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 ...

learn more… | top users | synonyms

0
votes
0answers
20 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?
0
votes
1answer
15 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.
1
vote
0answers
4 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 ...
1
vote
0answers
14 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
vote
1answer
15 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: ...
0
votes
0answers
20 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
23 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
54 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
36 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 $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
0
votes
1answer
51 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
35 views

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

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 ...
0
votes
0answers
28 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
64 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 ...
2
votes
0answers
12 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
50 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
20 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
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 ...
0
votes
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 $sl(n,\mathbb{C})$.
0
votes
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 ...
2
votes
0answers
30 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 ...
1
vote
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 : ...
1
vote
1answer
37 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 = ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
25 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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 = ...
0
votes
0answers
10 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
50 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 ...
0
votes
0answers
18 views

Finding the generators of $SU(3)$ different from the Gell-Mann matrices?

I want to find a set of generators of SU(3) different from the Gell-Mann matrices. How should I go about it? Can I construct it in such a way that at least three of the 8 generators when squared gives ...
2
votes
1answer
31 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})} ...
1
vote
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 ...
1
vote
0answers
28 views

Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]

Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...
5
votes
1answer
63 views

An intuitive way to understand the Jacobi's formula.

Suppose that $\mathbf A=\mathbf A(t)$ is a matrix whose entries are parametrized by a variable $t$. The Jacobi's formular states that $$ \frac d{dt}\left( \det \mathbf A\right)= \text{Tr}\left( ...
2
votes
1answer
21 views

a basic question in crossed product for compact group action

I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me. Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and ...
1
vote
0answers
20 views

Expression of the Laplacian of the reduced Heisenberg group?

Let $\mathbb C^n$ be the n-dimensional complex field endowed with a positive definite hermitian form $H(z,w)$. The corresponding symplectic form is $E(z,w)= \Im (H(z,w))$, where $\Im $ denotes the ...
0
votes
0answers
33 views

Matrix exponential between Lie algebra and Lie group (help with a proof)

Theorem 3.42 in Hall's Lie Groups, Lie Algebras and Representations is a key step towards proving that the matrix exponential maps a neighbourhood of zero in the Lie algebra to a neighbourhood of the ...
1
vote
1answer
24 views

Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example ...
1
vote
1answer
16 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 ...
2
votes
0answers
14 views

Closed subgroups of $\mathrm{SL}(n,\mathbb{R})$.

I have this question about closed subgroups of $\mathrm{SL}(n,\mathbb{R})$. So assume I have $H$ a (strict) closed subgroup of $\mathrm{SL}(n,\mathbb{R})$. It is therefore a Lie subgroup of ...
3
votes
1answer
29 views

What coset intuitively means in this case

Let $G=SO(3)$ and $K$ be the subgroup of $G$ .Let $K$ be the rotations around $Z$ axis . $$K = {k(ϕ) : 0 ≤ ϕ < 2π}$$ $$K(ϕ) =\begin{pmatrix}cos ϕ & − sin ϕ &0\\sin ϕ &cos ϕ &0\\ 0 ...
1
vote
0answers
17 views

Reduction of functions with Lie group symmetries

If I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with a Lie group G as a symmetry, $f(Ax)=f(x),\quad A\in G$ how might I go about obtaining a reduced function $\tilde{f}$ on ...
2
votes
1answer
39 views

Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
0
votes
1answer
20 views

Discrete action on Lie groups

Given a Lie group $G$ and a discrete subgroup $\Gamma$ of $G$. Why is the action of $\Gamma$ on $G$ properly discontinuously?
0
votes
2answers
41 views

Why is a Lie group homomorphism from SO(3) to SU(2) always trivial?

The Lie group $SU(2)$ is a double cover of $SO(3)$.$SU(2)$ is simply connected as a manifold,and $SO(3)$ is $RP^3$ .But why must a Lie group homomorphism from $SO(3)$ to $SU(2)$ be trivial? i.e. the ...
2
votes
0answers
46 views

How I could define a inner product in the characters in $SL(2, \mathbb R)$

I have homework in a course on Lie groups, in which I must show that $(\pi_m,\mathbb C_m[x,y])$ are the only irreducible representations of finite dimension in SL(2,ℝ). Here $ C_m[x,y])$ is the vector ...
0
votes
0answers
28 views

inner product of characters in $SL(2,\mathbb{R})$ [duplicate]

tengo una tarea en un curso de grupos de Lie, en la cual debo demostrar que $(\pi_m,\mathbb{C}_m[x,y])$ son las únicas representaciones irreducibles de dimension finita en $SL(2,\mathbb{R})$. Donde ...