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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Coadjoint representation $Ad^*$ of the Heisenberg group

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 $z=x+ i y \in \mathbb C$ ...
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18 views

The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
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56 views

Characters group and cocharacters group Hom duality

Let $T$ be an algebraic torus over $\mathbb{C}$. For brevity denote $C = \mathbb{C}^\times = \mathbb{G}_{m,\mathbb{C}}$ the multiplicative subgroup of $\mathbb{C}$. Define character group by $$ X^*(T) ...
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54 views

When are orbits maximal integral manifold

If G is a Lie group acting on a manifold M through $\Psi$, one can argue that orbit of any $p\in M$ is an integral submanifold of the generators of group action. Roughly the proof is : 1) Fixing p ...
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25 views

Root spaces for a Cartan subalgebra of $\frak{sl}(2,\mathbb{C})$

I'm having trouble computing this. If I take the Cartan subalgebra generated by $$\left(\begin{matrix}0&1\\-1&0\end{matrix}\right)$$ then which are the two eigenspaces for the nonzero roots?
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163 views

Is geodesic distance equivalent to “norm distance” in $SL_n(\mathbb{R})$?

Take any norm, $\|\cdot\|$on $\mathbb{R}^n,$ and consider the resulting norm on $SL_n(\mathbb{R})$: $$\|A\|:= sup\{\|Av\|: \|v\|=1\}.$$ Now take any left-invariant Riemannian metric, $g$, on ...
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55 views

Why is the Haar measure of a Lie group with finite abelianization both left and right translation invariant? (Moved from math.SE)

I'm reading Foundations of Hyperbolic Manifolds by Ratcliffe. On the way to proving Gromov's theorem on the proportionality of hyperbolic volume and simplicial volume, he states that "it is a basic ...
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1answer
33 views

Are group theoretic splittings of Lie groups automatically differentiable?

Suppose that $G$ is a Lie group, and that $N$ is a normal Lie subgroup of $G$. Then $G / N$ is also a Lie group. If $0 \to N \to G \to G/N \to 0$ splits as groups (i.e. $G$ is a semidirect product of ...
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22 views

Fixed-point free representation of a Lie group G, with non-trivial fixed points for proper subgroup

Assume we have a compact Lie group $G$ and a closed proper subgroup $H$. Is there an elementary way to see that there is a finite dimensional representation $V$ of $G$ with only trivial fixed points ...
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43 views

$\mathbb{R}$ action on a manifold $M$ induced by a vector field $X$

Suppose we have a manifold $M$ and non-vanishing vector field $X$ on it. That vector firld induces a flow $\phi^X$. If the domain of $\phi^X$ is $\mathbb{R} \times X$ than we have an action of ...
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29 views

Examples of quotient manifolds which are not locally trivial fibrations?

Let $X$, $Y$ be differentiable manifolds, and $f : X \to Y$ a smooth surjection. Then $Y$ is said to be a quotient of $X$ if 1) $Y$ has the quotient topology 2) A function $g : Y \to \mathbb{R}$ is ...
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13 views

Restricted root descomposition of $SL(n,R)/SO(n).$

I am trying to find the restricted root space descomposition asociated with the Rimannian symmetric space SL(n,R)/SO(n) but I am getting a little bit stucked. Could someone help me? Many thanks.
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1answer
58 views

On lifting an action of $G$ on $X$ to an action of $G'$ on $\tilde{X}$.

I am reading the section on covering actions from Glen Bredon's Tranformation groups. Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path ...
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23 views

Check that $u_4\bar{u_3}u_2\bar{u_1}=i$ and $\bar{u_1}u_2\bar{u_3}u_4=1$ so the product of the four reflections is indeed $q \to iq$

This is an exercise from "Naive Lie Theory" and $u_1, u_2, ...,u_4$ are the unit quaternions. I have read the section many times but still don't understand. Can someone explain the material and solve ...
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45 views

Understanding rotations of $\mathbb{R}^4$ and pairs of quaternions, showing a rotation is a product of reflections in hyperplanes

I am working through Stillwell's "Naive Lie Theory" and am completely stumped by the questions in this section. An example of one of the questions is Show that the rotation that sends $1$ to $i$, ...
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34 views

Lie Algebra of $\mathrm{SO}(2)$ and $\mathrm{O}(2)$ are the SAME - why?

If $G$ is a Lie Group (with identity element of $e$), then my definition of the Lie Algebra $\mathfrak{g}$ of $G$ is the tangent space of $G$ at $e$, so that $\mathfrak{g} = T_{e}G$. The Lie Algebra ...
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21 views

Is a finite volume Lie group compact?

