All Questions
Tagged with lie-groups group-actions
65 questions
2
votes
0
answers
201
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Manifolds as simultaneous coset spaces
Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
CommunityBot
- 1
6
votes
0
answers
427
views
Non invertibility of certain integral arising from group action
Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question.
Edit 2: I understand from the following post that some part of the previos version of my question has ...
CommunityBot
- 1
3
votes
1
answer
136
views
symmetric group of regular polyhedrons
Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $c:=c(...
- 56.9k
2
votes
1
answer
262
views
From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?
Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of $...
CommunityBot
- 1
9
votes
4
answers
1k
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Action of a Lie group with finitely many orbits
EDIT: Let a real Lie group $G$ act on a smooth manifold $M$ with finitely many orbits such that each orbit is locally closed ($M$, but not $G$, may be assumed to be compact in my case). Let $\mathcal{...
- 25.3k
2
votes
1
answer
141
views
Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space
Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...
- 4,729
4
votes
1
answer
1k
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Transitive action on the sphere
Hello!
From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...
- 6,546
4
votes
1
answer
163
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Representations of Finite Subgroups on Homology
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...
- 27.8k
15
votes
0
answers
637
views
"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
- 5,891
1
vote
1
answer
164
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Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$
There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$
Let $\mathbb{Z}/2=\...
- 31.2k
2
votes
1
answer
417
views
A question regarding Lie group actions
Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?
- 366
8
votes
1
answer
730
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 12.6k
9
votes
3
answers
2k
views
Lie group actions and f-relatedness
Background
Let $f: M \to N$ be a smooth map between smooth manifolds.
Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be $f$-related if for all $p \in M$, $(f_*)_p(X_p) = Y_{f(p)}$; ...
- 1,500
0
votes
0
answers
700
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Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 18.7k
7
votes
1
answer
944
views
When is a conjugacy class of matrices an embedded submanifold?
Let $M_{n\times n}$ denote the set of $n\times n$ real matrices and let $GL_n$ be the subgroup of invertible matrices. $GL_n$ acts on $M_{n\times n}$ smoothly by conjugation, which means that each ...
- 27k