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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canonical action of symmetric groups on orthogonal groups
There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...
- 543
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Does Maurer-Cartan form define surjection from Lie Group to Algebra-valued forms?
Let $G$ be a connected Lie Group of dimesion $m<\infty$ and let $g\in G$. The Maurer-Cartan form allows us to define a map from $G$ to the space of $\mathfrak{g}$-valued forms, via
$$g\rightarrow ...
- 99
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Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
- 141
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Geodesics on Sp(2,R)
Given the symplectic group $\mathrm{Sp}(2,\mathbb{R})$, represented as real $2\times 2$ matrices, I would like to compute the geodesic from the identity matrix $1\!\!1$ to the group element
\begin{...
- 285
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Four Sphere Fibrations
Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
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118
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Faithful representations of non-amenable Lie groups [duplicate]
Can a non-amenable connected Lie group have a faithful finite-dimensional unitary representation?
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180
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Cocompact (finite covolume) lattices in euclidean groups
1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)?
2) Also what is (if any) the ...
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Minimize matrix distance to tensor product
Minimize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product.
...
- 2,099
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finite upper half-plane model for the $\text{GL}_2(\Bbb{F}_q)$ Weil representation
Let $\Bbb{F}_q$ be a finite field with $q$ elements, let $\Bbb{F}_{q^2}$
be its quadratic extension, and consider the finite "upper" half space
${\frak{H}}_q := \Bbb{F}_{q^2} - \Bbb{F}_q$. Apeing a ...
- 2,137
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Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices
Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...
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Integrals of representations over geodesics
Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...
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547
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Does a symplectic group act on a tensor power of a spin representation?
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$.
Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
- 9,132
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Centralizer of hermitian matrices with zero trace
In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to $\mathfrak{su}(N)=...
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Is the space of volume-preserving maps path-connected?
This is a clarification of another post of mine.
Fix $n$ a positive integer. Let $SL(n)$ have its usual matrix representation, so that it really is the codimension-one subset of $M(n) = \mathbb R^{n^...
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Imbedding of $SU(2)$ in the exceptional Lie group $E_7$
We consider the classical Lie groups so that that center of the covering space is $\mathbb{Z}_2$. From the list of simple Lie groups it follows that they are:
$SO(2k+1)$ (case B) for $k\ge 1$ or $...
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Fréchet manifolds vs ILH manifolds
What is the precise relation between ILH manifolds and Fréchet manifolds? Specifically:
Does any ILH manifold has a canonical structure of a Fréchet manifold?
If so, is it true that any ILH ...
- 29.1k
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Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?
In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...
- 1,509
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How to embed $U(1)$ into a bigger group, using Dynkin diagrams
I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
- 153
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Decomposition of $L^2(\Gamma \backslash G)$
Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ (...
- 809
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spherical buildings for non-split groups
I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of ...
- 31.2k
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Large spin expansion of affine $\mathfrak{su}(2)_k$ characters
There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for
$$
\frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)...
- 71
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Formula for the Haar measure in the linear symplectic group
What is (or where can I find) an explicit formula for the Haar measure of the group of linear symplectic transformations of $\mathbb{R}^{2n}$?
Added 13/05/2014.
Some clarifying remarks:
(1) by ...
- 13.5k
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Euler-Poincaré equations with constraints
It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} \Xi(t)^{-1}...
- 2,099
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Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
- 397
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subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
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cartan killing metric [closed]
I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...
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Hasse diagrams of G/P_1 and G/P_2
in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous $G$-varieties for $G$ ...
- 1,479
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640
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Characterisation of Q-rank 1
I'm looking for a reference and/or the original source for the following fact:
An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
- 10.4k
3
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What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?
If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
- 803
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Mal'cev completions of finitely generated torsion-free nilpotent groups
There is some question from geometric group theory:
One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
$\Gamma$ and $\...
3
votes
1
answer
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Parallelizable nearly-Kahler manifolds
In this question, we have discussed how the following bundle:
$E_{d} = TS^{d}\oplus \Lambda^2 T^{\ast}S^{d}$
is always trivial, where $S^{d}$ is the $d$-dimensional standard sphere. Now, let us take ...
- 2,818
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Cohomology of lattice subgroups
I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly ...
- 96.4k
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The group of polynomial homeomorphism of the plane
Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that
both $f$ and $f^{-1}$ are polynomial maps.
We equip $G$ with the compact open topology and the obvious group ...
- 356
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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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Simply connected Lie groups homeomorphic to R^n are solvable
I have found the following claim in many proofs "Simply connected Lie groups homeomorphic to $\mathbb{R}^n$ are solvable". But the universal covering of $SL(2,\mathbb{R})$ satisfies the hypothesis of ...
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Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
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Is there a good definition of the universal cover for non-connected Lie groups?
It is well-known that the universal cover $\tilde G$ of a connected Lie group $G$ has a Lie group structure such that the covering projection $\tilde G\to G$ is a Lie group morphism. Of course $\tilde ...
- 2,140
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Stiefel manifolds and polar decompositions
The real Stiefel manifold $V_{n,k}$ of orthogonal $k$-frames in $\mathbb{R}^n$ can be viewed as the reductive homogeneous space $G/H=O(n)/O(n-k)$. If ${\frak{so}}(n)$ is the Lie algebra of $O(n)$, ...
- 1,378
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votes
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Plancherel expansion for Spin(n-1,1)
I am interested in the principal series (unitary irreducible) representations of $Spin(n-1,1)$, and in the generalized Pancherel's formula for the delta function on the group in terms of a sum (and an ...
- 129
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Holomorphic line bundles on smooth points of a quotient
I am an amateur algebraic geometer, so maybe this question is trivial and if this is the case, then I apologize. This is a question that came up while working on something completely different.
...
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How to calculate symmetric tensor products of SO(10) representations?
I just want to consider the simplest case:
Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?
My conjectured formula based on the results of LiE program for finite k values is:
$...
- 161
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votes
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549
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Special linear groups contained in symplectic groups
Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and $\...
- 533
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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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answer
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Invariant regular cones in Lie group representation
I am following Analysis and Geometry on Complex Homogeneous Spaces by Faraut et al. I'll set up all of what I need and then ask my questions.
Let $G$ be a connected semi-simple non-compact real Lie ...
- 165
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votes
2
answers
206
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Measurable representations of semi simple Lie groups
Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in
I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple
Lie groups, ...
- 53
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Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?
Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$)
Or is it known that it cannot be such a lattice ?
- 2,696
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Non-existence of projections in crossed product
If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...
- 1,903
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0
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173
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Levi decompositions of k-rational points of linear algebraic groups
Let $k$ be a field with characteristic zero and $G$ be a (connected or not) linear algebraic group defined over $k$. We know that $G$ has a Levi decomposition i.e., $G=R_u(G)\rtimes L$, where $R_u(G)$ ...
- 1,702
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1
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604
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Brownian bridge on a Lie group as a stochastic differential equation
Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
- 73
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1k
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Complex Lie group without faithful real representations?
Does there exist a complex analytic Lie group which doesn't have faithful representations in $GL(N,\mathbb R)$, viewed as a real Lie group?
There are examples of complex Lie groups which do not allow ...
- 1,050