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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how to determine the Weyl group of a diagonalizable subgroup?
Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(...
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A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
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Other than SU(3), SO(4), SU(2)xU(1), are there compact semisimple Lie groups which exactly two 3-dimensional representations that are dual to each other?
In my original question, I asked which compact Lie groups $G$ have a certain property. Jim and Dan showed that this property is equivalent to $G$ having exactly two irreducible 3-dimensional ...
user2529
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distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
- 342
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Lattices in $SL(n,\mathbb R)$
If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$?
I know that the ...
- 93
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How to find the normalizer of a finite subroup in a Lie group?
If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers $N_{G_i}(...
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Homology versus cohomology of Lie groups
A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that ...
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Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?
More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
- 118k
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How can one find generators of basic differential forms on homogeneous spaces?
Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable. ...
- 399
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Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
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Transformation of the fundamental group of Lie groups under group homomorphisms
Hello!
I have encountered the following problem while trying to solve a different one: let G, H be two semisimple Lie groups, and $G\to H$ a Lie group homomorphism. Does anyone know if the question ...
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Proof for which primes H*G has torsion
In 1960 Borel proved a beautiful result:
**Theorem**. Let G be a simple, simply connected Lie group. Suppose that *p* is a prime that does not divide any of the coefficients of the highest root (...
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"Modern" proof for the Baker-Campbell-Hausdorff formula
Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric (...
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Lorentz quotient and orientation
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U....
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Commutativity of the fundamental group of any Lie Group [closed]
How do we formally prove that the fundamental group of any Lie group is always commutative?
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Finite dimensional homogeneous spaces of $Diff(S^1)$
This question is a refined version of Representations of infinite dimensional Lie algebras as vector fields on manifolds
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. ...
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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
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Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
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Topologic or geometric mean of the structure constants of a semi simple lie algebra
Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by
$$k(X,Y,Z)=B([X,Y],Z).$$
with $X,Y,Z\in \mathfrak{g}$.
...
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How to deal with the singular reduction of the Hamiltonian n body problem?
I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.
...
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Can one understand the Kelvin transform conceptually?
Let $U = \mathbf{R}^n - \{ 0 \}$, $n > 2$ and consider for a function $f \in C^2(U)$ the Kelvin transform
$$f^\star(x) = r^{2-n} f\left(\frac{x}{r^2}\right),$$
where $r = \lvert x \rvert$. One ...
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Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
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Proper morphisms: Lie groups vs. group schemes
A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.
In particular, let $G$ be a Lie group acting on a ...
- 1,211
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Lie groups vs. algebraic groups and deformations
I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...
- 1,211
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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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Matrices generating non-discrete subgroups of SL(2,R)
Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
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Heights in reductive groups
Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
- 11.2k
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density of conjugate of arithmetic subgroup
$K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?
- 709
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equations over (some) lie groups
To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...
- 96.4k
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density in SU(2,1)
Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
- 709
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connected compact semisimple lie group finite fundamental group
I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact. Is there any source however that fleshes this out ...
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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 $...
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Examples in the vein of smooth manifold + group = Lie group [closed]
I am currently writing a thesis and got to thinking about the bigger picture of mathematics in the following sense. Both manifolds and groups have highly developed theories in their own rights. When ...
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answers
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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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Intertwining Integral defined on a Weyl group?
Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?
http://www.jstor.org/...
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Is $R(su_{4})\cong R(so_{6})$?
This is one of small the unsettled questions I had in my senior project. I want to prove for type $D$ we have $R(T)$ is a free module over $R(G)$ by finding a basis. I think we should have,$R(G)\cong ...
- 799
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Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 471
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Different Lie group structures on a vector space with the same Lie algebra structure
This is an eccentric question: recall that a smooth Lie group structure on $\mathbb R^n$ is uniquely identified by a triple $(\mu,\iota,e)$ where $\mu:\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ is ...
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Lie algebra "generated" by matrix-valued curve?
Let $A(t)$ be a $n\times n$-matrix-valued continuous (plus possibly other niceness conditions; see below) curve, with the matrix entries being complex in general. If I am not mistaken, $A(t)$ ...
- 784
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Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?
Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
- 965
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Free affine actions of Borel subgroups
Call an upper triangular $m\times m$ matrix $A$ admissible if the lowest non-zero entry of $A-I$ lies in the last column, and is strictly lower than any other non-zero entry of $A-I$. I'll also regard ...
- 1,089
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Orbit of the identity matrix under Lie group algebra actions
I would like an explicit description of $\mathbb{R} SO(n) I_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$ by left multiplication. Equivalently, what is an ...
- 4,466
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When are all centralizers in a Lie group connected?
Let $G$ be a compact connected Lie group acting on itself by conjugation,
$$ G\times G\to G,\qquad (\sigma,h)\mapsto \sigma h \sigma^{-1}.$$
The fixed point set of a closed subgroup $H\le G$ equals ...
- 35.9k
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An innocent looking subgroup of $U(n)$
Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
- 4,466
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identify a curious subgroup in $U(n)$
Consider the following element $A$ in $U(n)$:
$$ \begin{pmatrix} 1/2(1+z) & 1/2(1-z) & \\\\
1/2(1-z) & 1/2(1+z) & \\\\
& &I_{n-2} \end{pmatrix},$$
where $|z| = 1$.
Now ...
2
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3
answers
549
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Non-continuous representations of $\operatorname{SL}_2(\mathbf{R})$
How does one construct a non-continuous representation $\rho:\operatorname{SL}_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?
- 7,609
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Abel transform is an * isomorphism for SL(2, R)
Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group.
Then we have an isomorphism of $*...
- 11.2k
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Chains in $K\backslash G/B$ lying over a closed $K$-orbit
Let $G$ be a complex connected reductive Lie group, $\theta$ an
involution, and $K = G^\theta$ the fixed-point subgroup.
Then $K$ has finitely many orbits on $G/B$, one of which is open
and (quite ...
- 27.8k
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Non-linear Lie group [duplicate]
Possible Duplicate:
Complex Lie group without faithful real representations?
We know that for a matrix (linear) Lie group $G$, we define it to be a closed subgroup of $GL(n,\mathbb{C})$. But Lie ...
- 53
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3
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Linear subspaces in cones over orthogonal groups
Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...
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