All Questions
Tagged with lie-groups reference-request
298 questions
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107
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Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure
In this question, the following fact was used by the respondent
A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar
measure contains a coset of $G^0$, the connected component of
$G$ ...
- 2,118
12
votes
2
answers
849
views
Groups associated with infinite dimensional Lie algebras
There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
- 1,089
9
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2
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413
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Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$?
Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$?
I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?
- 183
2
votes
1
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217
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A variation of closed-subgroup theorem
$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...
- 14.3k
7
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3
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599
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Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 103
4
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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 ...
- 35.5k
8
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2
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482
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Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.1k
9
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1
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361
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Homotopy groups $\pi_{4n-1}(SO(4n))$
There is a very natural way to define generators of $\pi_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, On ...
- 641
4
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0
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552
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Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
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0
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142
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Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
- 11
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3
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577
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Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$
$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel ...
- 1,565
2
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0
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98
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Sublattices in the standard integral symplectic lattice
Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
- 427
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1
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1k
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Classification of (not necessarily connected) compact Lie groups
I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
- 473
2
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1
answer
85
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Closure of the set of topological non-generators in compact Lie group
Let $G$ be a (non-abelian) compact connected Lie group. Let $K\subseteq G^2$ be a set of pairs $(g_1, g_2)$ which $\textbf{do not}$ generate $G$ topologically.
For which Lie groups $G$ is it known ...
- 728
5
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1
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283
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Finite order automorpisms of affine Kac-Moody Lie algebras
It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan ...
- 59
2
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0
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134
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Homogeneous metric surfaces
I am looking for a reference for this result.
Let $S$ be a metric space such that
it is homeomorphic to a two-dimensional manifold,
it is 2-homogeneous: given two pairs of points
$(x,y)$ and $(x',y')...
- 18.7k
19
votes
1
answer
1k
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A result on Lie group actions on 15-dimensional spheres?
In this interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05., what result is the host talking about?
Transcription of the relevant part:
"If you have two sets of symmetries, known as ...
- 23.3k
6
votes
2
answers
342
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Positive genus Fuchsian groups
Let $G$ be a lattice in $SL(2,\mathbb{R})$. Is it always true that there exists a finite index subgroup $F$ of $G$ such that the quotient surface $\mathbb{H}/F$ has positive genus? Is the statement ...
- 63
18
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1
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631
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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
9
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1
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750
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Learning from unsuccessful attempts at the Poincaré conjecture
This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong.
Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
- 1,763
5
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1
answer
147
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Equivalence generated by Jacobian minors
Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
- 15.5k
7
votes
2
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669
views
Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914
Question 1.
Does Élie Cartan's paper
Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355
contain a classification of $\Bbb C$-linear involutions of simple ...
- 14.1k
4
votes
1
answer
249
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Regarding extensions of finite groups by Tori
I know how to prove the following result. However, my proof is a little bit long and complicated and only uses fairly low tech results in group cohomology. It would be nice if I could find a citation ...
- 318
4
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2
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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
3
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0
answers
65
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Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries
Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...
- 345
3
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0
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58
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Isoclinism for Lie groups: existing accounts of basic properties?
Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps
$$...
- 25.8k
1
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0
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461
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When two isomorphic subgroups are conjugate?
I would like where could I find a reference in which this question is answered.
Let us consider $H,H'\leq G$ two isomorphic subgroups of a Lie group $G\in \{\mathbf{SO}(n),\mathbf{SU}(n),\mathbf{Sp}(...
- 201
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0
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120
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Commensurability of arithmetic, irreducible, nonuniform lattices
Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
- 255
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0
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38
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Classification of involutions on $G_{2}$-homogeneous spaces
Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?
- 2,630
6
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1
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395
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Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$
Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the ...
3
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0
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133
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Weyl group stabilizer of semisimple element in adjoint group
Let $G$ be semisimple group over $\mathbb{C}$ of adjoint type. Let $T$ be a maximal torus, $s\in T$ semisimple element. Let $W$ be a Weyl group and $W(s)$ be a stabilizer of $s$ in $W$. I am ...
- 31
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0
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234
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Nascent formal group law
$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal ...
- 10.5k
4
votes
1
answer
511
views
Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
- 2,527
4
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2
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267
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Finite models for torsion-free lattices
Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...
- 1,255
1
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0
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76
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Nomenclature: does this coset space have a name?
in my work I tripped on a specific coset space and before starting thinking about it by myself, I wanted to check the literature. However, I do not know if the object has a name (which makes ...
- 111
2
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0
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86
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Homomorphisms from circle to $GL(k,\mathbb{R})$ [duplicate]
Example 3 at the website tricki proves that every measurable homomorphism of groups from the circle to the non-zero complex numbers is continuous. Is there any analogous (true) statement for ...
- 323
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1
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262
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What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?
Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
- 7,633
1
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0
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164
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Continuous vs $L^2$ homomorphisms from circle to non-zero complex numbers
Let $T:S^1\to C^\ast$ be a group theoretic homomorphism from the circle to the non-zero complex numbers.
Presumably it is true that if $T$ is $L^2$, then it is continuous. Is there a simple proof, or ...
- 323
7
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1
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429
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K-type in discrete series representation
The following result seems well known.
Let $G$ be a reductive Lie group with a maximal compact subgroup $K$. If $\mu$ is an irreducible unitary representation of $K$, then there exist only finitely ...
- 951
4
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0
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105
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Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$
Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
- 6,180
2
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1
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82
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Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
6
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0
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550
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Lattices in Lie groups
In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups.
Is there a result that gives a general description of a lattice in an arbitrary Lie group?
Something ...
user130903
12
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1
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684
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Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?
Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
I was told some fact along this line is true but could not find any ...
- 511
4
votes
1
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302
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On maximal closed connected subgroups of a compact connected semisimple Lie group?
Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra.
Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
- 2,446
3
votes
1
answer
261
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Subgroups of compact Lie groups generated by a subset of nodes of the Dynkin diagram
Where can I find a reference for the following fact, or as close as possible to it?
Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{g}...
- 32.5k
3
votes
1
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276
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For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
- 332
8
votes
1
answer
266
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When does $BG \to BA$ loop to a homomorphism?
If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
- 35.5k
4
votes
1
answer
119
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Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
- 991
3
votes
1
answer
138
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Frechet Lie groups and their subgroups
1) Let $G$ be a Fréchet Lie group. Let $H$ be a closed subgroup. Is it always true that the centraliser of $H$ is a Fréchet subgroup of the lie group?
2) Is the closed subgroup theorem valid for ...
- 755
6
votes
2
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788
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Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations
I recently got interested in representation theory in quantum mechanics and I read the following theorem:
Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\...
- 189