**-3**

votes

**0**answers

146 views

### Maps from $S^3$ to $S^3$ [on hold]

As a physicist, I apologize for imprecise language.
I am interested in maps from $S^3$ to $S^3$ (identical to the group $SU(2)$). Since $S^3$ is threedimensional, there is some similarity to maps ...

**2**

votes

**0**answers

31 views

### Are rays in Carnot groups straight?

A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...

**10**

votes

**2**answers

343 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**4**

votes

**1**answer

120 views

### Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...

**4**

votes

**1**answer

137 views

### A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...

**5**

votes

**2**answers

87 views

### References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...

**18**

votes

**0**answers

316 views

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...

**9**

votes

**2**answers

248 views

### Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$
Which in our case is an isomorphism since $G$ ...

**0**

votes

**0**answers

27 views

### Integrable modules and comodules

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...

**6**

votes

**1**answer

368 views

### what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs ...

**4**

votes

**1**answer

158 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any ...

**2**

votes

**0**answers

57 views

### Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane

Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix ...

**0**

votes

**0**answers

49 views

### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...

**6**

votes

**0**answers

123 views

### intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie ...

**9**

votes

**1**answer

276 views

### A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning ...

**4**

votes

**0**answers

55 views

### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...

**2**

votes

**0**answers

47 views

### Order of metaplectic operator

I have a weak background on this subject.
Suppose $S$ be a $2m \times 2m$ symplectic matrix of order $n$. Suppose $W_S$ be the corresponding metaplectic operator on $\mathcal{S}(\mathbb{R}^m)$, the ...

**1**

vote

**0**answers

54 views

### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...

**11**

votes

**4**answers

313 views

### Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...

**2**

votes

**1**answer

75 views

### multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$.
That means:
1)
There exists an ...

**1**

vote

**0**answers

69 views

### Notation clash between a representation and spectral radius

I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by ...

**3**

votes

**1**answer

75 views

### Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...

**-1**

votes

**1**answer

63 views

### Zariski open set in orthogonal grassmanian [closed]

I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form ...

**6**

votes

**1**answer

390 views

### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...

**4**

votes

**0**answers

141 views

### Do I understand the Chevalley Restriction Theorem correctly? [migrated]

Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map ...

**8**

votes

**1**answer

125 views

### Expressing $SO_8$ element as product of $L_u$ and $R_u$ for unit octonions $u$

Welcome octonions friends !
Long time ago when I travelled through octonion land, I conjectured that every $SO_8$ element can be expressed as product $L_a L_b R_c R_d$ for unit octonions $a$, $b$, ...

**4**

votes

**1**answer

126 views

### When this Ad-invariant function on a Lie algebra is zero?

Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ ...

**18**

votes

**6**answers

626 views

### Automorphism group of real orthogonal Lie groups

I would like to know what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$. My working definition of $Out$ is as follows:
Let us denote by $Aut(G)$ the ...

**3**

votes

**1**answer

81 views

### Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...

**4**

votes

**1**answer

179 views

### Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...

**2**

votes

**1**answer

75 views

### Projections of orbifolds

A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an ...

**7**

votes

**1**answer

98 views

### Symmetries of the flag variety

Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma ...

**3**

votes

**0**answers

140 views

### The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...

**5**

votes

**1**answer

211 views

### What is meant by a Lie group acting by affine transformations?

This question is a continuation of this one, where I did not receive a complete answer so am moving to MO. I am trying to understand a paper by JP Szaro that was referred to there, and in particular, ...

**2**

votes

**1**answer

62 views

### Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$

Consider the group $GL_n(\mathbb{R})$ with its standard topology.
It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...

**3**

votes

**0**answers

66 views

### False optima for control on Lie groups?

Consider the equation
$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$
evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:
$J:G \rightarrow ...

**1**

vote

**1**answer

83 views

### Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group $G$ by the left action of a maximal lattice , i.e. a discrete cocompact ...

**4**

votes

**1**answer

73 views

### Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...

**4**

votes

**1**answer

126 views

### Difference between the Laplacian and the sub-Laplacian of a Lie group

Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...

**-1**

votes

**0**answers

63 views

### A Lie group associated to a matrix via semi direct product [migrated]

Assume that $A \in M_{n}(\mathbb{R}) $ is a matrix. Then $A$ generates a one parameter (with parameter $t\in \mathbb{R}$) family of group automorphisms of $\mathbb{R}^{n}$ with $x\mapsto ...

**5**

votes

**1**answer

112 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

324 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

63 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

186 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

69 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

99 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

117 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

69 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 ...