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2
votes
0answers
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 ...
5
votes
0answers
181 views

Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
0
votes
1answer
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 ...
2
votes
1answer
75 views

Set of singular points for momentum map (with coisotropic action)

Let $G$ be a Lie-group acting on a connected symplectic manifold $M'$ in a hamiltonian way, with an $\operatorname{Ad}^*_G$-equivariant momentum map. Assuming $G$ acts properly on $M'$, we can ...
1
vote
1answer
46 views

multiplicity free actions - Guillemin&Sternbergy collective integrability

In this post I already ask a similar question. Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^*}$ the corresponding ...
1
vote
1answer
104 views

Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$. Assuming I can find ...
3
votes
1answer
93 views

coisotropic action on $TS^{2n+1}$

Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...
1
vote
0answers
54 views

Momentum Map on cotangentbundle as submersion

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
2
votes
0answers
89 views

Pulled back foliation is completely integrable

There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm". Assume $M$ is a symplectic ...
6
votes
2answers
176 views

Accuracy of the truncated Hausdorff moment problem

For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that $$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$ In other words, $M_s$ ...
13
votes
2answers
2k views

What is the significance that the Springer resolution is a moment map?

Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution $$ \mu: T^*\mathcal{B}\rightarrow \mathcal{N} $$ is the moment ...
4
votes
3answers
1k views

Why can we define the moment map in this way (i.e. why is this form exact)?

Given a symplectic manifold $(X, \omega)$ and a group $G$ acting on $X$ preserving the symplectic form, we define the moment map $\mu : X \to \mathfrak{g}^*$ so that $$ \langle d\mu(v), \xi\rangle = \...
2
votes
0answers
282 views

Are schematic fixed points of a torus action on an affinized twistor deformation flat?

This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...