**1**

vote

**1**answer

115 views

### Lift Lie group action on a small neighborhood

Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$.
Q: Why in a small neighborhood of $N$, $G$ also action ...

**3**

votes

**1**answer

130 views

### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem:
THEOREM. Let $M$ be a ...

**6**

votes

**1**answer

110 views

### Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...

**3**

votes

**0**answers

59 views

### Action of longest element of Weyl group on zero weight space

Let:
$G$ be a real semisimple Lie group;
$\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space;
$A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is ...

**0**

votes

**0**answers

24 views

### Super classical r-matrices and Poisson Lie supergroups

In classical case, given an r-matrix $r$ for $sl_n$, we can compute the corresponding Poisson bracket on $SL_n$ by using the formula $\{L \otimes L\} = [r, L \otimes L]$. For example, let $g=sl_2$ and ...

**2**

votes

**1**answer

75 views

### 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**

votes

**1**answer

102 views

### Maximize inner product of a tensor of unitary matrices

How can one maximize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$.
Both the maximum value of ...

**1**

vote

**0**answers

103 views

### cohomology ring of homogenous manifold

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :
$$
G/H ...

**0**

votes

**1**answer

64 views

### Is the toral component of a connected Lie group equal to the toral component of its radical? [closed]

Given a connected Lie group, define its toral component as the maximal connected and compact subgroup of its center.
Is the toral component of a connected Lie group equal to the toral component of ...

**-3**

votes

**0**answers

161 views

### The center of a group is equal to the center of its radical?

Given a linear algebraic group $G$, is the connected component of the identity of the center of $G$ equal to the connected component of the identity of the center of its solvable radical? If not, is ...

**5**

votes

**1**answer

234 views

### How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...

**2**

votes

**0**answers

102 views

### Distance metric on Riemannian quotient manifold

It's well known that if I have a Riemannian manifold $M$ and a Lie group of isometries $G$ that acts freely and properly on $M$, then the quotient $M/G$ is a manifold and inherits the Riemannian ...

**17**

votes

**2**answers

421 views

### When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?

A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...

**1**

vote

**1**answer

149 views

### A representation of Spin(9,1)

Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ).
Consider the induced ...

**2**

votes

**1**answer

66 views

### Decomposition into irreducible components of a representation of $Spin(9)$

It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey).
Consider the induced representation of $Spin(9)...

**3**

votes

**1**answer

66 views

### Explicit generators of the Lie algebra $spin(9)$

It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the ...

**3**

votes

**3**answers

572 views

### A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...

**1**

vote

**0**answers

104 views

### Generating $\mathfrak{so}(7)$

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...

**6**

votes

**1**answer

129 views

### Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...

**0**

votes

**1**answer

70 views

### Hamiltonian potential invariant under lie group action?

Let $(M,g)$ be a riemannian manifold and $G$ a Lie group acting on $M$ by isometries.
Taking a $G$-invariant function $f\colon M \to \mathbb{R}$, we have the riemannian gradient $\operatorname{grad} ...

**2**

votes

**0**answers

33 views

### $TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...

**2**

votes

**1**answer

107 views

### If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...

**6**

votes

**1**answer

97 views

### Involutions and Little Adjoint Representations of Simple Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$.
Is easy to see that ...

**6**

votes

**1**answer

153 views

### Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?

**1**

vote

**0**answers

111 views

### Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...

**4**

votes

**2**answers

209 views

### Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...

**14**

votes

**3**answers

544 views

### Is SO(2n+1)/U(n) a symmetric space?

I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408....

**4**

votes

**1**answer

146 views

### Differential structures on compact Lie groups

Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?

**2**

votes

**1**answer

211 views

### Irreducible representations of Sp(2)

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$.
I can ...

**1**

vote

**0**answers

58 views

### Symplectic gradients whose span doesn't intersect Lie group orbits

I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask:
Let $G$ be a ...

**0**

votes

**0**answers

61 views

### charts for some riemannian embedding

Let $(M,g)$ be some riemannian manifold with some Lie group $G$ acting properly, freely and by isometries. So $M/G$ is a manifold and using the projection $\pi \colon M \to M/G$, we get a smooth ...

**3**

votes

**1**answer

56 views

### Criterion for convergence of sums for non-continuous functions

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{...

**5**

votes

**1**answer

121 views

### Centreless semisimple Lie group that is not real algebraic

Let $G$ be a connected semisimple Lie group with trivial centre and $\mathfrak{g}$ its Lie algebra. The adjoint representation of $G$ defines an isomorphism of $G$ onto the connected component of the ...

**1**

vote

**1**answer

81 views

### slice theorem for proper actions

I'm trying to understand the slice-theorem for proper Lie-group actions.
Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each ...

**3**

votes

**1**answer

216 views

### Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...

**3**

votes

**0**answers

137 views

### Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...

**2**

votes

**3**answers

108 views

### How can one show G/T is a coadjoint orbit for G a compact lie group and T it's maximal torus

Let $G$ be a compact lie group and $g$ it's lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me ...

**14**

votes

**4**answers

700 views

### Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...

**0**

votes

**1**answer

54 views

### set of coisotropic orbits open and dense, iff group acts locally transitively almost everywhere

I worked now some time with coisotropic actions of Liegroups on manifolds.
But there is one key fact, that I don't understand, although it is very central in my considerations.
Let $(M,\omega)$ be a ...

**1**

vote

**0**answers

64 views

### Representation equivalent lattices

Suppose $G$ is a absolutely almost simple algebraic groups over a number field $K$. Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G(K)$. Then $\Gamma_1$ and $\Gamma_2$ are said to be ...

**0**

votes

**0**answers

36 views

### Bruhat decompositions for Lie supergroups

Are there some references of Bruhat decompositions for Lie supergroups? I searched on google but did not find any papers about Bruhat decompositions for Lie supergroups. Thank you very much.

**5**

votes

**2**answers

136 views

### Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...

**2**

votes

**2**answers

137 views

### Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra

Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...

**1**

vote

**0**answers

99 views

### Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by $\...

**4**

votes

**1**answer

97 views

### geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$

Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G_v$ is discrete (so iff $\dim G_v =0$). Lets assume $g$ acts properly on $T(G/K)...

**5**

votes

**1**answer

227 views

### Motivating the existence of Canonical Bases for Representations

In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the ...

**4**

votes

**0**answers

67 views

### Exceptional symmetric spaces embedded in exceptional Lie group

In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...

**6**

votes

**1**answer

355 views

### The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...

**25**

votes

**3**answers

1k views

### What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...

**3**

votes

**1**answer

123 views

### A more precise description of conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...