# Tagged Questions

**5**

votes

**1**answer

116 views

### Existence of lattices in reductive Lie groups

What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups ...

**11**

votes

**2**answers

331 views

### Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
...

**4**

votes

**1**answer

68 views

### Orientability of orbit type strata of Lie group actions

Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...

**1**

vote

**1**answer

190 views

### Connected subgroups of $SL(2,C)$

Where can I find a list of all connected real Lie groups inside the 6-dimensional real Lie group $SL(2,C)$, up to conjugacy? How can one verify that a partial list is complete?
I found on wikipedia a ...

**1**

vote

**1**answer

189 views

### What are the finite subgroups of $\operatorname{Sp}_{2n}(\mathbb{Z})$?

I've read the following question:
Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request)
and it made me wonder. It's easy to see that ...

**0**

votes

**1**answer

77 views

### full set of invariant functions on manifold

Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.
Is it always possible to construct $k$ functions $f_1, ...

**1**

vote

**2**answers

101 views

### Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...

**6**

votes

**1**answer

120 views

### The quantum group SUq(n) as von Neumann algebra

i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...

**2**

votes

**0**answers

80 views

### Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra

Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k ...

**2**

votes

**0**answers

59 views

### Diagonal invariants of $SO(n)$

Consider a Lie algebra $\mathfrak g$ (I am mostly interested in the case $\mathfrak g=so(n)$), its universal enveloping algebra $U$ and its center $C$. There is an adjoint action of $\mathfrak g$ on ...

**0**

votes

**0**answers

63 views

### Matrix representation of the Heisenberg quaternionic group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3= \mathbb C\times \mathbb R$ is given by
$$
\begin{pmatrix}
1 ...

**1**

vote

**0**answers

56 views

### Dimension of tangent space to manifold of cross section slices

Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...

**5**

votes

**0**answers

98 views

### 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 ...

**2**

votes

**1**answer

108 views

### Characterization of restricted weights of representations of real semisimple Lie groups

I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...

**1**

vote

**0**answers

191 views

### Siegel domains and the Baily-Borel compactification of $\mathcal{A}_2$

Consider the connected, almost simple, algebraic group $Sp_4$ over $\mathbb{Q}$ (embedded canonically in $GL_4$). For the following facts, I refer the reader to Murnaghan, Linear Algebraic Groups, ...

**6**

votes

**2**answers

164 views

### Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5

Up to topology, the 5D homogeneous space
$$
G_2/P
$$
of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space
$$
M:=J^1(2,1)=\{(x,y,u,p,q)\}
$$
of scalar functions in two ...

**30**

votes

**1**answer

549 views

### Is there a geometric construction of hyperbolic Kac-Moody groups?

Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...

**0**

votes

**0**answers

63 views

### Curves in $\mathfrak{su}(n)$ with specific property

Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...

**1**

vote

**0**answers

49 views

### Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$ [closed]

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...

**1**

vote

**1**answer

102 views

### 'Accidental' isomorphisms for $Spin^C(n)$

The complex spin groups $Spin^C(n)$ appear in the fibration
$Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$
which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence
...

**0**

votes

**0**answers

58 views

### Connectedness of Centralisers in Unitary group

I want to understand centralizers of semisimple elements in unitary groups.
Let us begin with example of $GL_n(k)$. Centralizers of semisimple elememts are a product of smaller $GL_m(k)$ thus ...

**1**

vote

**2**answers

215 views

### Nilpotency of Lie Algebra from Structure Constants

Suppose we have a Lie algebra with structure constants
$$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$
for some coefficients $a_{ijk}$.
In this setting, how may be checked (perhaps ...

**9**

votes

**0**answers

62 views

### On the proof by Chu-Kobayashi that transformation groups are Lie groups

Chapter I of Kobayashi's Transformation Groups in Differential Geometry contains a very general theorem on transformation groups, due to Palais. I have some questions about its proof (which I attach ...

**4**

votes

**0**answers

50 views

### Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...

**26**

votes

**1**answer

515 views

### Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...

**5**

votes

**1**answer

141 views

### If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?

Let $G$ be an absolutely simple simply connected and connected algebraic group defined over a global field $k$ with ring of integers $\mathcal{O}$. Fix an embedding of $G$ into $GL_n$. Given $v$ a ...

**11**

votes

**3**answers

387 views

### $A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...

**1**

vote

**0**answers

91 views

### Roots of matrices in $G_2(Z)$

Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...

**6**

votes

**3**answers

297 views

### Simple lie algebras, (almost-)simple groups of Lie type

Take an algebraic group $G$ defined over a finite field $K$. Suppose its Lie algebra $\mathfrak{g}$ is simple. It should follow that $G$ is almost-simple. (By this I mean not that $G(K)$ is simple -- ...

**4**

votes

**0**answers

171 views

### Compactly supported distributions as a projective G-module

For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...

**0**

votes

**1**answer

65 views

### Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of ...

**3**

votes

**1**answer

77 views

### Uniform lattice in semidirect product

A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact.
My question is whether a uniform lattice exists in the group
$$
G={\mathbb ...

**2**

votes

**0**answers

135 views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
Edit: The ...

**2**

votes

**0**answers

148 views

### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, ...

**1**

vote

**2**answers

133 views

### 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?

**2**

votes

**0**answers

140 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n ...

**3**

votes

**0**answers

87 views

### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
...

**2**

votes

**0**answers

96 views

### Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...

**4**

votes

**2**answers

278 views

### odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...

**0**

votes

**0**answers

47 views

### Generalized Gaussian Decomposition

Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...

**4**

votes

**0**answers

154 views

### Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...

**8**

votes

**1**answer

326 views

### Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...

**1**

vote

**0**answers

57 views

### Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...

**6**

votes

**0**answers

91 views

### Samelson Products in $SO(n)$

Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...

**17**

votes

**2**answers

651 views

### Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, ...

**20**

votes

**4**answers

908 views

### Technical issue in the approach to Lie groups taken in a book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...

**3**

votes

**1**answer

194 views

### Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every ...

**9**

votes

**3**answers

274 views

### Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...

**6**

votes

**0**answers

139 views

### An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory:
Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have
$G^F=GL_n(\mathbb{R})$. Consider the ``Lang ...

**4**

votes

**1**answer

145 views

### Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...