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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rotation along a coordinate axis and Kac random walk
With due respect, I am profoundly puzzled by an amazing paper of David K. Maslen on the eigenvalues of the Kac random walk on $SO(n)$.
The Kac walk is essentially given by choosing a random pair of ...
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Why is there a unique hyperbolic simplex of largest area?
Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...
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Maximal Tori and group structures on spheres
It is known that for any compact Lie group $G$ with maximal torus $T$, that any other maximal torus $T'$ is conjugate to $T$. This might be a bit of a stretch, but I was wondering if it is possible to ...
3
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Springer isomorphisms and parabolics
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $...
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Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
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2
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What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$?
Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $ \mathbb{Q}_p$ by a non archimedean field and $\mathbb{Z}_p$ by the ring of integers.
Note that the space ...
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Normalizers of maximal compact groups?
Consider a reductive group over a local field. What is the normalizer of a maximal compact subgroup?
If this is to general, what is the normalizer of $GL(n, \mathbb{Z}_p)$ in $GL(n, \mathbb{Q}_p)$, ...
3
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Transitive action on moduli space of holomorphic curves.
If $G$ is a complex semi simple Lie group and $P$ is a parabolic subgroup of it, the quotient $G/P$ is endowed with a Kahler structure and the action of $G$ on $G/P$ is holomorphic with respect to ...
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A Weyl invariance constructed from Clebsch-Gordan Coefficients.
Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition:
\begin{equation}
V \otimes \tilde{V} = \bigoplus_i U_i
\end{equation}
\noindent were $U_i$ are also irreps ...
6
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locally-free Lie group action not preserving any measure
I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a ...
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Haar measure on infinite dimensional Lie groups?
Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like ...
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Representing a product of matrix exponentials as the exponential of a sum
In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
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symplectic representations: when could the center act trivially?
I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
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Is an identity that is true for matrix Lie groups true for all Lie groups?
Many identities for Lie groups are more easily proved for matrix groups. A non-trivial example is the equation
$$
\frac{d}{dt}\vert_{t=0} \exp(-X)\exp(X+tY) = \frac{1-e^{-\operatorname{ad} X}}{\...
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one-parameter subgroup and geodesics on Lie group
Hi,
Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
4
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Non-integrable almost-complex structures for homogeneous spaces
Let $M$ be a smooth homogeneous $G$-space for a Lie group $G$, and let $J$ be a $G$-invariant almost-complex structure for $M$. Do there exist succinct sufficient (and necessary) conditions for $J$ to ...
17
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4
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Exceptional isomorphisms of Lie groups
It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment ...
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Extension of the Hall-Witt identity
Although this question is mostly out of curiosity (as of now), I hope it is nevertheless suitable for MO.
This very recent (and still open) question about the Hall-Witt identity led me to wonder:
...
4
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some weird relations among beta random variables
Let $X_i, Y_i, Z_i$ be three iid family of standard normal random variables. Then the following random variable is distributed the same as the first coordinate of a uniform $2$-sphere:
$$ \frac{X_1}{\...
0
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2
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real orbits of highest weight vectors
Let $G_\mathbb{C}$ be a complex simple Lie group and let $V_\lambda$ be its finite dimensional irreducible representation with highest weight $\lambda$. Define $\mathcal{H}\_{\mathbb{C}} \subset V_\...
5
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Invariants of co-diagonalizability in real symmetric matrices
This question has been mentionned to me by U. Frisch. He wanders whether it has ever been considered by algebraists.
In the vector space ${\bf Sym}_n({\mathbb R})$, two elements commutte to each ...
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The Schwartz Space on a Manifold
I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.
This question is also vaguely related (both questions arose ...
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Software for Computing Baker-Campbell-Hausdorff
Does anyone have a recommendation for software which can efficiently calculate the Baker-Campbell-Hausdorff series in classical Lie algebras?
Right now, I have a problem which boils down to ...
