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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Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
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answer
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A statement on a connected semisimple non-compact Lie group with finite center
In the book `Ergodic Theory and Semisimple groups' (1984 Edition, page 64, Proposition 4.1.8), Zimmer has made this statement "if G is a connected semisimple non-compact Lie group with finite center, ...
- 105
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2
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Covering relations in $K\backslash G/B$
Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\...
- 27.8k
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are the smooth vectors of a Frechet space dense?
Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
- 342
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topologies on globalizations
I am reading notes by David Vogan on Unitary representations and Complex analysis (pdf / dvi).
The setting is as follows (page 23): Let $X$ be a $(\mathfrak{g},K)$-module and let $X(\mu)$ denote its ...
- 8,597
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votes
1
answer
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Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups
The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid ...
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Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 511
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2
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A kind of orthogonal subtorus
Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup
$S = \{x \in \...
- 285
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votes
4
answers
2k
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Commutator of closed subgroups
Suppose we have a simply-connected Lie group $G$. Let $G_1$ and $G_2$ be two closed and connected subgroups of $G$. Is it true that the commutator $[G_1,G_2]$ is a closed subgroup of $G$?
- 468
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3rd homotopy group of a compact Simple Lie Group
Suppose $G$ is a compact simple Lie group with Lie algebra $\mathfrak g$. Then we know that $\pi_3(G)=Z$. Now suppose that $H_\alpha$ is a co-root vector in correspondence with a root $\alpha$. So it ...
- 131
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Simple discrete subgroups of Lie groups
Upon Ian Agol's suggestion, I separated this question from the one on non-residual finiteness in
Non-residually finite matrix groups
Question. Are there infinitely generated simple discrete ...
- 31.2k
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answer
656
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Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
- 657
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votes
3
answers
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Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
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answer
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Symmetric tensor product of bosonic/fermionic Hilbert space
Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\...
- 965
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What is the "positive part" of the unit ball in $M_n(R)$ ?
In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...
- 52.3k
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1
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Diophantine elements in SU(2)
Following notions from [1], call a set of elements $g_1, \dots, g_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W_m$ of ...
- 3,323
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2
answers
1k
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Borel–Weil theorem - reference request
I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
- 1,219
5
votes
3
answers
712
views
A followup on non-homogeneous spaces.
This question asks for an example of a manifold which is not a homogeneous space of any Lie Group, and many examples are given in the answers. However: is there a an example known with a metric of ...
- 96.4k
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Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ?
Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My ...
- 41
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3
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Peter-Weyl theorem as proven in Cartier's Primer
I'm reading Pierre Cartier's A primer of Hopf algebras to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. ...
- 33.8k
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Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
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Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
- 41
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0
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339
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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
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
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votes
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answers
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Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
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0
answers
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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
6
votes
4
answers
1k
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Polar decomposition for quaternionic matrices?
A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
- 3,979
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1
answer
976
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Character determines the representation?
Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
- 11.2k
13
votes
1
answer
731
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free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...
- 2,035
11
votes
2
answers
491
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Cohomology of $T^n/W$ for compact Lie groups
Let $G$ be a compact, connected and simply connected Lie group.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ ...
6
votes
4
answers
889
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On the determination of a quadratic form from its isotropy group
Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let
$$
O(F):=\{g\in GL_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\}
$$
be the isotropy group of $F$.
Q: So how ...
- 7,609
11
votes
1
answer
258
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Algorithmic Borel finiteness for hyperbolic manifolds
It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
- 96.4k
7
votes
1
answer
743
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schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups
I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
- 4,466
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2
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921
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Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
- 799
3
votes
2
answers
445
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Lie algebra version of principal bundle?
I am wondering whether there is a Lie algebraic version of principal bundle for Lie group over a given manifold $M$. The first thing I try to think of is group cocycle picture of principal bundle.
- 1,271
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votes
3
answers
4k
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Can every Lie group be realized as the full isometry group of a Riemannian manifold?
Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to ...
- 2,830
12
votes
5
answers
4k
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Weight lattice and the fundamental group
Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of $...
- 1,219
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votes
2
answers
4k
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Metric Connections on a Lie Group
A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...
- 1,378
3
votes
3
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467
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Generating a reductive real Lie group with finitely many maximal real tori
Let $G$ be be a connected real algebraic reductive Lie group. Is it always possible to find finitely many maximal algebraic $\mathbf{R}$-tori $\{T_i\}_{i=1}^n$ such that the group generated by the $...
- 7,609
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2
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669
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SO(p,q) and Howe Duality
I recently learned of a relationship between the representations of the groups $SO(p,q)$ and $SL(2,\mathbb{R})$ which is part of an apparently much larger set of ideas known as Howe Duality. My ...
- 295
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927
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How does duality of symmetric spaces explain the hyperbolic cosine theorem?
There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
- 10.4k
1
vote
0
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Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
4
votes
0
answers
184
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Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
2
votes
0
answers
224
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$(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$
I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.
...
- 8,597
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votes
1
answer
287
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rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
- 4,466
4
votes
2
answers
643
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Cartan-Hadamard Theorem
Can someone point out the gap in this argument. Consider a simply-connected Lie group with the (-)-connection. This connection is flat and so the sectional curvatures are zero. Then, by the Cartan-...
- 1,378
2
votes
1
answer
268
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Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group
I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy ...
- 4,466
2
votes
2
answers
551
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L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 4,466
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votes
3
answers
1k
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volume of compact simple Lie groups under the natural Euclidean embedding
I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I ...
- 4,466
12
votes
1
answer
2k
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unitary irreps of O(p,q)
I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in $\mathbb{...
- 295