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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Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
1
vote
1
answer
244
views
Irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$
I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of ...
- 13
2
votes
0
answers
558
views
What is the precise definition of connect semisimple Lie groups "without compact factors" in the literature?
I frequently see in the literature (of homogeneous dynamics, in particular) of the notion "semisimple Lie groups without compact factors" without seeing its precise definition.
I have three (...
- 1,565
2
votes
1
answer
182
views
Gelfand-Naimark and Peter-Weyl for the unitary group
Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
- 1,144
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50
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How to construct lattice points in bounded symmetric domain?
Consider the Hermitian bounded symmetric domain for $k \leq m$:
$$
C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \}
$$
where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
- 8,597
3
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0
answers
167
views
The subalgebras of $\mathfrak{su}(2^n)$
$\DeclareMathOperator\su{\mathfrak{su}}$I want to find out all the subalgebras of $\su(N)$, in particular, $N=2^n$, which is the Lie algebra of $n$-qubits.
I don't know whether this is a hard question ...
- 89
4
votes
1
answer
164
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Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation
We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...
- 73
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1
answer
125
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Any abelian Lie subgroup containes a connected Lie subgroup of codimension 1 [closed]
I am trying to understand the proof of the following claim (see A.L. Onishchik, E.B. Vinberg (Eds.) Lie Groups and Lie Algebras III, p.50, Theorem 3.1).
Theorem 3.1 (ii) If the Lie group $G$ is ...
- 19
3
votes
1
answer
176
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Example of a supersolvable Lie group/algebra whose nilradical does not have a complement
What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is ...
- 33
-1
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555
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Representation of Lie algebra $\operatorname{SE}(2)$
When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the ...
- 129
12
votes
2
answers
887
views
Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$
which in our case ...
- 7,789
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votes
0
answers
229
views
Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201
4
votes
1
answer
702
views
Centralizers of semisimple subgroups
$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....
- 491
1
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1
answer
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Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
- 317
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3
answers
2k
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How bad can $\pi_1$ of a linear group orbit be?
Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
- 31.5k
5
votes
1
answer
309
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Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it
I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is ...
- 2,533
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votes
1
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645
views
The group of isometries of Shahshahani metric
Edit: 28 January 2023 I just realized that this metric is frequently used in this paper
https://hal.science/hal-01382281/document
Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\...
- 356
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2
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923
views
Eigenvectors of random unitary matrices
Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$,
$$U=VDV^\dagger,$$
where $D={\rm diag}(z_1,z_2,...,z_N)$ is diagonal.
If $U$ is taken at random uniformly with respect to Haar ...
- 1,549
2
votes
0
answers
86
views
On the "integrality condition" of the bilinear form in the Chern-Simons action
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be a principal $G$-bundle over a smooth orientable manifold $\mathcal{M}$. Furthermore, let $\langle\cdot,\cdot\...
- 1,429
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7
answers
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Langlands Dual Groups
Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
- 16k
3
votes
1
answer
355
views
The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 10.5k
1
vote
1
answer
159
views
Holomorphic map to Möbius group
$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...
- 307
7
votes
1
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572
views
Does Aut(G) → Out(G) always split for a compact, connected Lie group G?
The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
- 473
2
votes
0
answers
199
views
Element conjugate to a maximal torus
It is well known that any element in a compact connected Lie group is conjugate to an element in a maximal torus. Let G be a Lie group that is not necessarily compact and/or connected. Let $x\in G$ ...
- 59
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0
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340
views
Can discrete groups be Lie groups? Are all finite groups Lie groups? [closed]
I was trained as a physicist, rather than a mathematician. So, I apologize if my question is naive.
As a physicist, I think of Lie groups as being continuous groups. I have begun studying the second ...
6
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2
answers
1k
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Non-faithful irreducible representations of simple Lie groups
For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...
- 835
5
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1
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199
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Coordinate-free description of an alternating trilinear form on pure octonions
Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$,
and ...
- 14.2k
8
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0
answers
228
views
What can we say about the homogeneous spaces $E_8/E_7$ and $E_7/E_6$?
