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Questions tagged [lie-groups]

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 (group-...

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Question about the diffeomorphism group of a Lie group.

I was reading about the diffeomorphism group of varying Lie groups. Wikipedia states that: When M = G is a Lie group, there is a natural inclusion of G in its own diffeomorphism group via left-...
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Regarding the complete reducability of the Lorentz group

I was just reading that the Lorentz group has the property of complete reducability, that is any representation can be written as the direct sum of irreducible representations. This reminds me ...
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Equality of Lie-Cartan coordinates of the first kind and Lie-Cartan coordinates of the second kind

For a set of basis $(w_1,w_2,w_3) \text{ of linear space } \mathbb R^3$, define Lie-Cartan coordinates of the first kind: $$ R_1 = \exp(\alpha_1 \hat{w_1} + \alpha_2 \hat{w_2}+\alpha_3 \hat{w_3}). $...
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8 views

Probability that $N$ matrices from GUE(d) form a trivial centralizer

Take $N$ matrices $G_i$, $i=1,2,\ldots,N$ sampled from GUE(d) (that is $N, \, d\times d$ Hermitian matrices). What is the probability $p(N,d)$ that the centralizer of the subalgebra generated by such ...
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46 views

Examples of non-abelian simply connected nilpotent Lie groups.

I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
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48 views

Understanding Lie algebra of matrix Lie group

In my lecture, we gave a very sloppy (physics people ...) proof of the fact that the Lie algebra $\mathfrak{g}$ of a matrix Lie group $G$ is a subspace of $\text{Mat}_n(\mathbb{F})$. I am not ...
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21 views

Why does the exponential map have this form?

I believe I understand what the exponential map does, in informal terms. However I cannot relate this to the equation I see before me in the papers. The exponential mapping function for a symmetric ...
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29 views

Compactness of Lie group and integration

During studying integration on Lie group, I got stuck on one statement. If the Lie group $G$ is compact, then $$ \int_G \omega = \int_G L^*_g \omega. $$ where $L_g:G\rightarrow G$ defined by $...
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25 views

Prove that an integration is left invariant.

$G$: Lie group of dimension $n$. $\tilde{\Omega}$: Orientation on $G$. $\Omega=\epsilon^1\wedge \epsilon^2\wedge \cdots \wedge \epsilon^n$ where $\epsilon^1\, \epsilon^2, \cdots ,\epsilon^n$ is ...
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23 views

If the stabilizers are compact, the action is proper

Suppose that a Lie group $L$ acts on a space $X$ transitively. Assume also that the stabilizers are compact, say $K$. Then, we have that $$X \cong L /K.$$ Then, I want to prove that the action of $L$ ...
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11 views

Is the action of $SO(1,2)$ in null cone proper?

It is known that the action of the group $SO(1,2)$ on the null-cone inside the $3$-dimensional Minkowski space is free and transitive. However, is the action also proper?
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Confusion about quotient of the Lie group $\mathbb{S}^1$

I have read that given a Lie group $G$ and a closed subgroup $H$ then $G/H$ is a smooth manifold. I cannot explain though the following example: take as $G = \mathbb{S}^1$ and as $H =\{\pm 1\}$, $H$ ...
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1answer
40 views

Interpretation of the Lie algebra of a Matrix Lie group

I'm looking for an intuitive explanation of the meaning of the Lie algebra for a matrix Lie group from a differential geometry perspective. Right now, the procedure I've been following is using the ...
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1answer
23 views

'Constant' vector field on a Lie group is smooth

I have a really basic question which I am struggling to articulate formally in differential geometry. I have a Lie group $G$, and a tangent vector $v\in T_{1_G}(G)$. I want to claim that the vector ...
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1answer
53 views

Proof of Logarithm map formulae from $SO(3)$ to $\mathfrak {so}(3)$

According to exponential map, there also exist a logarithm map $$\log:SO(3) \to \mathfrak {so}(3).$$ Suppose a vector $t \in\mathfrak {so}(3)$ and $t=\|t\|w$, according to exponential map $$R = \cos\|...
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1answer
60 views

Yamabe's theorem proof

*I'm trying to make the proof of Yamabe's Theorem that says that an arcwise connected subgroup of a Lie group G is a Lie subgroup of G. I found the proof in Goto's article (https://www.ams.org/...
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1answer
31 views

