Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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22 views
What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?
If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
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1answer
113 views
The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$
Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...
4
votes
1answer
99 views
connections on principal bundles over $S^1$
Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...
5
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1answer
253 views
Root space decomposition
What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...
4
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1answer
378 views
Goin' with the flow with Kummer and Pascal: Combinatorics and geometry underlying the logarithm of the derivative operator
In a MO-Q111165 and associated MSE-Q125343, I present a pair of raising / lowering (creation / annihilation) operators $R_x = log(D)$ and $L_x = -x·D$ with $D=d/dx$ (for a sequence of functions ...
3
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1answer
101 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 ...
4
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1answer
383 views
Topologie sur l'ensemble des sous-groupes de GL_n(R)
Bonjour,
Est ce qu'il existe une topologie naturelle sur l'ensemble des sous-groupes du groupe général linéaire ?
English translation: "is there a natural topology on the set of subgroups of the ...
4
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69 views
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 ...
3
votes
0answers
41 views
Closed form for 3j-symbol ratios
I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...
2
votes
0answers
88 views
On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?
Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
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0answers
69 views
Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$
Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...
2
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1answer
99 views
lifting group action
Let a Lie group $G$ act on a manifold $M$. Under which condition(s) we have a lifting action of $G$ on the universal covering of $M$?
By Bredon, I already know that there is a group $\widetilde{G}$, ...
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votes
1answer
202 views
Computing the fundamental group of a flag variety
Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
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5answers
350 views
Reference requested: Random walk on groups
I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
1
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1answer
129 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 ...
4
votes
1answer
74 views
Connection between degree of growth and return probabilities of random walks on Lie groups
Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...
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0answers
99 views
Mathematica package for Lie algebra computations?
I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...
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0answers
35 views
Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
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1answer
178 views
Subgroups with trivial Centralizers
Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?
...
4
votes
1answer
136 views
Tannaka–Krein duality
First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...
4
votes
0answers
75 views
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 ...
3
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0answers
143 views
What is a pure algebraic interpretation for this dynamical property?
According to comments of Yves Cornulier to the previous version of this question I revise the question as follows:
To what extent the following types of Lie algebras $A$ are classified? And what is ...
9
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0answers
293 views
How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?
Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
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1answer
176 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 ...
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0answers
177 views
Connections on a Lie Group
A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...
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51 views
lie groups and coset manifolds
Let $G$ be a lie group, $H\subseteq K\subseteq G$ be closed subgroups , and $H$ be normal in $G$. I wonder if the coset manifold $\frac{\frac{G}{H}}{\frac{K}{H}}$ is diffeomorphic to $\frac{G}{K}$.
...
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2answers
191 views
When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...
4
votes
5answers
387 views
How to characterize Dirac's gamma matrices in differential geometry?
I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 ...
5
votes
2answers
315 views
When is a subgroup of a Lie group itself a Lie group?
Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?
Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.
...
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1answer
94 views
Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space
Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...
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1answer
171 views
$SO(N^2-1)$ and the adjoint representation of $SU(N)$
It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
2
votes
1answer
114 views
Indefinite orthogonal groups over p-adics
Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
1
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1answer
110 views
Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure
I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...
6
votes
1answer
215 views
Can Galois conjugates of lattices in SL(2,R) be discrete?
Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
3
votes
1answer
145 views
Stationary curves on homogeneous spaces
Consider $M \cong G/K$ ($G$ a lie group with a transitive action on $M$ and $K$ a subgroup) and consider a Lagrangian $\mathcal{L}: TM \rightarrow \ \mathbb{R}$ (no time dependence). Consider also ...
6
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0answers
216 views
Injectivity of Lie group exponential function
If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...
4
votes
2answers
198 views
Summary of Lie-Algebra integration tactics
If this is in the scope of MO, I would like to gather here the known tactics of
Lie algebra integration, since it appear surprisingly hard to find such a
compendium, library or any other kind of ...
0
votes
2answers
150 views
A question on Lie algebras
To what extent, the following types of Lie algebras are classified :
Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.
2
votes
1answer
127 views
Distance measure for noisy $SE(3)$ transforms
I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...
6
votes
1answer
123 views
Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation
Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...
4
votes
0answers
90 views
Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
6
votes
1answer
279 views
Origin of symbols used for half-sum of positive roots in Lie theory?
The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...
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0answers
93 views
A question on lie groups( Lie algebras)
What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:
There are two non zero vector fields $X, Y \in ...
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1answer
55 views
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 ...
3
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0answers
80 views
Equivariant Poincare Series of Based Loop Group of SU(2)
Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
2
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0answers
71 views
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 ...
0
votes
1answer
121 views
Tensor product of generator of SU(n)
I'm doing research in quantum mechanics and got some trouble. Any help would be very much appreciated.
Let $\{\lambda_j\}$ be the set of generator of $SU(n)$. Consider the operator:
$K=\sum_j ...
2
votes
1answer
85 views
Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?
For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. ...
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1answer
103 views
the group of all biholomorphic group automorphisms of complex tori
My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let ...
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0answers
120 views
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 ...
