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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3
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Generating a reductive real Lie group with finitely many maximal real tori
Let $G$ be be a connected real algebraic reductive Lie group. Is it always possible to find finitely many maximal algebraic $\mathbf{R}$-tori $\{T_i\}_{i=1}^n$ such that the group generated by the $...
- 7,609
2
votes
1
answer
545
views
Characters separating points on Maximal Torus modulo Weyl group?
Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group.
Every finite-dimensional representation of G has a character, which is a function on G, T and T/...
- 585
6
votes
2
answers
815
views
Classification of real forms up to inner automorphisms
I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be real forms of a complex ...
- 63
3
votes
0
answers
170
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The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...
- 2,091
2
votes
2
answers
1k
views
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 ...
- 313
5
votes
0
answers
304
views
Decompositions of a compact Lie group into "fixed point set types"
Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
- 1,942
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vote
0
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247
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Fourier transform of a matrix represented compact lie group
In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...
- 141
2
votes
2
answers
1k
views
A question about the affine Grassmanian
For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as:
$$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$
Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
- 741
1
vote
0
answers
84
views
Integrable modules and comodules
Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
- 6,201
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votes
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465
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Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
3
votes
1
answer
235
views
Reference to definition of matrix log with domain SO(3) which is Borel measurable
I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...
- 203
2
votes
2
answers
311
views
Lie groups and NSS+LC group
Let $G$ be a locally compact group without small subgroups. Is $G$ a "finite" dimensional Lie group? (i.e, $G$ is not infinite dimensional Lie group.)
Are Lie groups precisely the locally Euclidean ...
4
votes
0
answers
173
views
Ring of SO(n)-invariant differential operators on M_n,m
I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...
- 41
2
votes
1
answer
319
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Reference request: Calculation in exceptional Lie groups
Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of $...
- 804
1
vote
1
answer
379
views
Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$
Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
user21574
3
votes
1
answer
687
views
pullback diagram of principal bundles
Let $G, G_1, G_2$ be compact Lie groups with homomorphisms $f_1:G_1 \to G$ and $f_2: G_2\to G$. Let $P_1, P_2$ be principal bundles for $G_1,G_2$ and assume that the bundles $P_i\times_{G_i} G$ are ...
- 1,282
4
votes
0
answers
83
views
Points of failure in definition of X- and A-moduli spaces for arbitrary G
In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...
- 1,131
6
votes
0
answers
697
views
distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
- 342
3
votes
2
answers
445
views
Lie algebra version of principal bundle?
I am wondering whether there is a Lie algebraic version of principal bundle for Lie group over a given manifold $M$. The first thing I try to think of is group cocycle picture of principal bundle.
- 1,271
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votes
2
answers
1k
views
Killing form vs its counterpart in a given represenation
Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
- 965
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votes
2
answers
833
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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 ...
- 33.3k
1
vote
1
answer
340
views
Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$
I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
- 561
3
votes
1
answer
187
views
Rational homogenous spaces and symmetric spaces
What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
- 838
2
votes
0
answers
115
views
Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
- 513
2
votes
2
answers
344
views
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 ...
- 11.2k
4
votes
2
answers
545
views
Is every group object in TopMan a Lie group?
Recall that a Lie group is a group object in the category of C∞ manifolds.
If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure ...
- 54.6k
2
votes
1
answer
414
views
Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain
Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
- 103
2
votes
0
answers
62
views
Codistal subgroups of locally compact groups
Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is codistal in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal ...
- 4,728
4
votes
1
answer
799
views
Transitive action on the sphere
Hello,
One of the subgrouops of $SO(n)$ which acts transitively on the sphere $S^{n-1}$ is the (compact) symplectic group $Sp(n/4)$. The center of $Sp(m)$ is isomorphic to $\mathbb{Z}_2$. Can we embed ...
user30435
5
votes
1
answer
472
views
Finite dimensional homogeneous spaces of $Diff(S^1)$
This question is a refined version of Representations of infinite dimensional Lie algebras as vector fields on manifolds
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. ...
- 548
0
votes
1
answer
296
views
Compact Lie groups with only 3 dimensional cohomology generators
Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.
For which $M$, $...
2
votes
2
answers
479
views
Non-trivial representation of second-smallest dimension
Hi,
The complex simple algebraic group $Sp_{m,\mathbb{C}}$ of $2m$-dimensional space $V$ has, for $m≥2$, an irreducible representation of dimension $m(2m−1)−1$ in a subspace of codimension $1$ of the ...
user30435
2
votes
0
answers
111
views
A question about the associative classes of parabolic subgroups
Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
P(\...
- 21
1
vote
1
answer
130
views
Structures on open surfaces
Let $\phi\in PSL(2,R)$ be hyperbolic and $\varphi\in PSL(2,R)$ be elliptic.
Is it possible to find a local homeomorphism $f:H^2\rightarrow H^2$ such that
$f(\phi(x))=\varphi(f(x))$ for all $x\in H^2$ ?...
- 31
2
votes
1
answer
283
views
Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$
Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...
- 101
2
votes
0
answers
125
views
Global decomposition of reductive spaces
Consider a reductive homogeneous space $M=G/H$ with corresponding Lie algebra decomposition $\frak{g}=\frak{m}+\frak{h}$. Then there is a local diffeomorphism
$$
(exp\, X, h)\mapsto (exp\, X) h\quad \...
- 1,378
1
vote
1
answer
178
views
Reducible reductive Lie subalgebras of so(p,q)
Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
- 601
1
vote
0
answers
184
views
How the exceptional simple Lie groups/ algebras were first discovered and by whom?
I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...
- 21.8k
8
votes
0
answers
408
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Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
- 52.9k
1
vote
1
answer
264
views
labeling state vectors in representation space of a simple lie algebra
Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation space....
- 85
6
votes
2
answers
315
views
Covering relations in $K\backslash G/B$
Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\...
- 27.9k
3
votes
0
answers
62
views
Reference request: table of representation rings and relations
Where can one find a table of generator–relator expressions for representation rings $R(G)$ of simple Lie groups $G$ and explicit maps between them? For example, given a maximal torus $T < G$ or a ...
- 2,995
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
7
votes
1
answer
721
views
Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups
The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid ...
- 295
1
vote
0
answers
324
views
Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation
Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
- 2,099
8
votes
1
answer
2k
views
Invariant Vector Fields for Homogenous Spaces
As we all know, the space of invariant vector fields on a Lie group can be identified with the tangent space at the identity (or any other point for that matter). My question is: How does this ...
- 751
8
votes
1
answer
730
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 1,414
5
votes
1
answer
224
views
Lattice in motion group
Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group
$$
G={\mathbb R}^d\rtimes O(d).
$$
Let $\phi:G\to O(d)$ the projection homomorphism.
Is it true that $\phi(\Gamma)$ is ...
user1688
0
votes
2
answers
448
views
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_\...
- 8,597
1
vote
1
answer
121
views
A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]
Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...
- 183