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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$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group
Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...
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20
votes
6
answers
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Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
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votes
3
answers
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Nonnegativity of an integral over the unitary group
For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
- 1,593
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vote
1
answer
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The identity connected component of centralizers of unipotent orbits
This is, in a way, a follow up question to Unipotent orbits and intersection with Levi and pseudo-Levi subgroups.
I was reading "A generalisation of the Bala–Carter theorem for nilpotent orbits&...
- 141
10
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1
answer
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Is it possible to average a riemannian metric over an action and preserve curvature bounds?
Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric
$$g'(X,Y)=\int_Gg(...
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votes
3
answers
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Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
1
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1
answer
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A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?
The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...
- 1,144
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Analogue of the special orthogonal group for singular quadratic forms
The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
user82886
18
votes
1
answer
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Best texts on Lie groups for number theorists
What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering ...
- 221
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$q$-analogs of total positivity
A real matrix $M$ is called totally positive if all of its minors are positive; these matrices have been extensively studied, and there are generalizations to other Lie types, for example by Lusztig.
...
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9
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Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
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votes
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answers
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Textbook or lecture notes in topological K-Theory
I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
6
votes
0
answers
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Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
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When is a smooth field's flow map volume preserving diffeomorphism
Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a (single) real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi_V$ defined as ...
- 43
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answers
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CW structure on $\mathrm{PU}(3)$/Heisenberg group
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where
$H$ is the Heisenberg group of order 27
$G_{81}$ is the No. 9 group of order 81 (...
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answer
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What is the subgroup of $\mathrm{SL}(n,\mathbb{C})$ which preserves the discriminant?
$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $\mathcal{P}_{n-1}$ be the space of complex polynomials in one variable, say $z$, of degree at most $n-1$. As a complex vector space, it is clearly $n$...
- 5,215
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1
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$G/T$ has finitely many $G^\theta$ orbits
Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.
I'm looking for ...
- 139
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Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$
Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$?
Apparently, there is an almost complete classification in ...
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1
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Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization
Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant.
...
- 385
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votes
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Are invariant forms on homogeneous spaces necessarily closed?
Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed ...
- 141
13
votes
3
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Probability of commutation in a compact group
It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...
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3
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Rank of a Lie subgroup generated by two Lie subgroups
$\DeclareMathOperator\rank{rank}$Let $G$ be a compact connected Lie group and $H$, $K$ be two closed connected subgroups. By Mikhail Borovoi's answer to In a compact lie group, can two closed ...
- 377
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votes
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An equivariant map from sphere to a Lie group of lower dimension which is not null homotopic?
Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This ...
- 356
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Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups
This is a question about the answer in this other post: Symplectic group over integers and finite fields.
In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
user341790
4
votes
2
answers
634
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How to describe the compact real forms of the exceptional Lie groups as matrix groups?
I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...
- 5,215
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vote
0
answers
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
- 1,121
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votes
1
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Why are root data a natural candidate for classifying connected reductive groups?
For the purpose of this question, you may assume that we are working over the complex numbers.
Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the ...
- 557
3
votes
0
answers
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Equivalent condition for compact group to act transitively
Let $ M $ be a connected manifold. Let $ \pi_1(M) $ be the fundamental group of $ M $.
Suppose there exists a compact group $ K $ that acts transitively on $ M $. Then $ \pi_1(M) $ must have a finite ...
- 4,081
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votes
1
answer
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Improved classification of compact Lie groups
This question is a follow-up to Classification of (not necessarily connected) compact Lie groups. In the answer to that question, @LSpice proved that any compact, not necessarily connected Lie group $...
- 473
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0
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When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?
Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
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3
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Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic
Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...
- 139
1
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1
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Group structure on the strip
Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits_{l\in \{a,b\}}\{d(x,l)\}$.
Let $(X,*)$ be a topological group with the following ...
- 107
1
vote
0
answers
188
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Is the group law for SO(2n, R) encoded in so(2n,R)?
Note that this is a partial duplicate of my math.stackexchange question here. In this post I am asking something slightly broader. Note that I am a mathematical physicist and not a representation ...
- 201
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votes
3
answers
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Integral cohomology of $SU(n)$ - looking for constants
I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$,...
- 653
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votes
0
answers
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Question on a remark in Speh's paper
I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
- 1,121
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votes
0
answers
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Product decomposition for intersection of a parabolic with a mirabolic of a closed subgroup
Let $G$ be a reductive group defined over $\mathbb{Z}_{p}$ and let $H$ be a closed reductive subgroup of $G$. Let $Q_{G}$ be a parabolic subgroup of $G$ with Levi decomposition $Q_{G} = L_{G} \ltimes ...
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votes
1
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Schubert cells in G/P for reductive G
All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
- 1,677
1
vote
0
answers
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Infinite-dimensional Lie group corresponding to $U\mathfrak{g}$?
Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U\mathfrak{g}$ is then an infinite-dimensional associative algebra which can be endowed with the structure of a Lie algebra. Is ...
- 101
1
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1
answer
358
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Unipotent orbits and intersection with Levi and pseudo-Levi subgroups
Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the ...
- 141
3
votes
1
answer
280
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Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 145
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votes
1
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The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
- 141
5
votes
1
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Elementary $p$-subgroups of a compact Lie group
How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?
Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}_p$, there is an ...
- 935
6
votes
3
answers
327
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$SO(m+1)$-equivariant maps from $S^m$ to $S^m$
Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.
Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.
Question 1: Is F the ...
- 63
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votes
0
answers
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The density of the image of a unitary irrep (a generalization of Burnside's theorem)
I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...
- 161
0
votes
1
answer
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Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 6,211
18
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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votes
0
answers
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Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$
$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
- 161
8
votes
1
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228
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Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at ...
- 10.5k
4
votes
2
answers
467
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What does this paper have to do with Hilbert's fifth question?
Apparently, there is a paper
M. Sablik, Final part of the answer to a Hilbert's question. Functional Equations - Results and Advances. Edited by Z. Daróczy and Zs. Páles, Kluwer Academic Publishers ...
- 51
4
votes
2
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315
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Low dimensional integral cohomology of $BPSO(4n)$
Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in
Hiroshi Toda, Cohomology of ...
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