How to block diagonalize a real skew-symmetric matrix of 3*3

Suppose $t = [t_1,t_2,t_3]^T\in \mathbb R^3,t \neq 0$. Then define $$t^{\land} = \begin{bmatrix} 0 & -t_3 & t_2 \\ t_3 & 0 & -t_1\\ ...
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43 views

Obtaining all structure constants of $ \mathfrak{su}(N) $

I want to understand certain properties of general curves $\textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-...
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14 views

Projective representations of Orthogonal groups

Does every orthogonal, continuous group have a projective representation? Also does SO(3) have any lower dimensional non-projective representation?
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1answer
22 views

Are the SO(3) defining representation and Spin 1 the same?

Since the equation $$\langle j,m^\prime|J_\pm|j,m\rangle =\sqrt{(j\mp m)(j\pm m+1)} \delta_{m^\prime,m+1}$$ holds both for spin 1 (the $\underline{1}$ rep for SU(2)) and angular momentum (SO(3)). Does ...
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1answer
33 views

Homomorphism from $p$-adic to $l$-adic groups

I have seen and heard the statement that the $p$-adic and $l$-adic topologies are incompatible. I would appreciate a proof or references supporting this statement. More precisely, I am interested in ...
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47 views

Subgroup generated by union of two maximal compact subgroups of $GL_2(\mathbb{Q}_p)$

Let us denote by $G:= GL_2(\mathbb{Q}_p), G_0:= GL_2(\mathbb{Z}_p), g:= \begin{bmatrix}0 & 1\\p& 0\end{bmatrix}$ and by $G_1:= g G_0 g^{-1}$. I want to know if we have a good description of ...
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1answer
36 views

Metric on $M/G$ which makes $\pi: M\to M/G$ a Riemannian submersion

Let $G$ be a Lie group acting isometrically, freely and properly on a Riemannian manifold $(M,g)$ and $\pi:M\to N:=M/G$ the natural projection. Show that there is a unique metric $h$ on $N$ which ...
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Upper-triangular subgroup is not unimodular

Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$. My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not ...
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2answers
29 views

Moving frame method with non-matrix Lie group

I am trying to understand the modern formulation of the moving frame method for Lie group acting on a manifold. I know the following theorem Let be $M$ a manifold, $G$ a Lie group and $\omega$ the ...
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22 views

Reference for the relation of the Casimir element to the Laplace Beltrami operator

Wikipedia says, "If $G$ is a Lie group with Lie algebra $\mathfrak {g}$, the choice of an invariant bilinear form on $\mathfrak {g}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. ...
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1answer
38 views

Continuous function $ f:\operatorname{SO}(3) \to \operatorname{SU}(2)$

Given the usual surjective homomorphism $ Φ:\operatorname{SU}(2)\to \operatorname{SO}(3)$ that maps a quaternion to a rotation matrix, Does there exist a continuous function $f:\operatorname{SO}(3)\...
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17 views

$\rho$-shift in parabolically induced representations

In order to define the principal series representations one takes an irreducible, unitary representation $\sigma$ of $M$ (here $G$ is a semisimple Lie group with Iwasawa decomposition $G=KAN$ and $M$ ...
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1answer
28 views

Why must these have integer coefficients?

We are considering diagonal subgroup of classical groups and their lie algebras. We then consider $l=a_1l_1 + a_2l_2 + ...$ where $l_i(H)$, H in the lie algebra, returns the ith entry of H. We then ...
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15 views

Adjoint orbits of semisimple Lie algebras

Let $\mathfrak g$ be a semi-simple (complex) Lie algebra and let $G$ be a Lie group with Lie algebra $\mathfrak g$. Denote by $\mathcal O$ the orbit of an element $H \in \mathfrak h$ under the adjoint ...
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1answer
57 views

Why special unitary group has the dimension $n^2-1$? [closed]

I’m studying the examples of Lie group.. there is special unitary group.. it is written the dimension of $SU (n)$ is $n^2-1$.. it is given as $\det (A)=1$ , so it is last dimensional we say dimension ...
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94 views

Choice of volume forms in the Weyl Integration Formula

The following proof is adapted from Bump's Lie Groups. I've tried to rewrite the parts I find unclear. However, I seem to end up with a discrepancy. Please help me complete the proof or point out any ...
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1answer
34 views

Does the Lie group $G_2$ contain any normal subgroups?