I know an example of a finite volume homogeneous space which is not compact, $SL_2(\mathbb(R)) / SL_2(\mathbb{Z})$. But what about a Lie group with this property? Can it happen? (The Lie group is ...
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1answer
31 views

Cartan subalgebras and Jordan Normal form

I'm stuck with Kac notes on Introduction to Lie Algebras. I logically understand all the definitions and everything is fine but I can't understand what's the thinking behind it. So I'm not asking for ...
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12 views

Inner automorphisms on the Lorentzgroup

Consider the Lie algebra at the neutral element of the Lorentzgroup $O_1(n)$, which is $\mathfrak{g}_0=\{ X\in R^{n\times n}|X^T\eta+\eta X=0 \}$ and its complexification ...
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22 views

When are stabilizers of the coadjoint action equal?

Let $G$ be a compact connected Lie group with Lie algebra ${\frak g}$. For $\lambda\in{\frak g}^*$ let $$G_\lambda=\{g\in G:{\rm Ad}_g^*\lambda=\lambda\},$$ i.e. $G_\lambda$ is the stabilizer of ...
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45 views

What is the relation between the usual topology of $S^1$ and its subspace topology in Homeo$(S^{n+1})$?

Let the set of all self homeomorphisms of $S^{2n+1}$ - $\operatorname{Homeo}(S^{2n+1})$, be given the compact open topology. Fix $a_0,\cdots,a_n\in\mathbb Z$ to be $n+1$ coprime integers. Let $S^1$ ...
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72 views

Relation of $G$-invariants and $g$ -invariants.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H ...
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Tits algebras of E_6

The general construction of Tits algebras of algebraic groups can be found in Knus, Merkurjev, Rost, Tignol - The book of involutions § 27 For every projective, homogeneous G variety $X:=G/P$, with ...
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1answer
30 views

Lie Algebra Homomorphisms for Lie Subgroups

Let $G$ be a Liegroup and $H$ a Lie subgroup of $G$. Then we find a Liegroup homomorphism $i \colon H \to G$ and the induced map $i_* \colon \mathfrak{h} \to \mathfrak{g}$ between the corresponding ...
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167 views

Proof of the Isomorphism between: $SL(2,\mathbb R) \times SL(2, \mathbb R) \cong SO^+(2,2)$

I want to do a proof that $SL(2,\mathbb R)\times SL(2, \mathbb R) \cong SO^+(2,2)$. My idea was to use the same Argument as in this Question. So I wanted to begin with the Basis of the Lie algebra ...
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2answers
74 views

Finding the basis of $\mathfrak{so}(2,2)$ (Lie-Algebra of $SO(2,2)$)

I want to calculate the Basis of the Lie-Algebra $\mathfrak{so}(2,2)$. My idea was, to use a similar Argument as in this Question. The $SO(2,2)$ is defined by: $$ SO(2,2) := \left\{ X \in ...
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1answer
31 views

Lie bracket of a semidirect product

I'm trying to solve problem 1.12 of chapter 1 from Duistermaat & Kolk' Lie groups. In the exercise you have a Lie group $G$ and a finite-dimensional vector space $V$, and a homomorphism ...
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66 views

Show that $ \exp \left(SL(2,R)\right)$ is the set of all matrices with positive trace $\geq -2$

Using the fact that every matrix in $SL(2,\mathbb{R})$ is conjugate in $SL(2,\mathbb{R})$ to one of the following matrices: $$ \left(\begin{array}{rr} a & 0\\ 0 & \frac{1}{a} ...
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1answer
33 views

Exact sequence for the Spin group

I read that the $Spin(n)$ is the double cover of $SO(n)$ such that the following sequence is exact $$ 1 \to \mathbb{Z}_2 \to Spin(n) \to SO(n) \to 1 $$ My first question is what information does this ...
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22 views

Good reference on the parametrization of SU(3) and SU(N)

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}cos(\theta) & -e^{-i\beta}sin(\theta) \\ ...
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80 views

Lie-group existence on universal covering manifold

Let $X$ be an n-dimensional smooth manifold with Lie group $G$ acting transitively on $X$, i.e. $X$ is a homogeneous space. Let $\tilde{X}$ be the associated universal covering space. To what extent ...
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38 views

Spheres as Homogeneous Spaces

Any odd dimensional sphere $S^{2n+1}$ can be expressed as an homogenous space of $SU(n)$ by $S^{2n+1} \simeq SU(n)/SU(n-1)$. Any even dimensional sphere $S^{2n}$ sphere can be expressed as an ...
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1answer
32 views

Proving continuity/smoothness for a special function on a Lie group.

