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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Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
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Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal
irreducible representation. It can be shown that the number of
$\mathfrak{g}$-...
- 59
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Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n?
I'm rather ignorant in both fields, but I would still like to endeavor asking this question. I've just learned that any Lie group is diffeomorphic to a compact Lie group cross $\mathbb{R}^n$. While ...
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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^...
- 1,282
1
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1
answer
173
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Representing elements of U(N) or SO(N) by elementary rotations exp(i phi_n sigma_n)
Given a (orthogonal) basis $(\sigma_n)_{n=1,\dots,K}$ of the algebra $u(N)$, we can represent any element $U$ of the corresponding group $U(N)$ in the form
$U=e^{i\sum_{n=1}^K\varphi_n\sigma_n}$.
Is ...
- 13
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563
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Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?
To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$
(vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching
an extra vertex to every old vertex in $Q_0$. Then ...
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Symmetry analysis of differential equations
What is the connected component of the identical transformation in the pseudogroup of local diffeormorphism on the real line?
similar question
Let $\tilde t=T(t)\quad T_t>0,$ be a local ...
1
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218
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Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I have a few questions on an application of the Weyl character formula.
To start with we work with the $\mathbb{Q}$ version of Hamilton's quaternions and consider the maximal order $\mathfrak{O} = \...
- 562
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Is the object we get when we quotient $U(N)$ by $U(N-k)$ familar?
If we quotient $U(N)$ by $U(N-1)$ we get the odd dimensional sphere $S^{2N-1}$. (Here the quotient is in the sense of embedding $U(N-1)$ in the bottom right hand corner (with 1 as the (1,1) entry and ...
- 1,776
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on Neron defect of smoothness for groups schemes
Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...
- 3,472
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1
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312
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Which covers of Lie groups will I get
Here is a question I get from sitting in my Lie algebra class:
Fix a Lie algebra $\mathfrak{h}$, we know there is a unique simply connected Lie group $H$ which serves as the universal cover of other ...
- 1,160
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155
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complex reductive Lie groups which are not defined over the real numbers
Hello
Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it possible,...
- 1
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700
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Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 2,709
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Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 8,512
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247
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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 ...
- 31
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350
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Dimension of Lie group embedding
Let $G$ be a compact Lie group of dimension $n$. Then we can embed $G$ (topologically) into a connected compact Lie group $H$. (One may choose $H=U(m)$, the unitary group, for example.)
The question ...
- 804
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answers
226
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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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341
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Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$
Hi,
I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ ...
- 6,131
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224
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$(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$
I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.
...
- 8,597
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386
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Is the quotient functor of points of a Lie group with the subfunctor of a closed subgroup a sheaf?
Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that $H$ has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds.
The ...
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608
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Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
- 157
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211
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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}{\...
- 4,466
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votes
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657
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For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
- 637
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answers
82
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Free S^1 action on a symmetric space of compact type
Consider a symmetric space G/H of compact type where rank(G) is greater than rank(H). The Euler characteristic of this space is known to be zero. What can be said about the existence of a fixed point ...
- 21
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359
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Does Branching in the Weight Diagram affect an embedding?
All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$.
Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
- 5,191
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vote
1
answer
443
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Sections and subgroups in a unipotent group
Let $U$ be a smooth connected unipotent algebraic group over an algebraic closure $k$ of $\mathbf{F}_p$. Let $U' := [U,U]$ be the derived (= commutator) subgroup, and assume it is central in $U$. Let $...
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When do reflection groups act discretely on quadrics in indefinite/semi-Riemannian situation?
The hyperbolic case seems to be well understood after work of Vinberg. Given a lattice $L$ with quadratic form $Q$ of signature $(1,n)$, the orthogonal group of $L$ acts discretely on the affine ...
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Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
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Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
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To what extent does (co)homology of groups made discrete depend on set theory?
There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...
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418
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Centralizers and Cartan involutions
This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, ...
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Construction of an algebra with prescribed representation of the automorphism group.
For this discussion, $G$ is a compact semisimple Lie Group.
For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some ...
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Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
- 5,264
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Adjoint orbits of small subspaces in Lie algebras
I've been studying isometric quotients (by compact Lie groups) of compact simply connected homogeneous spaces $G/H$ and their inherited curvature. One of the issues that continually arises is the ...
- 6,546
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374
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a question about centralizers in semi-simple groups
I have a question concerning centralizers in real reductive groups. I'd like to know if the following property is available in any references.
Let $L\subset H\subset G$ be an inclusion chain of ...
- 1,305
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Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups
In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
- 2,392
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votes
1
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121
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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^+$ ...
- 3,637
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1
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399
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correspondence between invariant forms and Lie groups
In Lie theory, one often asks about alternating forms on $\mathbb{R}^n$ which are invariant under some particular subgroup $G\subseteq GL_n(\mathbb{R})$, and there is always some algebra of invariant ...
- 6,069
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"locally" factoring subgroups of Lie groups
I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).
I start with a subgroup ...
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Orbits of semi-algebraic actions
Hello all,
I recently came across the following Theorem in Gibson (Singular points of smooth mappings, 1979). Since I haven't seen this result somewhere else and this reference is not so widespread, ...
- 461
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Finite subgroups (lattices) in the large N limit of SU(N)
I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
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Can we get fixed points sets of principal isotropy groups of orthogonal representations via iteration of involutions?
Consider an orthogonal representation of a compact Lie group $G$ on an Euclidean space $V$.
Denote by $H$ a fixed principal isotropy group, which we assume to be non-trivial.
Consider the fixed ...
- 4,729
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157
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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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164
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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 ...
- 4,466
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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 ...
- 161
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answers
105
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approximation in Lie algebras
Let $x_{1}$, $x_{2}$, $x_{3}$ three disctinct closed points of a curve $X$ over an algebraically closed field k.
Let G a connected reductive group and $\mathfrak{g}$ his Lie algebra.
I fix a Borel $...
- 3,472
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Constructing solutions to matrix equations
Let $k,n$ be integers, $u_1,\dots,u_n \in U(k)$, $d_1,\dots,d_n \in \mathbb Z$ with $\sum_{i=1}^{n} d_i =: d \neq 0$.
Consider the map $w:= U(k) \to U(k)$ with
$$w(v):= u_1 v^{d_1} \cdots u_n v^{d_n} ...
- 25.5k
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82
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decomposition lemma in adelic groups II
Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...
- 3,472
1
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0
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293
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Use Lie Sub-Groups of GL(3, R) for elastic deformation ?
I'm interested in representing elastic deformations (e.g. stretching) using Lie groups. There are a few references to using $GL(3,\mathbf{R})$ but I'm wondering if possible to use subgroups of $GL(3,\...
- 157
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189
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average Riemannian distance between Identiity and a random point in SO(n) or SU(n)
I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
- 4,466