2
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1
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Iwasawa decomposition and Cartan decomposition
The Iwasawa decomposition and Cartan decomposition for $GL(n)$ is available for local fields. This can be proven for totally disconnected fields and archimedian fields seperatly by hand.
Here is a ...
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1
answer
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Does the diffeomorphism group preserving a particular section act transitively?
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...
0
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Possible typos in Adams' Lectures on Lie Groups
In his proof of Lemma 5.39, that if $\theta_r$ is a simple root then $\phi_r$ permutes the positive roots except $\theta_r$, which goes to $-\theta_r$, I don't quite follow his second proof. He claims ...
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Continuous cohomology of semi-simple Lie group
Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_\text{cont}^q(G,M)$ vanish?
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Is there a "Cartan product" of Harish-Chandra modules?
If $\lambda,\mu$ are two dominant weights for a Lie group $G$, then
there is a canonical (up to scale, perhaps)
surjection $V_\lambda \otimes V_\mu \to V_{\lambda+\mu}$
of finite-dimensional ...
2
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1
answer
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$\omega$-Commuting matrices vs Stone-von Neumann Theorem
Let me first recall the Stone-von Neumann theorem that if two one-parameter groups of unitary operators $U_t$ and $V_s$ over a Hilbert space satisfy $U_tV_s=e^{ist}V_sU_t$ for every $s,t\in{\mathbb R}$...
0
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Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?
Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^...
2
votes
1
answer
467
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Theorem of Kuiper for Hilbert spaces with group action
Let $H$ be an infinite dimensional separable complex Hilbert space with Lie group action (I am mostly interested in the case $S^1$). Let $\text{Gl}_{G}(H)$ be the space of invertible, bounded and ...
8
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1
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Is $SU(\infty)$ amenable?
We can write the finitary special unitary group $SU(\infty)$ as the direct limit
$\varinjlim SU(n)$ of ordinary special unitary groups. These groups $SU(n)$ are compact, thus amenable. In other ...
7
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2
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Lie groups acting transitively (and isometrically) on anti de Sitter spaces
I hope this question is not deemed too localised.
Recall that anti de Sitter space is the lorentzian analogue of hyperbolic space; that is, a simply-connected lorentzian manifold of constant negative ...
4
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1
answer
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Decomposition of Haar measure other than Hurwitz's
Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
3
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criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group
By Givens' rotation $R(1,2;\theta)$ I mean a matrix which has the
$$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$
$2 \times 2$ block at ...
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3
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How to correctly generate uniformly distibuted random elements from SO(n)?
I already found some way to produce such matrices from SO(n) with a method called subgroup algorithm but I would like some advice on the method I used. Nowhere I could really find any paper relating ...
2
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2
answers
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a question about invariant volume forms on homogeneous spaces.
Here I consider $G$ a connected Lie group, which is assumed to be linear (i.e. embeddable in some $GL_n(\mathbb{R})$, and $X$ a homogeneous space under $G$. Fix a point $x\in X$, one considers the map ...
13
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What's the Lipschitz constant of the exponential map for $\mathrm{SO}(n,R)$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\so{\mathfrak{so}}$Consider the Lie algebra $\so(n)$ equipped with the metric $\langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}...
1
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1
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Dominant weights appear in Discrete Series
If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...
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subgroups with the same number of roots that the group.
When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10)...
6
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reference for the slice theorem for Banach Lie group actions on Banach manifolds
I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach ...
9
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1
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Principal congruence subgroups in higher rank
I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
4
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1
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is connected complex Lie group with a trivial center linear?
There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected ...
3
votes
1
answer
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Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
Background:
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
11
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1
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Lie $2$-groups and differential equations
I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: ...
33
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Isometry group of a homogeneous space
Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
4
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Taylor's series for Lie groups
Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) ...
6
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3
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Group cohomology of compact Lie group with integer coeffient
It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology ...
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Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$
I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...