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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1answer
212 views

restriction of $SL(2,R)$ representation to $SO(2)$

Let $d \geq 1$ and let $V$ be a $2d+1$ irreducible representation of $SL(2,\mathbb{R})$. We know the irreducible representations of $SO(2)$ are the $2$ dimensional spaces $V_k$ with the map $$\rho_k : ...
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1answer
29 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
4
votes
1answer
76 views

$E_8$ and theta functions

The root lattice $\Gamma_8$ of the exceptional Lie algebra $E_8$ is an eight-dimensional lattice which consists of lattice points in $\mathbb{R}^8$ which with respect to an orthonormal basis $e_1, \...
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1answer
53 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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3answers
52 views

If a Lie Algebra is solvable, is the corresponding Lie group solvable in the group theoretic sense?

I just started working with Lie Algebras with a professor. The way we defined them is probably the standard way; treat Lie Algebras as tangent spaces at the identity of the Lie Group. Now, consider ...
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0answers
28 views

Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
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1answer
34 views

A simple question for lie group

I know the following is wrong. But I don't see where is wrong. Let's begin by proving $\frac{d}{dt}(A(t)\cdot B(t))=(\frac{d}{dt}A(t))\cdot B(t)+A(t)\cdot\frac{d}{dt}B(t)$, where $A(t)$, $B(t)$ are ...
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0answers
22 views

real analytic structure on liegroup

Let $G$ be a Liegroup of $C^2$. I read, that if $G$ is compact, there exists a unique structure, such that $G$ is a real analytic manifold and the canonical action (left or right) is real analytic. ...
0
votes
1answer
40 views

Left and right invariant vector fields

I'm following Woodhouse's book on geometric quantization and I'm stuck with this problem. Let $R_A$ and $L_A$ be right and left invariant vector fields such that $R_A(e)=A=L_A(e)$, where $A$ is an ...
1
vote
1answer
30 views

How do we give G/T a symplectic structure

I am new in those staff and I can't find anything introductory explaining those stuff. I am interested in the following. Given $G$ a compact Lie group and consider it's maximal torus $T$. Then I have ...
0
votes
1answer
26 views

Why is the quotient map $G\rightarrow G/T$ a fibration?

I have just learnt about fibrations and I saw somewhere the following. Given a compact lie group $G$, one can consider a maximal torus $T$ of $G$ then the quotient map $G\rightarrow G/T$ is a ...
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1answer
24 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
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0answers
15 views

Laplacian in matrix form?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
3
votes
2answers
32 views

Möbius Band Bundle $(Mo,\mathbb{S}^1,\text{proj}_1,\mathbb{R}) $ is not a Principal $\mathbb{R}$-bundle

This is claimed in various places. The problem seems to be with finding a free and transitive group action that has the fibers of $Mo$ as its orbits. I construct $Mo$ as $$ Mo = \mathbb{S}^1 \times \...
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2answers
52 views

For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by A compact Lie group $G$ is semisimple if $[\...
1
vote
1answer
24 views

A basic question about this identity in Lie group setting.

$\textbf{Problem.}$ Let $G$ be Lie group. Let $F:G\times G\rightarrow G$ denote the multiplication map. Identify the space $T_{(e,e)}(G\times G)$ with $T_{e}G\oplus T_{e}G$ by $$v\in T_{(e,e)}(G\...
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1answer
98 views

Is $(\mathbb R^3\setminus \{0\})/\mathbb R^*$ a smooth manifold?

Let $G=\mathbb R^*$ act on $X=\mathbb R^3\setminus\{0\}$ by pointwise multiplication. That is for any $t\in\mathbb G$ and $(x_1,x_2,x_3)\in X$ we have $$t\cdot(x_1,x_2,x_3)=(tx_1,tx_2,tx_3)$$ Is ...
1
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1answer
30 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
1
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1answer
21 views

Convolution of matrix coefficients from inequivalent representations

Suppose $\delta_1,\delta_2$ are two inequivalent representations of a compact Lie group. Let $dy$ be the normalised Haar measure and define convolution for functions $f,g:G\rightarrow \mathbb{C}$ $$f*...
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0answers
41 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
1
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1answer
18 views

Every compact semi-simple Lie group has finite center

I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center. Here, semi-simple means, ...
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votes
0answers
21 views

Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
0
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1answer
33 views

reference request: lie algebra-lie group

I am looking for a reference where I can find a (relatively) elementary and self contained proof of the fact that all real, finite dimensional Lie algebras are the Lie algebra of some Lie group. ...
2
votes
1answer
38 views

Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
2
votes
1answer
22 views

Adjoint orbit of universal covering group

Let ${\frak g}$ be a complex semisimple Lie algebra and $G$ a connected Lie group with Lie algebra ${\frak g}$. Let $\tilde{G}$ be a universal covering group of $G$. Take $X\in{\frak g}$ and consider ...
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votes
0answers
19 views

Lie bracket and local group

How to prove this identity? X and Y and smooth vector field on smooth manifold M; $\theta_t$ is the local group (one-parameter group of diffeomorphism) of Y. $$\left.\frac{d}{dt}\right|_{t=0}\left((\...
0
votes
1answer
14 views

lie group jacobian function deduce

hi guys I have read the paper at http://www.ethaneade.org/lie.pdf , and regarding the equation (87) I have coded it for proven but found not correct , the code is like this ...
0
votes
0answers
16 views

Irreducible root system decomposition

I am looking for the name of and a good reference on the following theorem Theorem: let $G$ be a connected, compact and semisimple Lie group, and $T \subset G$ a maximal torus of $G$, there exists a ...
2
votes
0answers
24 views

Determining whether a Lie group contains more than one conjugacy class of subgroups of a particular isomorphism type

Suppose I have a Lie group $G$. How can one determine whether there is more than one conjugacy class in $G$ of subgroups isomorphic to a given Lie subgroup $H$? Put another way: Fix a Lie ...
2
votes
1answer
62 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{...
1
vote
1answer
28 views

canonical quotient map on lie group is proper?

