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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set of coisotropic orbits open and dense, iff group acts locally transitively almost everywhere
I worked now some time with coisotropic actions of Liegroups on manifolds.
But there is one key fact, that I don't understand, although it is very central in my considerations.
Let $(M,\omega)$ be a ...
- 501
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full set of invariant functions on manifold
Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.
Is it always possible to construct $k$ functions $f_1, \...
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Calculation with weights of $E_6$
Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
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Non invertibility of certain integral arising from group action
Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question.
Edit 2: I understand from the following post that some part of the previos version of my question has ...
- 356
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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: ...
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Conjugacy class of semisimple elements
Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak g$ and $G_1$ be a complex simple subgroup of $G$ with Lie algebra $\mathfrak g_1$. We may assume $G$ and $G_1$ to be of adjoint type. ...
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Infinitesimal equivalence of admissible representations
Let $G_0$ be the $\mathbb{R}$-points of a real reductive group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. What is the precise relation between the category of ...
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Dimension of Unipotent Radicals
A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
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How to get Haar measure on a compact Lie group, given the complexification?
This is the first in what may be a series of questions on the theme "a Banach algebraist/Bear Of Little Brain needs help with algebraic geometry".
$\newcommand{\Cplx}{{\mathbb C}}\newcommand{\fg}{{\...
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Exponential analogue of formal connections
Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...
- 2,751
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A question on Lie algebras, Lie groups and multiplets
I wonder if anyone can help me with this question regarding algebras and multiplets. In a nice review paper (McVoy, Rev Mod Phys 37(1)) the author states the following theorem: “Given any set of ...
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partial order on conjugate classes of subgroups
G is a group.
For a subgroup H of G, note $[H]$ the class of subgroups which are conjugate
to H.
Define the binary relation:
$[H] \leq [K]$
iff
$H_0 \subset K_0$ for some $H_0 \in [H]$ and $...
- 51
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Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
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How was this Lie algebra found?
In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.
...
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Lie Subgroups of SO(2)×So(n)
Hello,
I need to know (connected closed) Lie subgroups of SO(2)×So(n), indeed these are compact Lie subgroups of SO(2,n) which I am looking for. But I don't know what we can say about Lie subgroups of ...
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Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
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Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,920
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Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ?
Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My ...
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Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group
I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy ...
- 4,466
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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) ...
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Reference request: Basic H-Space properties of $SO(3)$
I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include
What H-maps are there $SO(3)\rightarrow S^3$...
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induced homomorphism of adjoint action on fundamental group
Let $G$ be a non-connected compact Lie group and $Ad_g: G \rightarrow G, x \rightarrow gxg^{-1}$ be the adjoint action. Look at the induced homomorphism $ (Ad_g)_* : \pi_1 (G) \rightarrow \pi_1(G)$. ...
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When is a homogeneous space a variety?
Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then $G/H$ may not be a group, but it will be a homogeneous space for $G$ with stabilizers conjugate to $H$. Sometimes, this is a ...
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The geometry of the holomorph of a Lie group
Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left invariant ...
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What does a homogeneous space of a linear algebraic group know about the group?
Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers
and $H\subset G$ is an algebraic subgroup.
In general, we can write the algebraic variety $X$...
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$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
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Is the endpoint map smooth
Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...
- 2,099
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tensor hierarchy for Lie groups from Maurer-Cartan form
The question is about the family of tensors that are naturally associated to any nice Lie group.
Take the Mauer-Cartan form, $\omega=g^{-1} dg$ and I would like to make the covariant index of this one-...
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Dimension of Span of Adjoint orbit in $\mathfrak{su}(n)$
Given two elements $A,B \in \mathfrak{su}(n)$ what is the dimension of the span of the following adjoint orbit: $\{Ad_{e^{sA}}(B) \ | \ s \in [0,t]\}$ for different values of $t$. Does it ever change ...
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Maximal subgroups of semisimple Lie groups
The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) ...
user22139
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line bundle on affine grassmannian and central extension
Let $G$ be a connected reductive group over $\mathbb{C}$, let $Gr$ be the affine grassmannian of $G$. On $Gr$, we know that there is a canonical line bundle $L$ (the generator of $Pic(Gr)$).
Now $G(\...
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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 ...
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Milnor's model of $EG$ and Kac-Moody groups
I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
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Lie Groups and Lie Algebras
What is the exact relationship between Lie groups and Lie algebras? I know it's not bijective because all commutative Lie groups have isomorphic Lie algebras.
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Integration over special unitary group
It is known that for $SU(N)$
$$
\int \chi_{\mu_1}(UV_1)\chi_{\mu_2}(U^{-1}V_2)\, dU = \delta_{\mu_1\mu_2}\frac{\chi_{\mu_1}(V_1V_2)}{\dim(\mu_1)}
$$
where $dU$ is Haar measure on $SU(N)$ normalized ...
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Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147
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Complex Lie inverse Galois problem
My question is about the inverse Galois problem for infinite dimensional complex manifolds. If $K$ is the field of meromorphic functions over a complex manifold $M$ and $G$ is a finite or infinite ...
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What's the story with the Hopf fibration and the Jacobi identity?
I like the Hopf fibration of the 3-sphere $S^3$ enough that I found a nice way to make a physical model of it. All you need is to combine a bunch of key rings in such a way that (ii) every pair of ...
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Are compact simple groups homotopically non-abelian?
Take a compact connected simple centreless Lie group $G$.
Can the commutator map $G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?
I am interested mostly in the case, ...
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How does duality of symmetric spaces explain the hyperbolic cosine theorem?
There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
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A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field
What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:
There is an $n$ dimensional sub vector space $V\...
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201
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Manifolds as simultaneous coset spaces
Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
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Split real form of $SL(2,\mathbb{C})$ description of the two sphere?
If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of $...
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A question on Grassmannian
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...
- 119
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Large and Small Conformal Groups
It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold ...
- 1,101
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Uniquely divisible neighborhoods of identity in topological groups
Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a ...
- 1,959
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Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?
I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
- 650
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468
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Complex symplectic reduction
Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...
- 1,347
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2
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921
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Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
- 799
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Homology of Lie groups
Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
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