So I asked this question, yesterday, forgetting the compactness requirement. Jack Lee commented shortly afterwards, noting that, if we take an inner product on the Lie algebra $T_eG$ of a Lie group ...
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1answer
22 views

Right unidimensional extension of Heisenberg Algebra

I'm reading a book on Mathemathical Physics and speaking of the Quantum harmonic oscillator says (this is a translation in english, hope is right): The commutation relations between operators are: ...
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3answers
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Why is $\mathbb{S}^1$ not a normal subgroup of $\mathbb{S}^3$

I know in general we try to show the left cosets are not equal to the right cosets but I need something more concrete to get me started in this example. How can I begin? (Here, $\mathbb{S}^3$ is the ...
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10 views

Term describing a restriction of a Lie group to a subset and its representation

I'm looking for the proper term for restrictions of Lie groups to subsets - I'm working with group invariants in data processing, but some data, like pixel representation of fonts is "partially" ...
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1answer
66 views

Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure

$\newcommand{\diff}{\mathrm{d}}$ TL;DR Having read this I know something about Haar measures, in particular that a left-invariant one exists and is unique on any Lie group $G$. I know that defining: ...
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1answer
52 views

How can I understand the isomorphism $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$ and compute the corresponding volumes?

I want to understand the above isomorphisms $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$. I seem to get some partial understanding but I miss a complete picture. For example I think that the last ...
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1answer
49 views

Are all compact subgroups of $GL(n,\Bbb C)$ in $U(n)$?

If $G$ is a compact subgroup of the multiplicative group $\Bbb C-\{0\}$, then it is easy to show that $G\subseteq S^1$. I wonder if this generalizes as follows: Question: If $G$ is a compact ...
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15 views

Analytic structures which induce same topology on $\mathbb{R}$

It an exercise in Cohn's book. Analytic structure on a Hausdorff space $M$ is a family of charts $\mathcal{F}$ satisfying At each point of $M$ there is a chart in $\mathcal{F}$ Any two charts of ...
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19 views

Let $PSL(3,q)$, $q$ odd. Then for $p \mid q(q^2 - 1)$ and $s \mid q-1$ ($p,s$ prime) we have a $p$-subgroup $X\ne 1$ such that $s$ divides $|N_G(X)|$

Let $G = PSL(3, q)$ with $q$ odd, the projective special linear group over a finite field of order $q$. Let $p, s$ be prime numbers. If $p$ divides $q(q^2 - 1)$ and $s$ divides $q-1$, then there ...
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1answer
58 views

Is a real logarithm of a special orthogonal matrix necessarily skew-symmetric?

The exponential map from the Lie algebra of skew-symmetric matrices $\mathfrak{so}(n)$ to the Lie group $\operatorname{SO}(n)$ is surjective and so I know that given any special orthogonal matrix ...
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16 views

Realizing a reflection group as a Weyl group

Question 1. Suppose given a compact, connected Lie group $G$ and a subtorus $S$ (not maximal) such that the effective image $N$ of the $\mathrm{Ad}$-action of the normalizer $N_G(S)$ on the Lie ...
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36 views

Showing that Heisenberg group is a Lie Group.

We define the Heisenberg group $H^{n}$ for $n\geq1$ as follows. As an analytic manifold $H^{n}=\mathbb{R}^{2n+1}$. We denote elements in $H^{n}$ by $(t_{i},q_{i},p_{i})$ with $t_{i}\in\mathbb{R}$ and ...
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1answer
22 views

Cartan subalgebra of ${\frak so}(5)$

I would like to find effectively the Cartan subalgebra of ${\frak so}(5)$. Does anybody knows how to proceed? Edit: I don't want to start from the simple roots and then derive it. I would like to do ...
3
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36 views

Is there an explicit left invariant metric on the general linear group?

Consider $GL_n^+$, the group of (real) invertible matrices with positive determinant. Is it possible to find an explicit formula for a metric on $GL_n^+$ which is left-invariant, i.e ...
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1answer
30 views

Matrix representation of Lie Algebra $B_2$

I'm writing some practical examples where to calculate the Killing form, the Cartan Matrix, Dynkin diagrams etc. Does anybody have on or two nice matrix representations of the $B_2$ Algebra? It would ...
2
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1answer
40 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
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35 views

Simply connected linear algebraic group [duplicate]

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
2
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1answer
63 views

If the Lie algebra is ${\frak g}={\frak a}\oplus{\frak b}$ then the Lie group is $G=AB$?

Let $G$ be a connected Lie group and suppose that its Lie algebra ${\frak g}$ splits into a direct sum of ideals $${\frak g} = {\frak a}\oplus{\frak b}.$$ Let $A$ be the connected Lie subgroup of $G$ ...