Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
2,938
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Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
5
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Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
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Trivial representation of a maximal torus
Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
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Expressing the union of principal orbits as a disjoint union of global slices for proper group actions
Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
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Mulitplicity one property for $\mathcal{D}'$ and $L^2$ over a homogeneous space
Let $G$ and $G_0$ be Lie groups, and suppose that a homogeneous space $X=G/G_0$ have a $G$-invariant measure.
It is known (E.G.F. Thomas showed) that there is an admissible parametrization $\{\mathcal{...
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1
answer
163
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Is the irreducible $ \mathrm{SU}(3) $ subgroup of $ \mathrm{SU}(6) $ maximal?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\S{S}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\...
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Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
3
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225
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Finite-maximal subgroups of orthogonal groups
I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite.
My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
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56
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Carnot–Carathéodory norm and the inner product norm
It is well-known that given the extended tensor algebra $T((\mathbb{R}^d))$ one may extract a separable Hilbert space by considering the subset
$$T^1((\mathbb{R}^d)) := \left\{h \in T((\mathbb{R}^d)) :...
6
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Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
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Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
4
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127
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Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
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Analyticity of the semigroup generated by the sublaplacian on unimodular Lie group
Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates ...
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Reliable literature with the list of centers of all simply connected simple real Lie groups
Wikipedia webpage https://en.wikipedia.org/wiki/Simple_Lie_group contains a full list of all simple (centerless) real Lie groups. One of the columns in tables (therein) contains fundamental groups of ...
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One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
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Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
6
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106
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Explicit representatives for Borel cohomology classes of a compact Lie group?
I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
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Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?
$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...
3
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2
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256
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Equidistribution on $\mathrm{SU}_2$
Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant ...
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A question on projective unitary representation of a Lie group
$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
7
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Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds
It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic ...
14
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472
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Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
2
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88
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Techniques for computing integrals on $G/K$
Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below
Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...
3
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115
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Embed exceptional non-compact simply connected simple Lie groups into classical simple Lie groups with preserving centers
It is well known that there are exactly 22 exceptional simple real Lie algebras (5 compact, 5 split, 5 complex and 7 others). To each of these algebras there corresponds a unique simply connected (...
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243
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Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?
$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here.
However, if I know right, this definition itself is known the "fundamental representation".
I wonder if there is any "...
4
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235
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Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
2
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Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group
Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...
7
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181
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Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
6
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1
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343
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All surjections onto trivial irrep split equivalent to being reductive
$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations
$$
0 \to W \to V \to k \to 0
$$
...
5
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1
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321
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Lattice generated by parabolics
Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
3
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155
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Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
4
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83
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Commutator-realisable connected simply connected Lie groups
Suppose that, for a connected simply connected real Lie group G, there is a Lie group H such that
G = H′ (the commutator subgroup of H). Can H always be chosen to be connected; that is, is there a ...
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40
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What is the form of the incomplete Eisenstein series on PGL_2(C)?
Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
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73
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Is the union of 1-dimensional pro-tori in a finite dimensional pro-torus dense?
Is the union of 1-dimensional compact connected abelian subgroups in a finite dimensional compact connected abelian group dense?
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145
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For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
4
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212
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Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?
Let $G\subset \operatorname{GL}(n)$ be a matrix Lie group. I am curious about curves $\gamma(t) = g \exp(tv)$, where $g \in G$, $v \in \mathfrak{g}$, and $\exp(.)$ is the matrix exponential. If we ...
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120
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A question about fixed point set of the compact group actions
Let $G$ be an infinite compact Lie group acting on a compact space $X$.
Denote $F=F(G,X)=\{x\in X$ : $gx=x$ for all $g\in G\}$.
Show that if $H^*(B_{G_x};\mathbb{Q})=0$ for all $x \notin F$ and $T^1$ ...
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149
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N(H)/H and the Weyl group
Let $ H $ be a connected subgroup of $ G=\mathrm{SU}(n) $ such that $ N_G(H)/H $ is finite. Is $ N_G(H)/H $ always a subgroup of the symmetric group $ \mathrm{S}_n $?
I just noticed this from the ...
2
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155
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Commensurability classes of subgroups of a nilpotent group
Here is a question I stumbled upon in my research.
Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?
Recall that two ...
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110
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A combinatoric identity for characters of reductive groups
Let $G$ be a reductive group over an algebraic closed field (of char 0 if necessary). Let $T\subset G$ be a maximal torus and $S=\mathrm{Sym}^*(X(T))$ be the symmetric algebra of characters of $T$. ...
0
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98
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Classification of all connected simple real Lie groups?
Is there an explicit(!) classification of all(!) connected real simple Lie groups up to isomorphism? Not just simply connected or adjoint, but all of them?
4
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198
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If a discrete and faithful representation of a surface group has proximal values, does the attracting points map have a continuous extension?
For some context, I'm studying the paper Anosov Representations and Proper Actions [GGKW].
$G$ denotes a non-compact real reductive Lie group of rank greater than $1$, $\Gamma$ denotes the fundamental ...
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78
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Lattice not contained in any connected subgroup is not contained in any positive dimensional subgroup
Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any ...
2
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151
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A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$
Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
2
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116
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The double quotient of SU(N) by its diagonal maximal torus
$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...
3
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1
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142
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Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
4
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1
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115
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How are Lie groups and polynomial resolvents related?
I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:
Nikolai's interest in [polynomial] resolvents led him to study Lie ...
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110
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How to build a representation of the diffeomorphism group of $U(n)$?
Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
12
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2
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466
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Coordinate ring of universal centralizer (BFM space)
In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
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76
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Character of principal series representations of $\mathrm{GL}(n,\mathbb{R})$
I am looking for an explicit form of the character of principal series representations of $\mathrm{GL}(n,\mathbb{R})$.
At the moment I am particularly interested in the case $n=2$.
A reference would ...