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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5
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1answer
50 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 ...
3
votes
1answer
35 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 ...
1
vote
0answers
11 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 ...
0
votes
0answers
17 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. ...
2
votes
1answer
32 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 ...
3
votes
2answers
372 views

Calculating the lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$ where this is defined as: $SO(2,1=\{X\in Mat_3(\mathbb{R})|X^t\eta X=\eta, \det(X)=1\}$ where $\eta$ is the matrix defined as: $$\left ...
4
votes
1answer
84 views

One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear. It is easy to see that a real matrix is complex linear if ...
0
votes
0answers
21 views

Prove that ${\frac{d}{{dt}}_{t = 0}}(\det ({e^{tX}})) = Tr(X)?$ [on hold]

How to prove $${\frac{d}{{dt}}_{t = 0}}(\det ({e^{tX}})) = Tr(X)?$$
8
votes
1answer
60 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 $A$. ...
0
votes
1answer
118 views

Basic Lie Algebra Question

Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector ...
15
votes
1answer
126 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
4
votes
0answers
44 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 ...
3
votes
1answer
29 views

Need some help with this exercise: why $\operatorname{Aff_n}$ is not closed

I ran into trouble while working on this exercise: Prove that $\operatorname{Aff_n}{(\mathbb K)} \subseteq GL_{n+1}(\mathbb K)$ is a matrix group where $$ \operatorname{Aff_n}{(\mathbb K)} = \left ...
0
votes
1answer
32 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 ...
3
votes
0answers
42 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 ...
6
votes
2answers
246 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 $\mathbb R^3$. ...
4
votes
1answer
29 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 ...
10
votes
2answers
59 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
1
vote
1answer
44 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 $GL_n$: ...
0
votes
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 ...
3
votes
0answers
29 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 ...
1
vote
1answer
563 views

Compactness of orthogonal Matrices

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.
3
votes
1answer
61 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 ...
3
votes
1answer
66 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
0answers
25 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 ...
4
votes
2answers
61 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 ...
2
votes
2answers
199 views

Proof that this diagram commutes

This is an exercise in a book I'm reading: Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ ...
1
vote
0answers
34 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 ...
2
votes
0answers
29 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\\ ...
1
vote
1answer
70 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 ...
4
votes
2answers
69 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
8
votes
1answer
160 views

One parameter subgroup that leaves every compact set is a proper map

If a one parameter subgroup $\phi:\mathbb{R}\rightarrow G$ of a Lie group $G$ comes back infinitely often to a compact set $K$, is it all contained in a compact set? I think $\phi(\mathbb{R})K\subset ...
3
votes
2answers
293 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
3
votes
1answer
39 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.
0
votes
1answer
37 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 ...
1
vote
1answer
17 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 ...
7
votes
2answers
82 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
-2
votes
0answers
60 views

Show that G is a Lie group and find a adjoint representation for G

$$G = \{ A \in GL(2,R): AA^t = p^2I, p>0, \det A >0\}$$ Show that G is a Lie group and find the explicity expression for their elements. And find a adjoint representation for G. Hi, I tried to ...
2
votes
2answers
41 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 ...
0
votes
2answers
30 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 ...
2
votes
2answers
20 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 ...
0
votes
0answers
14 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 ...
1
vote
2answers
50 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 ...
3
votes
1answer
49 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
3
votes
1answer
30 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 ...
0
votes
0answers
12 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 $n>4$?
1
vote
1answer
22 views

How can I show that this matrix is a flip

I am trying to show that $$ F = \left(\begin{array}{cc} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array} \right ) $$ is a flip about a line through the origin. What I ...
0
votes
0answers
15 views

Decomposition of SU(n) anticommutator

In $SU(N)$, the special unitary group, the algebra generators $T_a$ are hermitian and traceless. The structure constants are fixed with $[T_a,T_b]=i f_{abc}T_c$. In the fundamental representation of ...
2
votes
0answers
22 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
2
votes
1answer
96 views

Finding $J$ such that this diagram commutes

DISCLAIMER: This is not homework. I did this exercise here and need someone to check if my work is correct: Is it possible to find a matrix $J\in M_{2n}(\mathbb C)$ such that the following diagram ...