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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24 views

lie group and lie algebra: confusion about the tangent space

I am reading a document about performing 2D and 3D transformations in space specifically rigid body transformations which can be represented using Lie groups. For example, the rigid body ...
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1answer
25 views

Confused about classification of simple lie algebras

In the classification of simple lie algebra, I learnt that $\mathfrak{sl}(n + 1)$ has a root system $A_n$. For example http://stacky.net/files/written/LieGroups/LieGroups.pdf, page 82. But I found ...
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1answer
47 views

Compact Lie group with non discrete center?

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?
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1answer
42 views

Tensor product over Lie group isomorphic to that over its Lie algebra

Hi: I have a question about the tensor product of representation of Lie groups as follows: Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are ...
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30 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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102 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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63 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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1answer
103 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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1answer
36 views

Periodic geodesics in special unitary groups

Looking at the solution of the equation of motion for the action using a bi-invariant metric on a Lie group. If I take the lie group as $SU(2)$ the geodesics are always periodic because $SU(2)$ is the ...
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1answer
45 views

Is my proof that $\operatorname{Isom_n}{\mathbb R^n}$ is compact right?

I previously did this exercise: Prove that $\operatorname{Isom_n}{\mathbb R^n}$ is a matrix group. where $$\operatorname{Isom_n}{\mathbb R^n} = \left \{ \left ( \begin{array}{cc} A & ...
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1answer
43 views

The orthogonal group $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$

I tried to do the following exercise: Prove that $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$ The definition of $\mathcal O_n (\mathbb K)$ is ...
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29 views

Could someone check my work on why $\mathrm{Aff}$ is a matrix group

I solved the following exercise, please could someone check my work? Prove that $\operatorname{Aff_n}{(\mathbb K)} \subseteq GL_{n+1}(\mathbb K)$ is a matrix group where $$ ...
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1answer
30 views

Why are root spaces of root decomposition of semisimple Lie algebra 1 dimensional?

I'm trying to understand root system of semisimple Lie algebra but having trouble following one of the step in the note which explain why each root spaces are 1-dimensional. According to the note, ...
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32 views

on isometric group

Let $G_n$ be a Lie group, $g$ denotes the left invariant Riemannian metric on $G$. I want to ask for help that how to prove this conclusion: if all principal Ricci curvature of $(G_n, g)$ are ...
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1answer
71 views

Help understanding the argument used in this proof that $SO(n)$ is path-connected

Background: I am reading Tapp's matrix groups for undergraduates and I am in the process of showing that $SO(3)$ is path-connected. While working on my own arguemnt I found a proof online in these ...
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1answer
119 views

Algebra of matrix coefficient over a compact group is isomorphic to its dual.

Let $K$ be a compact group. Then we have the following definition of matrix coefficient: Definition: $f: K \rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite dimensional ...
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2answers
69 views

Need help with proof of $SO(3)$ is path connected

I am working on an exercise in Tapp's matrix groups for undergraduates. It is a proof that $SO(3)$ is path-connected. $SO(3)$ is the group $$ SO(3) = \{A \in O(n)\mid \det A = 1 \}$$ where $O(n)$ is ...
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1answer
37 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 ...
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2answers
68 views

Shouldn't this be “injective” rather than “well-defined”?

Consider the top of page 6 here: "...since different motions might place the globe in the same position think about why this group operation is well-defined..." He is talking about the group of ...
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1answer
60 views

Help needed in understanding the basics of Cartan decomposition of a Lie algebra

I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example. Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. ...
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1answer
23 views

Dimension of image of Lie bracket

Is there a method to calculate the dimension of the set of vectors in $\mathfrak{su}(n)$ $\{\ [A,B] \ \text{s.t} \ B \in \mathfrak{su}(n)\}$ for some fixed $A$. Is the dimension the same for all $A$?
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21 views

Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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1answer
44 views

Proving a relation for representations of gauge groups

Let ${\cal G}$ be a Lie group - possibly disconnected. Let ${\mathfrak g}$ denote the corresponding Lie algebra. Let $R_k$ be a general unitary representation of ${\cal G}$ and $R$ be the adjoint ...
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1answer
29 views

$B$-action on $U$.

Let $G$ be an algebraic, $B$ Borel subgroup, and $U$ unipoent subgroup of $G$. For example, we take $G=GL_n$, $B$ the subgroup of lower triangular matrices, and $U$ unipoent upper triangular matrices. ...
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1answer
47 views

What's wrong with this trivial proof that every element of a compact Lie group is contained in a maximal torus?

