-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
0
votes
0answers
64 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
0
votes
1answer
69 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
4
votes
0answers
72 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an ...
1
vote
0answers
79 views

Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
1
vote
1answer
54 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
57 views

A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...
1
vote
1answer
76 views

Finding the lie algebra of the symplectic lie group

I am having difficulties completing my proof that $\text{Lie}(\text{Sp}(2n)) \equiv \mathfrak{sp}(2n) = \{ X \in Gl(2n)\; |\; X^TJ + JX = 0 \}$ Where $J \equiv \begin{bmatrix}0 & \mathbb{1}_n ...
2
votes
2answers
191 views

When does the $\mathfrak g$-invariance of the symplectic form imply $G$-invariance?

Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $M$ be a smooth manifold. Suppose $G$ acts on $M$, $G \to \text{Diff}(M)$. This naturally induces an action $\mathfrak g \times M \to M$ ...
7
votes
1answer
199 views

Moment map of the action of $\operatorname{SO}(3)$ on the sphere

The moment map of the action of $\operatorname{SO}(3)$ on the sphere can be thought of as inclusion from $S^2$ into $\mathbb R^3$ by identifying $\mathfrak{so}(3)$ (the Lie algebra of ...
5
votes
2answers
733 views

symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...