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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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
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Topology on extensions of topological groups
Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups.
Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...
- 125
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Spin and SO groups associated to a degenerate symmetric bilinear form
In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...
- 4,312
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Relation of Lie Groups and Cohmology Theories via Formal Group Laws
There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.
For a Lie group G one can choose coordinates at the unit and expand ...
- 1,273
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313
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Symplectic structure on $Sym^kG^{\mathbb{C}} $
Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{...
user21574
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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 ...
- 1,060
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154
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Nilpotent orbits and subspaces
Let ${\mathbb g}$ be a simple complex finite dimensional Lie algebra, $X\subseteq{\mathbb g}$ a nilpotent orbit. Did anyone study maximal vector subspaces of the closure $\overline{X}$?
In particular,...
- 12.3k
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456
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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 $\frac{G}{K}$.
...
- 377
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101
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G-invariant functions on manifold for G compact
In a paper I saw the following statement:
Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
- 501
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2
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Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$
What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups ...
- 125
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Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
- 3,244
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Space of polynomially growing harmonic functions on a Lie group
There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads:
If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...
- 527
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Representations of Finite Subgroups on Homology
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...
- 4,920
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Normal form for trace-free real cubic forms in 3 variables under SO(3)-action?
I'm looking at irreducible, real representations of $SO(3)$. The 5-dimensional irrep is isomorphic to the space of trace-free quadratic forms on $\mathbb{R}^3$, and we all know that any such ...
- 818
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Support of an infinitely divisible measure.
Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of $\...
- 31
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Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]
Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$?
Cross posted from ...
- 2,099
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73
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False optima for control on Lie groups?
Consider the equation
$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$
evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:
$J:G \rightarrow [0,...
- 2,099
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Unitary dual of $Sp_4(\mathbb{R})$
Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
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The Gysin Sequence for an Associated Bundle over a Partial Flag Variety
Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the ...
- 4,920
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Harmonicity on semisimple groups
I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
- 41
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504
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Local version of a slice (for a Lie group action)
Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$.
Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such ...
- 5,824
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Action of the isometry group of the hyperbolic 5-space
We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the ...
- 1,692
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1
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316
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Decomposition of Lorentz-like groups
When studying the Lorentz group $O(1,3)$, one can decompose it into four parts... physicist usually called these
Proper-orthochronuos $\mathscr{L}^{\uparrow}_+$,
Proper-asynchronous $\mathscr{L}^{\...
- 690
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Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
- 1,719
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124
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query about Jacques Tits' "Homorphismes `abstraits' de groupes de Lie"
I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and he seems to be making a claim that if you have a simply connected Lie group then the derived subgroup is always ...
- 2,125
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Sections of inverse image sheaf of sheaf of sections of vector bundle
Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-...
- 1,368
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Invariant Finsler Metrics on Homogeneous Spaces
Given:
1) a Finsler metric $F_p : T_p G \rightarrow \mathbb{R}$ on $SU(N+1)$
2) a $U(N)$ subgroup of $SU(N+1)$ which is a stabilizer subgroup of some point on $CP^N$ (complex projective space) in ...
- 2,099
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158
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Kernel of the Weil homomorphism for compact symmetric spaces
Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...
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Automorphism groups of symmetric cones
Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...
- 954
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Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups"
First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn'...
- 339
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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 ...
- 1,282
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What is the intersection of Spin(7) and U(4)?
I'm just curious from Berger's classification of Riemannian holonomy, how do Spin(7) manifolds intersect the other types of Riemannian manifolds?
In particular, what is the intersection of Spin(7) ...
- 1,347
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730
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different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
- 1,305
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Is every irreducible unitary class one representation induced?
Let $G$ be a connected semi simple Lie group with finite center.
Fix a maximal compact subgroup $K$.
An irreducible representation $(\pi,V)$ of $G$ is called a "class-one representation", if it ...
- 147
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346
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Orbits of Root Vectors
Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let $$\frak{g}=\frak{t}\oplus\bigoplus_{\alpha\in\Delta}\frak{...
- 4,920
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Analogies, Riemann surfaces and Algebraic groups
Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...
- 3,020
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Reduction of antisymmetric complex matrices
Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...
- 4,123
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how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?
how many injective homomorphism between two lie algebra $sl_2 $and $sp_6$ up to conjugate by$Sp_6$ ?
- 709
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Invariant symmetric bilinear forms and H^4 of BG
I am reading this paper of Teleman and Woodward.
On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}_k$. Why is this ...
- 21k
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How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,019
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how to find the induced metric on an orbit?
Hi, Let $M$ be a pseudo-Riemannian manifold and $G$ a (Lie) subgroup of $Iso(M)$ which acts on $M$ smoothly and properly. Suppose we know the orbits up diffeomorphism. Is there a systematic way to ...
- 129
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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 $G$. ...
user47125
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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 ...
- 27.8k
5
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214
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Explicit generators for homotopy groups of Lie groups
I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$.
It is ...
- 17.4k
17
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What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
- 47.3k
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rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
- 4,466
4
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1
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502
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Action of $ax+b$ with compact support
I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at ...
- 8,098
1
vote
0
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150
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Symplectic structures on the grassmannian model of the based loop group
$\newcommand{\Ad}{\operatorname{Ad}}$
In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...
- 627
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83
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Rigidity of lower-dimensional lattices in Euclidean groups
Informal intro / motivation:
Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer ...
- 13.6k
5
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1
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404
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determining symplecticity (if that's a word)
Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the ...
- 96.4k