0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
2
votes
2answers
33 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...
0
votes
0answers
14 views

How can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space..

how can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space? Thanks..
1
vote
1answer
50 views

an identity related to moment map

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra, and $X,Y\in \mathfrak{g}$ and also let $\mu:M\to \mathfrak{g^*}$ be moment map($M$ smooth manifold) then prove the following equality ...
0
votes
1answer
30 views

concerning coadjoint representation

Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
3
votes
0answers
55 views

Concrete description of $T^*(G/B)$

Let $G=GL(n,\mathbb{C})$ and $B$ be a Borel in $G$. The variety $G/B$ is known to be a moduli of flags (i.e., a variety of Borels) with a bijection between a class of Borels and points of $G/B$. ...
6
votes
0answers
262 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
13
votes
3answers
544 views

Conditions for a smooth manifold to admit the structure of a Lie group

As we know, Lie group is a special smooth manifold. I want to find some geometric property, which is only satisfied by the Lie group. I only found one property: parallelizability. Can you show me ...
3
votes
0answers
238 views

Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given. But what are examples ...