Tagged Questions
0
votes
1answer
33 views
Invariant subspaces of tensor product of SU(2)
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
I know that irreducible representations of $\varphi_2 \otimes \varphi_3 = \varphi_5 \oplus ...
1
vote
0answers
17 views
Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)
This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups.
More precisely, given discrete groups below (a), (b), (c):
...
2
votes
1answer
31 views
Irreducible representation of tensor product
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?
3
votes
0answers
53 views
Discrete subgroups of SU(n) and SO(n).
Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.
I would like to know whether there is a complete understanding of discrete ...
0
votes
0answers
23 views
Conjugacy classes for su(2)
I am wondering how to calculate the conjugacy classes of the Lie algebra su(2). My guess is that they can be easily evaluated under the similarity transformations but I am not sure it that is all to ...
2
votes
1answer
44 views
Show that the SU(2) group is a Lie group
How can I prove that the SU(2) is a Lie group with the Pauli matrices as generators?
4
votes
1answer
86 views
How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?
Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
4
votes
1answer
22 views
Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.
I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
2
votes
0answers
28 views
To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?
In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
4
votes
1answer
46 views
Analogues of $SU(2)$ and $SO(3)$
The groups $SU_2(\mathbb{C})$ and $SO_3(\mathbb{R})$ are interesting in geometry, and there is a $2$-to-$1$ map from $SU_2(\mathbb{C})$ to $SO_3(\mathbb{R})$. There are finitely many finite groups in ...
1
vote
1answer
37 views
Isomorphisms of the Lorentz group and algebra
I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
0
votes
0answers
16 views
concerning coadjoint representation
Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
0
votes
1answer
41 views
Projective linear group - solvable
Let $q\geq 5$ and let PGL(2,q) be the projective general linear group.
Question
Do there exists a $q$ such that PGL(2,q) is solvable?
1
vote
0answers
30 views
Usage and determination of “rank” and “dimension” of groups & representations
Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately.
A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
2
votes
0answers
37 views
Finding the dimension of the symplectic group
How do you find the dimension of the symplectic group $Sp(2n,\mathbb{R})$?
$Sp(2n,\mathbb{R})\subset Gl(2n,\mathbb{R})$ is the group of invertible matrices $A$ such that $\omega = A^T\omega A$, where ...
3
votes
1answer
83 views
Proof that $U(n)$ is connected
I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ ...
3
votes
0answers
70 views
How many discrete subgroups does the Heisenberg group have?
Is there an easy way to describe an arbitrary discrete group in the Heisenberg group?
I figured that at least the family
$$
\begin{pmatrix}
1 & x\mathbb Z & z\mathbb Z\\0&1&y\mathbb ...
1
vote
1answer
43 views
Sp(1) and SO(3) Locally Isomorphic - Trouble with part of proof
My apologies if this is fairly basic. I'm trying to understand the proof that $Sp(1)$ and $SO(3)$ are locally isomorphic but are not isomorphic but have run into some trouble. Here's what my ...
1
vote
0answers
16 views
How to describe spatial tranformations that contain singularities?
I am a researcher in the field of medical image processing. A popular topic in this field is the search for spatial transformations that align two images from two different patients in order to find ...
3
votes
2answers
50 views
Action of $\text{SL}_2\mathbb{C}$ on $\mathbb{C}^3$ induces a 2:1 covering $\text{SL}_2\mathbb{C}\to \text{SO}_3\mathbb{C}$
Exercise 7.17 in Fulton's Representation Theory reads,
Identify $\mathbb{C}^3$ with the space of traceless matrices in $M_2\mathbb{C}$ so that $g\in \text{SL}_2\mathbb{C}$ acts by $$A\mapsto ...
2
votes
0answers
55 views
Lie Theory in Finite Groups
Currently, I have interest in Finite groups. I also want to learn Lie Groups, Lie algebras, and their representations. But I do not have any motivation for it.
Question What are simple but ...
2
votes
1answer
50 views
Frattini and Derived Subgroup of $SL(n,\mathbb{Z})$
It is known that the derived subgroup of $SL(2,\mathbb{Z})$ is subgroup of index 12 in the group.
1. What is known about derived subgroup of $SL(n,\mathbb{Z})$, for $n\geq 3$? (is it finitely ...
3
votes
0answers
28 views
Fundamental group of $Spin^c(2)$?
Is the fundamental group of $Spin^c(2)$, the second complex spin group, also $\mathbb{Z}$?
If so, how does one see this?
Just to avoid any confusion, my definition is:
$$Spin^c(2) = (SO(2) \times ...
1
vote
1answer
76 views
Determining which maps are isomorphisms
1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$.
My solution:
1-1: Suppose ...
