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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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 ...
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37 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})} \text{Mat}_n(\...
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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 $x \in k^n, x\neq 0$...
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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 ...
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68 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( \text{...
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28 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 ...
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21 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 ...
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39 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 ...
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33 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 3....
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1answer
20 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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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 $\mathrm{...
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34 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 ...
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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 $\mathbb{R}^...
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45 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 ...
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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?
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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 ...
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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 ...
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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 $\...
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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 ...
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1answer
27 views

Existence of Certain Lie Groups

Let $\mathfrak{h}$ be a Lie algebra (not necessarily finite dimensional). Does there necessarily exist a Lie group $G$ such that for the Lie algebra corresponding to $G$, denoted $\mathfrak{g}$, we ...
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40 views

Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
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Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
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148 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 ...
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54 views

Smoothness of the inversion map is redundant in the definition of Lie groups

The question I want to ask is different from this one. Let $M$ be a smooth manifold which admits a group structure such that the multiplication map $m:G\times G\to G$ defined as $m(g, h)=gh$ for ...
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2answers
85 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 $SL_2(\...
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1answer
54 views

Clarifying notation used in Lie groups

Suppose that $G$ is a Lie group and $g \in G$ is a generic element. What does the notation "$dg$" refer to? Is it the differential of the function $G \to G$ given by left (or right) multiplication by $...
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16 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in \mathfrak{...
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1answer
35 views

A lie group $G$ is compact iff $G/H$ is.

Suppose that $G$ is compact Lie group and $H$ is closed subgroup. Than $G/H$ is compact since canonical projecton is continuous. Is the inverse true, namely can one prove that: G is compact iff $G/H$...
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203 views

Left Invariant vector field on SO(3)

I have a Lie group, namely on SO(3), i.e. $SO(3,\mathbb{\mathbb{R}})=\left\{ A\in GL\left(3,\mathbb{R}\right)\mid A^{T}A=\mathbb{1},\,\det\left(A\right)=1\right\}$. I have a Left action $L_g$ and I ...
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61 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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19 views

Showing that the commutator subgroup of a Lie group is a Lie subgroup

I'm learning about Lie groups and Lie algebras independently, and I'm trying to show that the commutator subgroup, $H=[G,G]$, of a Lie group, $G$, is a Lie subgroup. My first instinct was to take a ...
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1answer
46 views

Is the size of the conjugacy class of a given element in a compact Lie group always finite?

Let $G$ be a compact Lie group and $g\in G$ be any given element in it. Consider the conjugacy class of $g$ in $G$, denoted by $[g]=\{hgh^{-1}:h\in G\}$. Our question is that: Could you find a $g_0\...
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2answers
32 views

Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular groups?...
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33 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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1answer
30 views

Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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1answer
19 views

Isotropy algebra for $U(n)$? [duplicate]

Let $G = U(n)$ be the Liegroup of $n \times n$ unitary matrices and $\mathfrak{g}$ the corresponding Lie algebra. Now $G$ can act on $\mathfrak{g}$ by the Adjoint-action. Since $G$ is a subgroup of $\...
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1answer
16 views

Show that we have a smooth path in $T_1(G)$, the tangent space of a matrix group

Consider the path $D_s(T)=A(s)B(t)A(s)^{-1}B(t)^{-1}$ in $G$ for some fixed value of s. Then the Lie bracket $[X,Y]$ can be related to the commutator of $A(s)B(t)A(s)^{-1}B(t)^{-1}$ of smooth paths $A(...
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55 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
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45 views

Relationship between invariant and harmonic forms

Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, ...
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1answer
25 views

Every orthogonal matrix represents a rotation around an axis

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 ...
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SU(N) tensor product decomposition

Let's consider the group SU(N). The adjoint representation is $\textbf{Adj}= $ $\textbf{N}^2\textbf{-1}$. The following decomposition holds generally ( have a look at this ref ) $$ \textbf{Adj}\otimes\...
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The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
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38 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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Exponentiating an ``affine subalgebra''

Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group $U(...
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1answer
38 views

Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
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1answer
27 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point $...
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Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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1answer
22 views

What's its use of the nonsingular 2-step nilpotent Lie algebras

What's its use of the nonsingular 2-step nilpotent Lie algebras which form an a class of 2-step nilpotent Lie algebras ? Recall: A $2$-step nilpotent Lie algebra $N$ is non-singular if $ad X : N \to ...