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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A specific question regarding a proof in Knapp's book
I got stuck in an apparently trivial point within the proof of Lemma 3.13 on p. 55 of Knapp's Representation Theory of Semisimple Groups. The author concludes in the first paragraph that $f_v$ must be ...
- 486
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213
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Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$
Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on $\...
user21574
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Sum relation for Clebsch-Gordan-Coefficients?
In the context of (numerically) calculating reduced density matrices in the Lipkin-Meshkov-Glick model (a model introduced to describe atomic nuclei, which has however found many other applications as ...
- 11
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285
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A lie Subgroup of SO(4n)
Hello everyone
thanks to all of you
I have two questions and I hope to get some guide:
1. One of the Lie groups in the Berger's list of holonomy groups of locally irreducible Riemannian manifolds is $...
- 129
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Geometrizing the Third Cohomology of a Complex Lie Group
If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\...
- 23k
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Riemannian metric on complexification of Lie group
Let $G$ be a compact linear group and $G^c$ be its complexification. Then there is a diffeomorphism $f: G^c \to G \times Lie(G) $ given by $$ x e^{iA} \to (x,A).$$
Let $h$ be the pull back metric of ...
- 157
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2
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641
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Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
- 7,609
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Is U(n) a normal subgroup of SO(2n)? [closed]
U(n) is the group of n by n unitary complex matrices and SO(2n) is the group of 2n by 2n real orthogonal matrices with determinant 1.So far I can show that how to get an injective group homomorphism ...
- 831
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1
answer
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An innocent looking subgroup of $U(n)$
Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
- 4,466
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473
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Lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type f(n)*delta^{n(n-1)/2}
Is there a lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type
f(n) * delta^{n(n-1)/2}? For which f(n)? How can it be proven?
n(n-1)/2 is the number of degrees of ...
- 145
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Is a Poisson Group a group object in the category of Poisson Manifolds?
I realized that I am very confused about a certain sign in the definition of a Poisson group. I will give some definitions, and then point out my confusion.
Definitions
Group objects
Let $\mathcal ...
- 54.6k
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162
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Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian?
Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian?
If possible I'd also like to know if the right translation of the Shatten $p-norm$ on the Lie algebra gives rise to a bi-...
- 2,099
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Compact homogeneous spaces that admit a self map of degree >1
It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly ...
- 14.3k
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168
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Automorphisms of Nilmanifolds
Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
- 419
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Abel transform is an * isomorphism for SL(2, R)
Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group.
Then we have an isomorphism of $*...
- 11.2k
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p-adic Lie theory
It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases.
...
- 1,706
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417
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A question regarding Lie group actions
Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?
- 21
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235
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The fundamental in the tensor square of a complex representation of $SO(N)$
I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...
- 173
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Words in two infinitismal rotations
I asked this as subquestion in a comment pursuant to my Banach-Tarski
question. I think it is worth promoting here to a question in its own right.
Consider these two matrices over ${\Bbb R}[[\...
- 17.6k
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evaluating an integral related to the volume of Hessenberg orthogonal matrices
Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm d}...
- 4,466
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1
answer
358
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Proper morphisms: Lie groups vs. group schemes
A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.
In particular, let $G$ be a Lie group acting on a ...
- 1,211
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votes
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426
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lowest weight representation of loop groups
I am trying to understand lowest representations of loop groups as developed in Pressley and Segal's book. Specifically I want to be able to compute the weight spaces that appear in a lowest weight ...
- 3,968
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0
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210
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Adjoint action of semi-direct product
Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions
$\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request
All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...
- 3,051
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answers
572
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How to find the normalizer of a finite subroup in a Lie group?
If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers $N_{G_i}(...
- 31
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Extending a discrete sub group to a lattice in unimodular Lie groups
Given a unimodular Lie group $G$ and a discrete subgroup $\Gamma\subseteq G$, under what conditions does there exists a discrete subgroup $H$ s.t. $\Gamma\subseteq H$ and $G/H$ has finite volume? Also,...
- 85
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
- 3,541
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differentiating positive energy LG reps
Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
- 43.2k
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question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
- 1,305
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Which orbits of a separable representation of the infinite unitary group are closed?
Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...
- 965
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325
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identify a curious subgroup in $U(n)$
Consider the following element $A$ in $U(n)$:
$$ \begin{pmatrix} 1/2(1+z) & 1/2(1-z) & \\\\
1/2(1-z) & 1/2(1+z) & \\\\
& &I_{n-2} \end{pmatrix},$$
where $|z| = 1$.
Now ...
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answer
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Periodic automorphism of nilpotent Lie algebra
Are there a non-abelian nilpotent Lie algebra $\mathfrak{n}$ over $\mathbb{R}$ and an automorphism $\alpha: \mathfrak{n} \to \mathfrak{n}$ such that:
$\alpha$ is periodic,
the fixed subspace of $\...
- 687
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196
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Reference Help: Matsuki duality Orbits
I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
- 111
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Adjoint orbit of two vectors
Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...
- 1,016
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Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
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Which maximal closed subgroups of Lie groups are maximal subgroups?
Which maximal closed subgroups of Lie groups are maximal subgroups?
- 17.6k
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$G$-invariant part of products of determinants of minors
Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for $\...
- 4,991
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Wang's C-subgroups and M-manifolds
Let $K$ be a semisimple compact Lie group.
In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer ...
- 585
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364
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Is $R(su_{4})\cong R(so_{6})$?
This is one of small the unsettled questions I had in my senior project. I want to prove for type $D$ we have $R(T)$ is a free module over $R(G)$ by finding a basis. I think we should have,$R(G)\cong ...
- 799
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172
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Generating Set for $O(V)$ over $\mathbb Z_2$
I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
- 105
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Topologic or geometric mean of the structure constants of a semi simple lie algebra
Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by
$$k(X,Y,Z)=B([X,Y],Z).$$
with $X,Y,Z\in \mathfrak{g}$.
...
- 1,111
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1
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Transformation of the fundamental group of Lie groups under group homomorphisms
Hello!
I have encountered the following problem while trying to solve a different one: let G, H be two semisimple Lie groups, and $G\to H$ a Lie group homomorphism. Does anyone know if the question ...
- 31
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1
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Hermitian symmetric structure on a homogeneous subspace
Let $G$ be a semisimple group over $Q$ and $K$ a maximal compact subgroup of $G(R)^+$.
I am assuming that $G(R)^+/K$ has a structure of a non-compact Hermitian symmetric domain.
Let $g= p + k$ be ...
- 547
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Semi-simple lie groups and their fundamental representations
I've got a really basic question on the representation theory of semi-simple Lie groups. I know that a rank-R semi-simple Lie group possesses R fundamental representations. But is the relation ...
3
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1
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559
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unitary representation of semisimple lie groups in view of Moore's ergodicity thm
Let $G=G_1\times\ldots\times G_n$ be a product of (connected) simple Lie groups and $(H, \pi)$ be a unitary representation of $G$. In a proof of Moore's ergodicity thm it uses the following fact $$\pi=...
- 853
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on z-extensions
Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a $...
- 3,472
1
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1
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164
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Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$
There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$
Let $\mathbb{Z}/2=\...
- 9,019
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votes
0
answers
148
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Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
- 6,351
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0
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304
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Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)
For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...
- 3,399
1
vote
1
answer
517
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Linearization of actions of semi-simple groups
What is known about local structure of actions of semi-simple groups? More precisely, suppose I have a semi-simple group $G$ acting on a variety $V$, and $x\in V$. Assume that the stabilizer of $x$ is ...
- 1,590