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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Riemannian symmetric pair $(G,H)$ with H non-compact

Let $G$ denote a connected Lie group and $H$ a closed subgroup. Suppose that $\sigma$ is an involutive automorphism of $G$. Assume that $(G,H,\sigma)$ is a Riemannian symmetric pair. So far I have ...
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36 views

How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
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12 views

The Weyl group of $E_6$ acting on embedded circles

I want to know the number of components of the normalizer of an arbitrary circle subgroup $S$ of (the compact real form of) the exceptional Lie group $E_6$. This number will always be $1$ or $2$. ...
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1answer
35 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
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26 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 ...
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1answer
80 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
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29 views

$ L^{2} $ convergence should imply convergence in infinity norm

My situation is the following: suppose we have a Lie group $ G=G_{1} \times G_{2} $ and let $ X = \Gamma \backslash G $ a homogeneous space arising from a lattice, i.e. we have a $ G $ invariant ...
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1answer
35 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
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23 views

Homeomorphism between SU(4) and SO(6)

http://www.mat.univie.ac.at/~westra/so3su2.pdf said that $\mathrm{SU}(2)$ acts homeomorphism to $\mathrm{SO}(3)$, via $$ \begin{pmatrix} z & w \\ -\bar w & \bar z \end{pmatrix} \mapsto ...
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76 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 ...
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1answer
22 views

Scalar multiple of one lattice contained in another

I believe my question boils down to the following question: Given lattices $L$ and $L'$ in $k^{n}$, does there exist $\lambda \in k^{\times}$ so that $\lambda L' \subseteq L$ and $\lambda L' ...
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1answer
35 views

Textbook literature on Lie groups

I'm a student that wants to get to know Lie groups. I know a bit about manifolds and a bit about groups, but nothing about topological groups or such things. Can you suggest a textbook that covers the ...
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37 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
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2answers
71 views

Why is $\mathfrak{sl}(n)$ the algebra of traceless matrices?

I'm studying Lie algebras as purely algebraic objects, without much of a background in the differential geometry surrounding Lie groups. The definition of $\mathfrak{sl}(n)$ has been given to me as ...
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1answer
65 views

Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group?

$\DeclareMathOperator{\isom}{Isom}$A discrete subgroup of the group of isometries in euclidean space is almost abelian. By this I mean that for each $n$ there exists $m$ such that for any discrete ...
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1answer
75 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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56 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
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1answer
43 views

Non-standard complex structure on $\mathbb R^{2n}$

First let me give some relevant information: For every $n$ every subgroup of $GL_n(\mathbb C)$ is isomorphic to a subgroup of $GL_m(\mathbb R)$ for some $m$. Let $\rho_n: M_n(\mathbb C) ...
2
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1answer
28 views

The dimensions for Lie Groups

How can I find out which is the dimension for $SU(n)$, $SO(3)$, etc? Can you explain me, please? thanks
2
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1answer
43 views

An $n$-dimensional subgroup of $GL_{n+1}(\mathbb R)$

Could somebody please tell me if my answer to the following exercise is correct?: Describe a subgroup of $GL_{n+1}(\mathbb R)$ that is isomorphic to $\mathbb R^n$ under vector addition. It's ...
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0answers
25 views

Measure in dual group - Kirillov theory

Let $G$ be a nilpotent connected, simply connected lie group. With the orbit method Kirillov describes the classes of equivalence of all irreducible unitary representations. Hence one identifies the ...
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2answers
55 views

The Weyl group of A_3

Could someone please list all elements of the Weyl group of the root system $A_3$ in terms of simple reflections. In this case the Weyl group is $S_4$. Its strange that GAP failed to list all elements ...
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1answer
11 views

Computing a centralizer in an orthogonal group

Consider a skew-symmetric $(4n+2) \times (4n+2)$ block-diagonal real matrix $A$ in normal form: $$A = \begin{bmatrix} \Lambda_1 & 0 & \cdots & 0\\ 0& \Lambda_2 & \cdots & 0\\ ...
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1answer
34 views

Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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1answer
36 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
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20 views

Set of quaternions that anti-commute

I tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $x,y \in \mathbb H$ such that $xy = -yx$. ...
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23 views

Why does the exceptional Lie group $G_2$ have dimension 14?

In ''Compact manifolds with special holonomy" by D. Joyce, on p. 242, the group $G_2$ is defined to be the subgroup of $GL(7,R)$ preserving the $3$-form: $$ \varphi_0 := dx_{123} + dx_{145} + ...
3
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1answer
44 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
0
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0answers
11 views

Integrating over a flag manifold

I need to calculate an integral over the flag manifold $U(4)/U(1)\times U(1)\times U(1)\times U(1)=U(4)/T^4$. How can I derive the correct Haar measure to use?
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1answer
84 views

Linear algebraic group [closed]

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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35 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
3
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1answer
80 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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39 views

Frobenius subgroups of Sz(8)

Does the Suzuki group Sz(8) has any Frobenius subgroup except $D_{14}$, $F_{20}$ and $F_{52}$?
2
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31 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
3
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1answer
36 views

Subgroup of $GL_{n+m}(\mathbb K)$

Let $G$ be a subgroup of $GL_n(\mathbb K)$ and $H$ a subgroup of $GL_m(\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$. I want to prove that there exists a subgroup of ...
2
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0answers
24 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of DEs. I'm presently doing research on identifying financial indicators/phenomena that can be ...
3
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3answers
73 views

Why not $SL_n (\mathbb R)$ in this exercise

I just solved the following exercise: Let $SL_2(\mathbb Z)$ denote the set of $2\times2$ matrices with integer entries and determinant $1$. Prove that $SL_2(\mathbb Z)$ is a subgroup of ...
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0answers
25 views

$SO(n)$ is connected, alternative form

I have the following exercise: Show that $SO(n)$ is connected, using the following outline: For the case $n = 1$, there is nothing to show, since a $1\times 1$ matrix with determinant one must be ...
1
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1answer
32 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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0answers
21 views

Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
2
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1answer
43 views

Cohomology of classifying spaces of exceptional Lie groups

Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong ...
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0answers
28 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
4
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1answer
107 views

A question on the unit tangent bundle of the sphere and $SO(3)$

Let the unit tangent bundle be defined as follows: $$T^1S^2=\{(p,v)\in \mathbb R^3 \times \mathbb R^3 | |p|=|v|=1 \text{ and } p \bot v \}$$ Let $SO(3)$ be the group of rotations of $\mathbb R^3$. ...
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1answer
54 views

Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
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1answer
25 views

Induced Lie algebra homomorphism from Lie group homomorphism: left translation

A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$. Which Lie algebra homomorphism ...
7
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1answer
150 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
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1answer
13 views

Generalization of Euler angle decomposition for unitary operators?

it is well known that a general operator from SU(3) can be effectively represented as a composition of at most three "one-parameter" ones. For instance in the form $\exp(i\, \alpha\, \sigma_z) ...
2
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2answers
85 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
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1answer
32 views

Is a Lie group connected?

I want to know if a Lie group is connected in general situation.I also want some example of Lie group. I will appreciate your help.
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1answer
47 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...