Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
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votes
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Representations of reductive Lie group
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie ...
- 671
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votes
0
answers
134
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Finite subgroups of compact simple Lie groups [duplicate]
The finite subgroups of $SU(2)$ consist of the symmetry groups of the Platonic solids plus the finite subgroups of $O(2)$. I would like to know if there are any results concerning $SU(3)$. In ...
- 351
1
vote
1
answer
166
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Directed graphs and Compact Lie Groups
Is there a method for associating the edges and vertices of an arbitrary directed graph with the irreducible representations of a compact Lie group and the intertwiners
of the adjacent edge ...
- 11
3
votes
0
answers
78
views
Can Bruhat cells in semi simple groups be induced from matrices?
Let $G$ be a semisimple Lie group. Embed it as a subgroup into a special linear group of suitable rank, $SL(n)$ (real or complex). The question is: is it always possible to find such an embedding, ...
- 31
2
votes
1
answer
295
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Sg: How to Show this Sequence is Exact?
Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg ...
- 105
1
vote
1
answer
489
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generalisation of GL(3,R) polar decomposition
Does polar decomposition work when the 'orthogonal' matrices are not orthogonal wrt to the identity ($O^TO=Id$), but wrt to some other symmetric matrix $K$ (i.e. $O^TKO=K$)?
Specifically, $GL(3,R)$ ...
- 11
1
vote
1
answer
558
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Understanding manifold GL+(3,R)/SO(3) ?
I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
- 157
1
vote
1
answer
390
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nullity of the second fundamental group of a Lie group
Can anyone tell me why it is that Lie groups seem to have their second fundamental group $\pi_2(G)$ equal to $0$, or provide me with a link to an article or a book reference?
I came across this fact ...
- 2,751
2
votes
0
answers
91
views
Is this functional maximized by SU(2) coherent states?
Let $D : SU(2)\mapsto\mathbb{B}(\mathbb{C}^{d})$ be an unitary irreducible representation of SU(2).
Denote the spin of this representation by S (i.e. $d = 2S+1$).
Define the functional $F$ by
$$F(\...
- 2,753
0
votes
0
answers
65
views
irreducible representation of a simple Lie group where each element has a fixed point
I was wondering if it possible that a simple Lie group $G$ could have a continuous irreducible representation on a finite-dimensional real vector space $V$ in which each element of $G$ has a non-zero ...
- 2,125
3
votes
0
answers
242
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criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group
By Givens' rotation $R(1,2;\theta)$ I mean a matrix which has the
$$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$
$2 \times 2$ block at ...
- 4,466
2
votes
1
answer
420
views
Compatible Iwasawa decomposition for embedding of the orthogonal Lie group
I am looking for an embedding of the orthogonal Lie group
O(n,C) into GL(m,C) such that the standard Iwasawa
decomposition (also known as the QR-decomposition) for the
group GL(m,C) induces an ...
- 21
5
votes
1
answer
561
views
What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?
Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(...
- 4,898
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answers
686
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Is endoscopy interesting in simply-laced cases?
Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...
- 4,949
2
votes
1
answer
295
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Inverting the integration along a subgroup
Given a locally compact group $G$ and a closed subgroup $H$, one often uses an operator of the form
$$P: C_c(G) \rightarrow C_c(H \backslash G), \qquad Pf(Hg) = \int_H f(hg) d_H h,$$
where $d_H h$ ...
- 11.2k
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votes
0
answers
153
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Can class in $H^4(BT)$ be realized as the second Chern class of a principal SU(2) bundle?
The question in the title, to which I add some clarification. Can every class in $H^4(B\mathbb{T})$ be realized as the second Chern class of a principal $SU(2)$ bundle?
$B\mathbb{T}$ is the ...
- 1,331
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votes
2
answers
559
views
Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?
Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
- 8,167
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vote
2
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563
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Reps of groups and reps of algebras
I've got what might be a couple of very basic questions on the fundamental representations of locally isomorphic semi-simple Lie groups and their relationship to representations of the corresponding ...
4
votes
0
answers
119
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Topological obstruction to icosahedral symmetry?
Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$
where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...
- 3,761
3
votes
1
answer
515
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Integral subgroup theorem for Banach-Lie groups
Let $G$ be a Banach-Lie group with Lie algebra $\mathfrak g$
and $\mathfrak h$ a closed subalgebra. Using the exponential
map and the Baker-Campbell-Hausdorff-formula one constructs a local
Lie group $...
- 4,729
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votes
0
answers
281
views
Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?
More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
- 118k
1
vote
0
answers
268
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how to determine the Weyl group of a diagonalizable subgroup?
Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(...
- 11
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votes
2
answers
355
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Terminology for nilpotent groups
I have a nilpotent lie group $N$ with upper central series
$$1 = N_0 \triangleleft N_1 \triangleleft \dots \triangleleft N_k = N$$
which induces the filtration $$0 = \mathfrak{n}_0 \subset \mathfrak{n}...
- 4,014
4
votes
1
answer
473
views
A question on random walks on semisimple groups
Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...
