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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Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
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0answers
15 views

Determining Matrix Group from Connected Component

I am interested in finding a method for determining all the matrix subgroups of a matrix group that have a specific connected component. This is what I thought would work from what I have read so far ...
3
votes
1answer
33 views

How to obtain a Lie algebra homomorphism from a Lie group homomorphism

In class we learn a theorem tells us one can cook up a Lie algebra from a Lie group: If $f: G\to H$ is a homomorphism of a Lie group then $T_I f: T_I G\to T_I H$ is a homomorphism of Lie algebra. ...
9
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1answer
75 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 ...
2
votes
2answers
47 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
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1answer
25 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
4
votes
1answer
45 views

Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
3
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2answers
63 views

Differential of the inversion of Lie group

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
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1answer
26 views

Lie algebra for SO(3) as a skew symmetric matrix

How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
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45 views

Embedded Lie subgroups are closed.

This is Exercise 2.1 from Kirillov's Lie theory book. Let $G$ be a Lie group and $H$ a closed Lie subgroup. Show that the closure $\overline{H}$ of $H$ in $G$ is closed in $G$. Show ...
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24 views

What do the right and left translation maps on $GL_n(R)$ look like?

Yesterday I asked this question: What does $GL_n(R)$ look like? (And got some good answers.) Since I would like to have some models of noncommutative lie groups in my head, I was wondering if anyone ...
3
votes
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117 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 ...
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vote
1answer
12 views

Commutative lie groups - how is $(R, >, 1)$ a $T^q \times R^p$

I just found out that the connected, commutative lie groups are all products of the form $T^q \times R^p$, where T is the circle and R the real numbers. Is the set of positive reals under ...
4
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57 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 ...
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43 views

SAGE vs. Mathematica for Lie algebras / groups?

What math software is better for working with Lie algebras and Lie groups, SAGE or Mathematica?
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1answer
30 views

Implied relationships between Lie groups and Lie algebras.

Suppose $\mathcal{L}$ is a finite-dimensional Lie algebra, and $\mathcal{G} = e^{\mathcal{L}}$ is it's compact, connected Lie group. Given a closed sub-algebra $\mathcal{L}' \subset \mathcal{L}$, it ...
2
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0answers
33 views

How to calculate the Maurer-Cartan form in the adjoint representation?

While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as $[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$ with $G_a\in\mathfrak g$. Under the ...
2
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2answers
44 views

Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
4
votes
1answer
97 views

Almost all subgroups of a Lie group are free

I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions. Let $w\colon G\to H$ be an analytic mapping between ...
0
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0answers
15 views

Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
3
votes
0answers
31 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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0answers
18 views

Affine connection defined by a quotient manifold?

Suppose $G$ is a Lie group with affine connection $X,Y \mapsto\nabla_X Y\in C^{\infty}(G,TG)$, and $Q$ is a subgroup of $G$ such that $G/Q$ is also a nontrivial Lie group. Does this quotient manifold ...
3
votes
0answers
12 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
3
votes
1answer
50 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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Campbell Baker Hausdorff formula for exp(X+Y)exp(X-Y)

Original Question: Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\exp(X)\exp(Y)$ (wiki), what is the corresponding formula for $\exp(X+Y)\exp(X-Y)$? The main ...
5
votes
1answer
59 views

why can i differentiate this term-by-term?

What's the best way to justify the following computation: For $A, B$ symmetric real matrices, $$\frac{d}{dt}|_{t=0}e^{A+tB}= \frac{d}{dt}|_{t=0}(1+(A+tB)+\frac{1}{2!}(A+tB)^2+...) = ...
4
votes
1answer
58 views

Looking for proof that $SO(3)$ is a submanifold of $\mathbb R^3$

It seems to be taken for granted in all sources that $SO(3)$ is a submanifold of $\mathbb R^9$. However, the one proof of this that I have been able to find has a step or two that doesn't make alot ...
2
votes
1answer
44 views

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 ...
2
votes
2answers
55 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, ...
2
votes
1answer
37 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 $2$. ...
0
votes
0answers
42 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
2
votes
1answer
45 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 $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
2
votes
0answers
33 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 ...
2
votes
1answer
102 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 ...
1
vote
1answer
38 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 ...
0
votes
0answers
26 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 ...
2
votes
0answers
85 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
26 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' ...
3
votes
1answer
46 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 ...
0
votes
1answer
46 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 ...
0
votes
2answers
94 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 ...
1
vote
1answer
68 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 ...
2
votes
1answer
87 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 ...
4
votes
2answers
67 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 ...
1
vote
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
votes
1answer
30 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
votes
1answer
44 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 ...
0
votes
0answers
27 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 ...
2
votes
2answers
66 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 ...
0
votes
1answer
19 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\\ ...