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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14 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. ...
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1answer
51 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\\ ...
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1answer
12 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 ...
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9 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 ...
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21 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 ...
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25 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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1answer
27 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 ...
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22 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 ...
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23 views

Form on $S^2$ that is invariant under $O(3)$ [on hold]

Let $\omega$ be a $1$-form on $S^2$ that is invariant under $O(3)$, can $\omega$ be non-trivial?
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8 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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1answer
25 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 $se(3)$, ...
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1answer
20 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: ...
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21 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 ...
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24 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 ...
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1answer
26 views

Decomposing representations

The problem I am trying to do is the following: Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 I ...
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1answer
58 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) ...
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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 ...
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37 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 $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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1answer
54 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 ...
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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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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 ...
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1answer
21 views

Determining if two given matrices in the symplectic Lie group $Sp(2)$?

Define the following quaternionic matrices $1=\pmatrix{1&0\\0&1}, i=\pmatrix{0&-1\\1&0}, j=\pmatrix{0&-i\\-i&0}, k=\pmatrix{i&0\\0&-i}$ I am given that the symplectic ...
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1answer
67 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 ...
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13 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 ...
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1answer
35 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 ...
3
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1answer
52 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 ...
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4answers
1k views

$SO(n)$ is connected

The question really is that simple: Prove that the manifold $SO(n) \subset GL(n, \mathbb{R})$ is connected. it is very easy to see that the elements of $SO(n)$ are in one-to-one correspondence with ...
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1answer
21 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 ...
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2answers
32 views

Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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1answer
61 views

Exponential map for the Lie group of upper triangular matrices

Let $G$ be the Lie group of all upper triangular real matrices (over $\mathbb{R}$) with positive diagonal elements. Denote $\mathfrak{g}$ its Lie algebra. Do we have surjectivity of $\exp : ...
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1answer
41 views

Classification Problems in Lie algebra

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras ...
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1answer
448 views

Map from $SL(2,\mathbb C)/SU(2)$ to $SL(2,\mathbb C)$

I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$. Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, ...
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1answer
28 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 ...
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1answer
19 views

Maple: How to use partial differential operators?

I am trying to calculate the commutator $[v,w]=vw-wv$ for given infinitesimals $$v=\dfrac{\partial}{\partial x}$$ and $$w=x\dfrac{\partial}{\partial t}$$ I know how to calculate the commutator by ...
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0answers
17 views

If $H$ is a diagonal matrix then its adjoint representation is also diagonal for any $H$ [closed]

I am not sure why this is true, is there a simple explanation? To be more precise I am trying to show that the set of traceless diagonal matrices form a Cartan subalgebra of $sl(n,\mathbb{C})$.
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1answer
33 views

Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} ...
2
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0answers
31 views

Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
3
votes
1answer
68 views

Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
1
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2answers
45 views

Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
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1answer
39 views

Foliation dense if $G = \textbf{R}$, where $G$ is a subgroup of a Lie group $G'$.

I have the following statement: Let $G$ be a subgroup of a lie group $G'$, and the action is left multiplication. The leaves are then the left cosets of $G$ in $G'$. If for example, we let $G = ...
4
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1answer
136 views

Relationship between Möbius transformations and flows/vector fields

I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For ...
3
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1answer
32 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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1answer
19 views

Lie algebra homomorphism and representation

I am solving a multiple part problem on Lie algebra representations. I have done the first three parts, but am stuck on part (iv) as follows: Define a linear map $\phi : \mathbb{g} \rightarrow ...
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0answers
25 views

What does it mean for a representation to be one-dimensional?

For example, take the Heisenberg Lie Algebra with the following basis: $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ $Y=\begin{bmatrix} 0 ...
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1answer
45 views

Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
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2answers
46 views

The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = ...
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0answers
11 views

Restriction of an isogeny is still an isogeny?

Given that $E \times F \twoheadrightarrow G$ is an isogeny where $E,F$ are both subgroups of $G$, is its restriction to a subgroup $H$ of $G$, $E \times (F \cap H) \twoheadrightarrow (G \cap H) = H$, ...
2
votes
1answer
51 views

Finding lie algebra of a group by using exp map and tangent space [duplicate]

I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and ...
6
votes
2answers
81 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
1
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1answer
53 views

elments of a linear algebraic group agreeing on a vector

Let $G \subset \mathrm{GL}_n(k)$ be a connected affine algebraic group over a field $k$ with the following property: for any two distinct elements $g,h \in G$ there exists a vector $x \in k^n, x\neq ...