- 41
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votes
0
answers
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Has the Lie group preserving a probability distribution been used in Bayesian statistics?
For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...
- 15.4k
6
votes
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answer
403
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Orbits for homogenous complex polynomials under unitary rotation of variables
Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.
...
- 1,612
2
votes
0
answers
117
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integral stable conjugacy classes
Let $G$ be a semisimple simply connected group over $k$ algebraically closed field .
Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$.
We assume that ...
- 3,472
5
votes
0
answers
567
views
Computing centralizers in Lie groups
Let $G$ be a real semisimple Lie group. Really, I only care about $\text{SL}(n,\mathbb{R})$ and $\text{Sp}(2n,\mathbb{R})$.
I'd like to perform a computer search for a finite group with a certain ...
- 101
1
vote
1
answer
132
views
Lorentz quotient and orientation
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U....
- 25.7k
8
votes
0
answers
388
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Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
1
vote
0
answers
158
views
Zariski dense subgroup of $SL(3,\mathbb{R})$
Let $\Delta$ be a Zariski dense finitely generated subgroup of $SL(3,\mathbb{R})$. Assume that $\Delta$ contains no element of finite order. Then, does there exist a finite-order element $A \in SL(3,\...
- 133
1
vote
0
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174
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Automorphism of a Lie group which preserves a maximal torus is necessarily an inner automorphism?
Let $G$ be a connected Lie group with a maximal torus $T$. Suppose $\sigma$ is an automorphism of $G$ so that $\sigma(T)=T$. Then can we conclude that $\sigma$ is an inner automorphism of $G$? (i.e. ...
- 351
2
votes
1
answer
467
views
Difference between action of group element and Lie algebra element in smooth representation
Let $G$ be a real reductive Lie group, $P$ its parabolic subgroup with Levi decomposition $P=MN$, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Suppose given a smooth representation $(\pi,V)$...
- 2,709
7
votes
1
answer
360
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Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
- 2,210
2
votes
0
answers
339
views
volume form in a symmetric space of real rank one
I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...
- 2,446
2
votes
0
answers
115
views
The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893
4
votes
1
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446
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Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.
This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the ...
- 3,399
2
votes
1
answer
176
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Chains in $K\backslash G/B$ lying over a closed $K$-orbit
Let $G$ be a complex connected reductive Lie group, $\theta$ an
involution, and $K = G^\theta$ the fixed-point subgroup.
Then $K$ has finitely many orbits on $G/B$, one of which is open
and (quite ...
- 27.8k
2
votes
0
answers
54
views
Conjugacy classes of a triple
In the paper " THE COMPONENT GROUPS OF NILPOTENTS IN EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS" by D. King I am unable to proof the lemma 3.7 which is omitted there.
The lemma is following:
Let $\...
- 116
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vote
0
answers
85
views
Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
- 1,582
5
votes
1
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384
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Conjugacy classes with elliptic limit points
Let $G$ be a reductive algebraic group over $\mathbb R$ and $K$ a maximal compact subgroup. Then we refer to the conjugacy class in $G$ of some $k \in K$ as an elliptic conjugacy class.
Question: ...
- 538
2
votes
1
answer
138
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Closed subgroups of $Tr_1(d,\mathbb{Q}_p)$
For a field $K$ let denote by $Tr_1(d,K)$ the nilpotent group of all upper triangular $d\times d$-matrices over $K$ with each diagonal entry equal to 1. Let $\mathbb{Q}_p$ the field of $p$-adic ...
- 657
1
vote
0
answers
360
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symplectic representations: when could the center act trivially?
I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
- 313
2
votes
0
answers
294
views
Numerical integration on manifolds
Hi all,
I need to integrate a system of coupled ODE in a manifold (SU(N)). I know that Runge-Kutta methods do not translate "automatically" to a integration scheme that preserves the manifold ...
- 163
7
votes
0
answers
509
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Small sum of group elements acting as rank 1 matrix.
I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some $c_1,...
- 2,179
0
votes
0
answers
153
views
Nontrivial copies of SO(r) in SO(n)
If $G=SO(n)=SO(\mathbb R^n)$ and $r\leq n$, it is easy to find a closed subgroup $H\leq G$ that is isomorphic to $SO(r)$, just let $S\subseteq\mathbb R^n$ be an $r$-dimensional subspace and let $H=\{g\...
- 1,987
9
votes
1
answer
266
views
Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
- 13k
1
vote
1
answer
358
views
What does the weights of Lie group mean?
Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system
of the exceptional Lie group $G_2$
with $\alpha_1$ is short and $\alpha_2$ is long,
so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\...
- 139
2
votes
0
answers
165
views
Reference request: injective homomorphisms between unitary groups
Let $U(n)$ be the group of unitary $n\times n$ matrices over $\mathbb{C}$. Is there a classification of the continuous, injective group homomorphisms $U(m)\to U(n)$? If so, is there a modern account ...
- 1,336
9
votes
0
answers
391
views
Reflection groups in O(n+1,n) arising `in nature'?
For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...
- 1,859