For the three exceptional compact Lie groups $E_8, E_7, E_6$ we have the inclusions
$$
E_6 \subseteq E_7 \subseteq E_8.
$$
What can we say about the the homogeneous spaces
$$
E_8/E_7, ~~~~ E_7/E_6?
$$
...
- 305
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votes
0
answers
85
views
The closure of the subgroup generated by a vector field may not be compact
Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group:
$$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$
where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
- 111
2
votes
1
answer
473
views
Explicit automorphism map of ${\rm Spin}(8;\mathbb{R})$, ${\rm SO}(8;\mathbb{R})$, ${\rm PSO}(8;\mathbb{R})$
$\DeclareMathOperator{\SO}{\mathrm{SO}}\DeclareMathOperator{\Spin}{\mathrm{Spin}}\DeclareMathOperator{\Inn}{\mathrm{Inn}}\DeclareMathOperator{\Out}{\mathrm{Out}}\DeclareMathOperator{\Aut}{\mathrm{Aut}}...
- 2,147
7
votes
1
answer
371
views
The space of skew-symmetric orthogonal matrices
Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \...
- 12.4k
2
votes
2
answers
446
views
Relation between two homomorphisms from $SO(3)$ to the Möbius group $PGL(2,\mathbb{C})$ [closed]
Let $f, g$ be two homomorphisms from $\mathrm{SO}(3)$ to $PGL(2,\mathbb{C})$. Does there exist $S \in \mathrm{PGL}(2,\mathbb{C})$ such that for all $X\in \mathrm{SO}(3)$ we have $g(X)=Sf(X)S^{-1}$?
...
- 2,286
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
5
votes
0
answers
199
views
Outer and inner automorphism of $\mathrm{Pin}$ groups
$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
- 317
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votes
1
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views
What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
- 1,111
1
vote
1
answer
156
views
Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]
Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
- 317
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votes
6
answers
2k
views
How do I stop worrying about root systems and decomposition theorems (for reductive groups)?
I apologize for this being a very very vague question.
Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ...
- 229
2
votes
1
answer
719
views
Classification of isometries of hyperbolic 3-space
Denote the upper half space by $\mathcal{H}_{3}=\Bbb{C}\times (0,\infty)$. A point $P \in \mathcal{H}_{3}$ is given as, $P=(z, t)=(x, y, t)=z+t j$ where $z=x+i y$ and $j=(0,0,1) .$ The group $P S L_{2}...
- 33
3
votes
0
answers
547
views
Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$
Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...
- 10.5k
1
vote
1
answer
372
views
What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...
- 23
6
votes
1
answer
551
views
Two definitions of automorphic forms on Lie groups
My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group.
The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
- 391
3
votes
2
answers
418
views
Homomorphism from noncompact semisimple Lie group to compact Lie group
Is it true that there is no homomorphism from a semisimple Lie group without compact factor to a compact Lie group?
10
votes
1
answer
590
views
Involutive automorphism of simple Lie algebra
I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...
- 951
5
votes
1
answer
978
views
Existence proof of Bourbaki, Differentiable and Analytic Manifolds
I am reading through Chapter III of Bourbaki, Lie Groups and Lie Algebras, and many proofs cite the Bourbaki volume Differentiable and Analytic Manifolds. I can't find this book anywhere. Does it ...
- 6,180
3
votes
0
answers
53
views
Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
- 563
2
votes
1
answer
217
views
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
3
votes
3
answers
629
views
Real representations of SO(n) and U(n)
I would like to get some references where I can find the theory of the real representations of $\mathbf{SO}(n)$ and $\mathbf{U}(n)$.
In particular, I would like to know for which dimensions there ...
- 201
2
votes
0
answers
187
views
Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
- 1,061
3
votes
0
answers
90
views
Closedness of subgroup corresponding to semi-simple real Lie subalgebra
I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
- 21.8k
1
vote
0
answers
86
views
Isomorphism between $T_{[g,X]} (G \times _H \mathfrak{g}/\mathfrak{h})$ and $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h}$
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $H$ be a Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$. Consider the action of $H$ on $G$ by right multiplication and the ...
- 139