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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Complex Finite Dimensional Representation of GL(N,C)
What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?
We already know all the complex finite dimensional linear representation of SU(N).
- 3,804
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2
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487
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Symmetric and Exterior products of sl(n,C)-module
Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$.
Let q be a symbol.
$f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$
$g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$
...
- 585
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answer
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Compact homogeneous spaces that admit a self map of degree >1
It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly ...
- 14.3k
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Invariants of a set of real unit vectors in 3d space, under SO(3)
I have a set of $n$ real unit vectors, in 3-dimensional space.
(It is a follow up of Sets of vectors related by a rotation.)
Is there a construction providing a complete set of independent*) ...
- 1,612
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1k
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Representation ring of SU(n)?
What's the structure of representation ring of SU(n)?
What are the representations of generators?
- 3,804
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674
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Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
- 3,399
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1
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652
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ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$
I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...
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Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$
I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \...
- 548
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Relation between Almost simple Lie groups and semisimple Lie groups?
Hello everyone,
What is the relation between almost simple Lie groups and semisimple Lie groups? (Especially in the case of subgroups of $SO(2,n)$.)
Recall that:
Def1: A Lie groups $G$ ...
- 129
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votes
2
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741
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Measuring how far from being cocompact a lattice is
Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-...
- 9,486
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Definitions of Reductive and Semisimple Groups
I'm a graduate student. I've been reading Knapp's two books Representation Theory of Semisimple Groups and Lie Groups Beyond an Introduction. He seems to give wildly different definitions for the ...
- 151
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1
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166
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Directed graphs and Compact Lie Groups
Is there a method for associating the edges and vertices of an arbitrary directed graph with the irreducible representations of a compact Lie group and the intertwiners
of the adjacent edge ...
- 11
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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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174
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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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Left invariant metric on ${\rm SL}_n(\mathbb{R})$
I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...
- 109
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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 ...
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answer
1k
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Are certain simple Lie groups linear algebraic groups?
Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)
Such a group should automatically ...
- 196
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2
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418
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About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
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Computational complexity of multiplication in a nilpotent group?
What is the computational complexity of multiplication in a Carnot group ?
Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition
...
- 785
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1
answer
484
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Sobolev norm of distance function on Riemannian manifold
$\DeclareMathOperator\SL{SL}$Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and ...
- 23
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638
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Curvature of the Cayley projective plane
The Cayley projective plane can be realized as the compact homogeneous space $F_4/\mathrm{Spin}(9)$. In this way one can compute the curvature of this symmetric space in terms of a suitable ...
- 403
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795
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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 ...
- 41
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256
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The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,019
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answer
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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}{{\...
- 25.8k
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votes
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291
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Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group
Hi All,
I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:
I am trying to understand the structure (e.g., decomposition) of the unitary ...
- 955
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votes
2
answers
721
views
Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure?
Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$).
My questions is: it is always true that we have ...
- 9,019
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Is SL(2,C)/SL(2,Z) a quasi-projective variety?
Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
Is $SL(...
- 18.7k
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votes
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505
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comprehensive presentation of the unitary dual of $SO_0(n,1)$
The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(...
- 2,446
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3
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1k
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$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
- 17.5k
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votes
1
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894
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Unusual decomposition of 3x3 real symmetric matrices - is this possible?
If $M$ is a 3x3, real symmetric matrix, then I know there are a few ways to decompose $M$ as
$M = A^T D A$,
where $D$ is a real diagonal matrix: e.g., this can always be done for some $A \in SO(3)$, ...
- 818
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1
answer
269
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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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1
answer
571
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algebraic closure of Lie groups in
Let $G$ be a connected, simply connected, solvable, complex Lie group with a discrete subgroup $\Gamma$.
Let also $G_a$ be Hochshild-Mostow hull of $G$, i.e., there exists a solvable linear algebraic ...
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"No Small Subgroups" Argument
What is the "no small subgroups" argument for $GL(n,\mathbb R)$? That is, how do we show that in $GL(n,\mathbb R)$ there exists a neighborhood of the identity which contains no subgroup other than the ...
- 486
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About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$...
- 9,019
5
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1
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393
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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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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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3
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787
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Nilpotent Lie Algebras
Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. Given $\xi\in\frak{g}$, what is known about the intersection of $im(ad_{\xi})$ (the image of $ad_{\xi}:\frak{g}\rightarrow\frak{g}$...
- 4,920
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164
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Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$
There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$
Let $\mathbb{Z}/2=\...
- 9,019
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337
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Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups?
Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.
Does every (...
- 1,089
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2
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714
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Infinitesimal rigidity vs. local rigidity
I am thinking about homomorphisms $\mathrm{Hom}(\Gamma,G)$, where $G$ is a Lie group and $\Gamma$ is a discrete, finitely generated subgroup.
This question talked about the difference of ...
- 1,211
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1
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502
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What is the importance of $\pi_{i}G$?
I often see conditons like $\pi_{2}G\not=0$ in reading old papers on Lie groups(no, my memory is wrong, they asked if $\pi_{1}G$ is free). I want to ask why we need this condition and how the higher ...
- 799
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2
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399
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Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)...
- 1,211
5
votes
1
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653
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Reference request for the list of maximal subgroups of SU(3,1)
Is there a reference with the list of maximal subgroups of SU(p,q) for "small" values of p and q? (such as SU(3,1) as suggested in the title of the question)
- 1,675
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3
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752
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How can I tell whether a manifold is homogeneous?
I have been influenced by this question with many beautiful answers.
Are there any useful practical criteria to say positively that a real connected paracompact smooth manifold $X$ is homogeneous?
I ...
- 12.3k
7
votes
1
answer
1k
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G-equivariant Whitehead's Theorem
Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...
- 8,529
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123
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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 ...
- 21
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2
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393
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Have derivatives of determinants along 1-psg's ever been 'coherently' computed via Jacobi's formula?
Suppose $\mathfrak{p}$ denotes all the symmetric matrices in $\mathfrak{sl}_{2n} \mathbb{R}$.
Then for each parameterized 1-dimensional linear subspace $\xi=\xi(t)$ of $\mathfrak{p}$ we get a 1-...
- 2,274
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votes
2
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496
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Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?
I need to answer (affirmatively, I hope) the following question:
In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset
...
- 1,804
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0
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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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2
answers
1k
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Why limit of discrete series representation?
In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?
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