Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...
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96 views

Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes: The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a parallel subbundle $Q$ which is locally spanned by $3$ ...
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4answers
912 views

Topological structure of SO(n) as a product

I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product $$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$ Allen Hatcher [1, p. 293 f.] ...
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Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
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What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...
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155 views

Is the restricted root system of a simple real Lie group irreducible?

As the title asks, is the restricted root system of a simple real Lie group irreducible? I believe this is true but I need a reference to cite.
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185 views

Intertwiners and Clebsch-Gordan coefficients

Consider two unitary irreducible representations on vector spaces $V_1$ and $V_2$ of a Lie group $G$. For $G$ is compact and $V_1$ and $V_2$ finite dimensional there is a unique decomposition of $V_1 ...
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145 views

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 ...
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1answer
143 views

Canonical class of partial flag variety

Let $F(d_1,d_2,\ldots,d_k;n )$ be the variety of all flags $\mathbb A^{d_1} \subset\mathbb A^{d_2}\subset\ldots\subset \mathbb A^{d_k}\subset \mathbb A^{n}$. This variety has the natural maps to ...
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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 ...
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114 views

symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example, $\Delta^2$: $\Delta^3$: Let ...
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286 views

Flag varieties and orbit of highest weight vector

I asked this here http://math.stackexchange.com/questions/1441408/orbit-under-lie-group-and-projective-variety but did not get any answers, so I am asking here. I apologize if this is not appropriate ...
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59 views

symmetric points on symmetric spaces

Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on ...
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142 views

Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ...
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138 views

Invariant polynomials with respect to group actions on matrices

Let $\mathfrak{gl}_n(\mathbb{R})$ be the Lie algebra of matrices with real entries and $GL_n(\mathbb{R})$ its associated Lie group. Recall that a linear subgroup $G \subseteq GL_n(\mathbb{R})$ acts by ...
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How stable is the top cell of a Lie group?

It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable ...
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Level sets on $SU(4)$

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not ...
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262 views

Riemannian metric of hyperbolic plane

I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector ...
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0answers
43 views

Equivariant exponential map on Hilbert manifolds

Let $M$ be a Hilbert Riemannian manifold and $G$ a finite-dimensional Lie group acting on $M$. It is well-known that when $M$, too, is finite-dimensional $$\exp_p: U \subset T_pM \rightarrow \exp_p(U) ...
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Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have $$ \mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*, $$ where $\lambda$'s are dominant weights. Let $U^-$ be the ...
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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 ...
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397 views

Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...
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49 views

Automorphism groups of symmetric cones

Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...
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206 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
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131 views

Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that $$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
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Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
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115 views

Global decomposition of reductive spaces

Consider a reductive homogeneous space $M=G/H$ with corresponding Lie algebra decomposition $\frak{g}=\frak{m}+\frak{h}$. Then there is a local diffeomorphism $$ (exp\, X, h)\mapsto (exp\, X) h\quad ...
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Parameterizing rotations of a cube

For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...
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119 views

Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$ be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
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231 views

Generalising the parametric transversality theorem to a foliation

The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all ...
3
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1answer
272 views

Infinite groups of finite exponent inside of SL(2,C)

Fix an integer $n>0$. Are there infinite subgroups of $SL_2(\mathbb{C})$ such that every element is $n$-torsion?
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258 views

Generalization of a theorem of Burnside to non-compact groups

The following two theorems are often attributed to Burnside: Theorem Let $G$ and $H$ be compact groups. Then the irreducible representations of $G\times H$ are precisely the representations ...
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Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$. Let $\rho$ be the standard 7-dimensional complex representation $$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$ We ...
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1answer
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Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
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Manifold approximations to $BO(3)$

We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$ Similarly $BO(2)$ can be approximated by closed, ...
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1answer
105 views

How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$. Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...
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Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads: If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...
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1answer
182 views

Check symplectomorphism property on infinitesimal generators

I stumbled over the following question: First, let me give the basic definition of a symplectic group action: Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G ...
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81 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
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106 views

Poincaré inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is: symmetric, adapted (in the sense that there is no proper subgroup $H$ such ...
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Permutable (Lie) subgroups

Let's recall that, a group $G$ being given, two subgroups $A,B\subset G$ are called permutable iff $AB=BA$ for the Minkowski law. It is straightforward to see that $(A,B)$ are permutable iff $AB$ ...
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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 $G$ ...
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Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]

Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Cross posted from ...
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A question about the maximal subgroup of SO(n+1)? [closed]

Is SO(n) a maximal subgroup of SO(n+1)?
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108 views

Gradient on $SU(n)$

I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
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Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ...
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Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...
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183 views

Does the Laplacian commutes with elements of the basis of the Lie algebra?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...
5
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1answer
177 views

Infinite-dimensional admissible representations of SL(2,C)

I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
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283 views

Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...