2
votes
0answers
58 views
Conditions for a group to admit the structure of a Lie group
This question is motivated by a previous one:
Conditions for a smooth manifold to admit the structure of a Lie group
and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
2
votes
1answer
133 views
proof by nuke of the fact that fundamental group of topological group is abelian
"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
6
votes
2answers
103 views
An example of a Lie group
I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a 2D blanket or a circle/curve or a ...
2
votes
1answer
33 views
How can I show that $ASL_n(F)$ is acting 2-transitively?
One of my friends asked me to ask this question here. This is a question from his last exam:
Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
1
vote
1answer
67 views
Quotient group $S^3/\{+I,-I\}$
How can I prove that the quotient group $S^3/\{+I,-I\}$ is isomorphic to $SO_3$ and that the group $S^3$ is not isomophic to $SO_3$?
Here $S^3$ is the subgroup of the quaternion group: ...
1
vote
1answer
32 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
votes
1answer
223 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 ...
1
vote
2answers
121 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 ...
6
votes
2answers
220 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, $\text{GL}_ n(\mathbb{R})$?
6
votes
2answers
78 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, ...
8
votes
2answers
134 views
Relation between $SU(4)$ and $SO(6)$
This is more of a particle physics question than maths. Since $SO(6)$ and $SU(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) ...
5
votes
0answers
124 views
Generating function for characters of representations
One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
4
votes
2answers
177 views
Subgroups of $\Bbb{R}^n$ that are closed and discrete
I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$.
Base ...
5
votes
3answers
495 views
how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$
Could any one give me hint for this one?
how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$,
well, Is it the same: there is a 2-fold covering map from $SU(2)$ to $SO(3)$?
what is that map will be?
2
votes
1answer
110 views
What is an “invariant form” of a group?
I have often seen this phrase used in at least two frequent contexts,
One uses the notation of $\omega_{AB}$ (the matrix $\{ [0 , I],[-I,0]\}$) to denote the symplectic form for $USp$ group.
One ...
3
votes
1answer
294 views
Some questions about representations of $SO(6)$
I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$,
What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
4
votes
1answer
191 views
Fundamental and the anti-fundamental representation of $U(n)$
I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate.
...
1
vote
2answers
140 views
$SU(2)$ subgroups of $SU(3)$: Is my reasoning correct?
When looking at the standard $3\times3$ representation of $SU(3)$, one immediately recognizes some subgroups isomorphic to $SU(2)$:
There is the subgroup acting on the first two elements of the ...
1
vote
2answers
123 views
Antisymmetric powers of $SO(n)$ representation.
I am particularly interested for $SO(3)$.
Let us say that I start with the natural/defining $3$-real-dimensional vectorial representation of $SO(3)$ and I choose the generator of rotation in the ...
3
votes
2answers
125 views
Presentation of discrete upper triangular group
Let $G$ be the nilpotent Lie group consisting of matrices
$$
\begin{pmatrix}
1 & a_{12} & \cdots & a_{1,n}\\
0 & 1 & \ddots & \vdots\\
\vdots & \ddots & \ddots & ...
2
votes
2answers
96 views
Is $\mathop{Hom}(O(1),\mathbb{C}^*)$ isomorphic to $\mathop{Hom}(\mathbb{C}^*,O(1))$?
I have an elementary question on homomorphisms.
Let $O(1)=\{A: A^t A=1 \}$ and let $\mathbb{C}^*= \{ z\in\mathbb{C}:z\not=0\}$.
Then what is the character group ...
3
votes
3answers
136 views
the transformation which rotates a matrix by a half turn
Consider
$$T_{2}: \left[ \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right] \rightarrow
\left[ \begin{array}{cc}
d & c \\
b & a \\
\end{array} \right]
$$
$$T_{3}: \left[ ...
0
votes
1answer
124 views
What transforms under SU(2) as a matrix under SO(3)?
A vector $\boldsymbol{r}$ in $\mathbb{R}^3$ transforms under rotation $\boldsymbol{A}$ to $\boldsymbol{r}'=\boldsymbol{Ar}$. It is equivalent to an SU(2) "rotation" as
$$\left( ...
4
votes
3answers
108 views
When does the Commensurator of a subgroup of a group $G$ not equal $G$?
Let $H\leq G$ be two groups. I'm interested in the Commensurator $$\mathrm{comm}_G(H)=\{g\in G : gHg^{-1} \cap H \text{ has finite index in both}\}.$$
Obviously, $\mathrm{comm}_G(H)\leq G$. I read on ...
2
votes
2answers
105 views
Is this a one dimensional Lorentz Boost? And can you have a 1-d Boost without group structure?
Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost.
His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( ...
0
votes
2answers
178 views
Is this proof that SU(2) cannot be isomorphic to SO(1,3) valid?
It seems intuitively obvious to me that there cannot be an isomorphism between $\mathrm{SU}(2)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ where SU(2) is the Lie Group with the Pauli matrices as ...