The Lie groups book I'm reading (Knapp, Lie Groups Beyond an Introduction, page 255) goes to some trouble to prove that every element of a compact Lie group is contained in a maximal torus. Why isn't ...
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1answer
84 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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87 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
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1answer
34 views

These subsets of $O(n)$ are clopen

Please could someone check my work on this exercise (from a book I am reading). Thanks! Exercise: Prove that $SO(n)$ and $ O(n)^- = \{ A \in O(n) \mid \det(A) = -1 \}$ are both clopen in $O(n)$. My ...
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44 views

Proof that $GL_n, SL_n$ are not bounded

Please could someone check my work on this exercise (from book I am reading). Thanks! Exercise: Prove that $GL_n (\mathbb K)$ is non-compact when $n \ge 1$. Prove that $SL_n (\mathbb K)$ is ...
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1answer
64 views

representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
4
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1answer
31 views

How to check if a representation of su(2) is irreducible

I have found a representation $\rho$ of the group $G=Su(2)$. I want to show that this representation is irreducible but I don't know how. Finding all invariant subspaces seems very difficult. I ...
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1answer
28 views

Confusion about change of basis matrix

This video here seems to suggest that if a vector $v = (c_1, \dots, c_n)$ is given with coordinates in some basis $b_1, \dots, b_n$ and $B$ is the matrix with columns $b_1, \dots, b_n$ then $Bv$ is ...
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1answer
47 views

Open (but not closed) subgroups of $GL_n$

The book I am currently reading states: "...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds." Prompted me to think about open subgroups of $GL_n$: ...
3
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32 views

Sets of orthogonal matrices are bounded

I have already shown that $O(n), SO(n), U(n), SU(n)$ and $Sp(n)$ are closed. Now I want to show that they are bounded. But when I tried, I noticed I need a metric or a norm on these sets. But there ...
2
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2answers
70 views

permutation group, lie group

Let $S$ be any set, and denote by $G$ the collection of all subsets of $S$. For $A, B \in G$ let be $AB = (A - B) \cup (B - A)$. I know how to show that this set $G$, with this product operation is a ...
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0answers
38 views

Heisenberg group as a suspension

I'm working on Heisenberg group and I want to understand the suspension viewpoint. Let me be more precise. Let us denote by $\mathbb{H}^3(A)$ the set of matrix \begin{equation} \begin{pmatrix} 1 ...
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34 views

Decomposiom of the representation of $SU(N)$

Let $T$ be the "fundamental" representation (I mean the one in which the matrices representing the group elements are simply themselves) of $SU(N)$ group. I have \begin{pmatrix} SU(N-1)& 0\\ ...
3
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1answer
41 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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1answer
43 views

Soft question: A good book for introduction to Lie group book

I am taking next semester introduction lie groups. I was wondering what do you guys think what book should I use for this course.
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1answer
18 views

Dense and integral zero.

Let $G$ be a compact Lie group and $u\in C^{0}\left(G\right) $. If $\int_{G} u\left( g \right)v \left(g \right)dg= 0$ for every $v\in V $, a subset which is dense in $C^{0}\left(G\right)$, then ...
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1answer
45 views

Are there “interesting” examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?

Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a ...
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1answer
74 views

Could someone check my work on this exercise

I solved the following exercise, could someone please check my work? Exercise: Let $$ A = \left ( \begin{array}{cccc} 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 ...
2
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2answers
45 views

Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?

An exercise in a book I'm reading is to show that $O(2)$ is not isomorphic to $SO(2)\times \{-1,1\}$. The problem is, I don't believe the statement. Let me elaborate why: $O(2)$ consists of ...
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2answers
31 views

Isn't this $f$ always a group isomorphism

Consider the following exercise from a book I'm reading: If $n$ is odd show that $$ f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$ is an ...
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2answers
23 views

Product of two arbitrary elements in $O(2)$

An exercise in a book I'm reading asks to describe the product of two arbitrary elements in $O(2)$. I would like to solve the exercise but I got stuck. I know that an element in $SO(2)$ can be written ...
3
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1answer
68 views

Induced Lie group action on a tangent bundle $TG\times TM\to TM$ and an example concerning Adjoint action

Suppose I have a Lie group action $$ G\times M\to M, (g,m)\mapsto g\cdot m. $$ which is transitive on $M$, then the tangent functor $T$ induces a corresponding map: $$ TG\times TM\to TM, (\delta ...
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26 views

What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points)

I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ...
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0answers
14 views

Topology of orthogonal groups when $n > 4$?

On Wikipedia I read that the topologies of $O(1)$ and $SO(1)$ to $SO(4)$ are known topological spaces. What about $O(n), SO(n), U(n)$ when $n>4$?
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1answer
34 views

Proof help: $SU(2)$ is a double cover of $SO(3)$

I am reading a proof that $SU(2)$ is a double cover of $SO(3)$. My source is this set of notes: http://www.damtp.cam.ac.uk/user/examples/D18S.pdf. The proof begins near the bottom of page 4. I have ...
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2answers
53 views

How can I show that these matrices don't commute

I want to show that $A\in O(2) \setminus SO(2)$ and $B \in SO(2)$ don't commute. To prove it I wrote $$ B = \left ( \begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta ...