Halls Book, I am not getting the definition of Matrix Lie group, as he says : A matrix Lie Group is any subgroup $G$ of $GL_n(\mathbb{C})$ with the following property: If $A_m$ is any ...
6
votes
2answers
255 views

Why is the name general “linear” group?

Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?
1
vote
1answer
155 views

Matrix Lie group counter-example: $e^X$ in the Lie group, but $X$ is not in the Lie algebra

What's an example of a Lie group $G$ and matrix $X$ such that $e^X \in G$ but $x \notin \mathfrak{g}$, where $\mathfrak{g}$ is the associated Lie algebra? This is the same as problem 2.10 in Bryan ...
3
votes
1answer
332 views

understanding adjoint representation

I need to understand what is meant by "tangent space at identity of a Lie group is canonically isomorphic to its Lie algebra" to understand the definition of adjoint representation. Could any one ...
1
vote
1answer
137 views

Definition of differential of Adjoint representation of Lie Group

Let $g$ be an element of Lie Group $G$, and $\gamma(t) : \mathbb{R} \rightarrow G$ be a path in $G$ such that $\gamma(0) = e$, the identity element of $G$. Denote the tangent space at $e$ as $T_eG$, ...
6
votes
2answers
378 views

Is there a non-matrix Lie group?

I'm new to Lie Groups, but all the examples I found are matrix groups. Can someone show a non-matrix Lie group?
3
votes
1answer
80 views

From $\mathfrak{so}(16)$ to $\mathfrak{su}(11)$?

The two compact real form Lie algebras $\mathfrak{so}(16)$ and $\mathfrak{su}(11)$ have the same dimension (120). They are certainly not isomorphic, but does there exist some kind of algebraic ...
2
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0answers
254 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
8
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0answers
165 views

Weyl character formula for locally compact Lie groups.

I was just wondering if there exists such a formula. Specifically I need to calculate characters of irreducible representations of GSp$(4,\mathbb{C})$. I know how to do it for the compact Lie group ...
1
vote
1answer
62 views

example of a nilpotent lie algera

I want some example of nilpotent lie algebras, here I also want to see how the set of all $n\times n$ matrices $(a_{ij})$ where $a_{ij} = 0\ \forall\ i\ge j$ forms a nilpotent lie algebra under the ...
4
votes
3answers
509 views

Prove that $O(n)$ is a maximal compact subgroup of $GL(n,\mathbb R)$

The indication is: Let $P$ is a symmetric positive definite matrix such that the norm of $p^k$ is smaller than a constant $C$ for every integer $k$, then $P=I_n$.
0
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1answer
39 views

an equation in a component of identity in a lie group

could any one help me how to solve : prove that there exist solution for the equation $x^2=y$ in identity component of a lie group. I dont know how to start this one, what is the specia; about ...
7
votes
1answer
877 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 $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
3
votes
2answers
479 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 ...
2
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2answers
1k views

Lie algebra of Heisenberg group

To find the Lie algebra of the Heisenberg group $H$, which we know to consist of upper triangular matrices, we see that exponentials of all strictly upper triangular matrices are in $H$. I do not get ...
7
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2answers
101 views

Why is positivity of first entry sufficient for a matrix in $\mathrm{SO}(1,3)$ to be in $\mathrm{SO}^{+}(1,3)$?

This is a result physics books tell all the time, that the branch of proper Lorentz transformations with positive first entry forms the identity component of Lorentz group. In mathematical language, ...
3
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1answer
109 views

Fraction of SO(3) within an angle

SO(3) describes a space of rotations. These rotations can be described in axis-angle representation. I would like to know what fraction of SO(3) has an angle less than 30 degrees.
0
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1answer
47 views

Local Isomorphism on Lie Groups II

Yesterday I asked about an example in Chevalley's book "Theory of Lie Groups I". Well, the second example on pages 38 and 39 made me work a lot to understand. ...
3
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0answers
46 views

How do evaluate the trace of a member in $SU(n)$?

In Clifford Taubes' differential geometry(if you doubt I might be wrong in my transcription, please check section 5.6) , he seems to be claiming that ...
3
votes
2answers
403 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 ...
27
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10answers
5k 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 ...