Let $G$ be Lie group and $K \subset G$ a compact Lie subgroup of $G$. Let $\pi \colon G \to G/K , \quad g \mapsto g.K=[g]$ denote the canonical projection on the quotient and endow $G/K$ with the ...
3
votes
0answers
28 views

Spherical Harmonics and $L_+$ and $L_-$ operators

I have the spherical harmonics $Y_{m}^{l}\left(\theta,\varphi\right)$ and I want to show that the operators $L^{\pm}$ act as "creation" and "annihilation" operators such that $$L^{\pm}Y_{m}^{l}=\sqrt{...
0
votes
2answers
36 views

Compact Lie subgroup of $GL_n(\mathbb{R})$

Let $K\leq GL_n(\mathbb{R})$ be a compact Lie subgroup. I need to prove that $K$ is a conjugate of a subgroup of $O(n)$. The hint is to use the Haar measure, but I really don't see how to do this.
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0answers
13 views

Determining the compact roots of the Cartan subalgebras of $\mathfrak sp(2,\mathbb R)$

I want to understand the notions of real vs imaginary roots and compact vs noncompact roots (among the imaginary ones) in the theory of Cartan subalgebras (CSA's) of real semisimple Lie algebras. I ...
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vote
0answers
23 views

Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
1
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1answer
24 views

Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: $$\varrho\left(e\right)=\left(\begin{...
0
votes
0answers
26 views

Reference Request: Monologues on Lie Groups/Algebras and Differential Geometry

I find that before really diving into a subject, I prefer to get a general idea of it. For instance, before studying ergodic theory through a standard textbook I enjoyed Paul Halmos' lecture notes on ...
1
vote
1answer
28 views

the physical significance of the Lie Algebra of SE(3)

as we all know, the Lie group of $SE(3)$ can be written in the form of $4\times4$ matrix, say $$ \begin{pmatrix} R & t\\ 0 & 1 \end{pmatrix},\tag{1} $$ and its Lie Algebra, denoted as $se(3)$, ...
0
votes
0answers
15 views

How to prove that the killing form is unique up to scalar multiple? [duplicate]

For complex simple lie algebra, how to prove that the killing form is the unique adjoint invariant bilinear form up to a scalar multiple. I know we have to use schur's lemma somewhere but don't see ...
3
votes
1answer
63 views

The tangent map of multiplication - Maurer-Cartan form

Question: Consider the multiplication map $\mu : G \times G \to G$ of a Lie group. So on the tangent level we have a map $T(G \times G) \to TG$. Making the proper identification $T(G\times G) \...
2
votes
0answers
42 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
1
vote
1answer
56 views

Is $SO(n)$ actuallly the same as $O(n)$?

$SO(n)$ is defined to be a subgroup of $O(n)$ whose determinant is equal to 1. In fact, the orthogonality of the elements of $O(n)$ demands that all of its members to have determinant of either $1$ or ...
0
votes
1answer
37 views

Question about calculating Lie bracket given a three dimensional Lie algebra [closed]

Suppose we have $\frak{g}\in\mathbb{R^3}$ spanned by $X, Y, Z$ such that $[X,Y]=Y, [X,Z]=Y+Z$. What is $[Y, Z]$? I tried to expand the bracket, $[X, Y]=XY-YX=Y, [Y, X]=YX-XY$, but don't see how to ...
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votes
0answers
30 views

Prove that two matrices in $SO(3)$ are conjugate if and only if they have the same trace

The matrix $SO(3)$ is the group of all $3\times 3$ matrices with determinant=+1. I showed that if the trace is equal then they are conjugate but don't know how to show conjugacy implies equivalent ...
0
votes
1answer
83 views

Prove that two Lie groups have homeomorphic universal covers if and only if their corresponding Lie algebra are isomorphic

Two Lie groups $G_1, G_2$ have homeomorphic universal covers $\tilde{G_1}, \tilde{G_2}$ respectively if and only if the corresponding Lie algebras $\frak{g_1}, \frak{g_2}$ are isomorphic as Lie groups....
2
votes
0answers
16 views

What does a maximal torus in GSpin$_{2n}$ look like?

I am interested in spin groups for a project at my university, and I was wondering: what would a maximal torus in GSpin$_{2n}$ look like, and how does one come to it? Does anyone maybe have a ...
3
votes
1answer
55 views

Why is Lie algebra a real vector space?

Let the set $\mathcal{g}$ be the Lie algebra of a matrix Lie group $G$. Then my book asserts that $\mathcal{g}$ is a real vector space because it's closed under real scalar multiplication. My question ...
1
vote
1answer
36 views

proper action on homogeneous space

Let $M = G/K$ be a homogeneous space. It is easy to show, that the left action of $G$ on itself by multiplication is a free and proper action. My question is, if the induced action $$G \times G/K \...
1
vote
1answer
24 views

maximal torus by dimension count?

Suppose $T$ is a maximal torus of $G$ with dimension = $n$. If there is another torus $H \subset G$ of the same dimension, could I then conclude that $H$ is also a maximal torus? In other words once ...
0
votes
1answer
30 views

Why is the tangent space to the orbit through $p\in\mu^{-1}(0)$ an isotropic subspace of $T_pM$?

I'm reading symplectic geometry notes by Ana Cannas da Silva. The set up is a Hamiltonian action $G\curvearrowright(M,\omega)$ of a Lie group $G$ on the symplectic manifold $(M,\omega)$, with moment ...