A nonabelian Lie group is called a simple Lie group if it contains no nontrivial connected normal subgroups. On the other hand, a group is called simple if it contains no nontrivial normal subgroups. ...
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Calculate the generators of symmetries from uniform sampling of commutant of a matrix

Let $X\in \mathbb{R}^{d\times d}$ a matrix. Suppose I can uniformly sample (at random) the comutant set of $X$, i.e. the set: $$ \mathcal{C}=\{S\in \mathbb{R}^{d\times d},\;s.t.,\;[X,S]=0\}. $$ ...
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Uniform grid of points on SO(3)

I would like to place $N$ equally spaced points on $SO(3)$ (or approximately equally spaced points). Here, every "point" refers to a rotation, and $N$ is user-specified. I came across the Fibonacci ...
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35 views

When can compact group actions be complexified?

Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the ...
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1answer
19 views

Implications of a local isomorphism on discrete subgroups and Kazhdan property (T)

In the book "Discrete groups, expanding graphs and invariant measures" by Alexander Lubotzky, page 37, the author says that all finitely generated discrete Kazhdan subgroups of $SO(3)$ are finite, and ...
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10 views

$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
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27 views

Elements close to the identity in real Lie groups

I am having trouble with the following. Let $G=\mathrm{SL}(2,\mathbb{R})$. Let $\Gamma<G$ be a non co-compact (arithmetic) lattice. Is it true that there exists $\epsilon>0$ such that if $\{u_1,...
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1answer
24 views

Existence of one parameter subgroups in homotopie classes of Lie groups

Let $G$ be a Lie group and $\alpha: [0,1] \to G$ a smooth path, connecting the neutral element $n_G$ of the group with a group element $g$, i.e. $\alpha(0)=n_G$ and $\alpha(1)=g$. Can we find a ...
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1answer
20 views

Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
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12 views

Branching of $SU\left(5\right)$ under $SU\left(2\right)\times SU\left(3\right)$

In the context of branching rules, what is the meaning of the projection matrix, and how/what do I use it for? For instance, for the branching of $SU\left(5\right)$ under $SU\left(2\right)\times SU\...
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17 views

What is the weight system for these $su\left(5\right)$ representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The 5 representation is ...
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18 views

General question of Lie groups results

I've been trying to self study Lie groups and right now i have a question about the general theory, so far i have seen that For example, in Peter's book (Applications of Lie Groups to Differential ...
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1answer
28 views

Subgroups of $E_8$ by using extended Dynkin diagrams

I need to show that the following are subgroups of $E_8$ using extended Dynkin diagrams. $$SU\left(5\right)\times SU\left(5\right)$$ $$SU\left(3\right)\times E_6$$ $$SU\left(4\right)\times SO\left(...
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1answer
33 views

Ergodic action of dense subgroup

Let $G$ be a group acting ercodically on a probability measure space $(X, \mu)$. Let $\Gamma$ be a countable dense subgroup of $G$. Is the action of $\Gamma$ also ergodic? The case I am interested in ...
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1answer
59 views

Do every two orthogonal matrices in $\text{SO}(n)$ lie in the same coset of $\text{SO}(2)$?

Let $A,B \in \text{SO}(n)$. Does there exist a homomorphism of Lie groups $\phi:\text{SO}(2) \to \text{SO}(n)$, such that $A,B$ lie in the same coset of $\phi(\text{SO}(2))\le \text{SO}(n)$?
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isomorphism between subset of SU(2) and SO(3)

I know that there is a surjective map $\Phi : SU(2)\to SO(3) $. My question is if there is a subgroup $A \subset SU(2)$ such that $\Phi_{|A}:A\to SO(3)$ can be a (group) isomorphism. What would $...
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11 views

foliations with diagonal leaves

The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a ...
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1answer
57 views

Does orthogonal-invariance of a differential imply invariance of the function?

Let $U:\text{Hom}(\mathbb{R}^d,\mathbb{R}^d) \to \mathbb{R}$ be a smooth function . If $U$ is orthogonally-invariant, i.e. $U(QA)=U(A)$ for every $Q \in \text{SO}(n),A \in \text{Hom}(\mathbb{R}^d,\...