All Questions
Tagged with lie-groups sg.symplectic-geometry
25 questions with no upvoted or accepted answers
8
votes
0
answers
285
views
Connection between integrable systems and group actions
An integrable system can be defined as a symplectic manifold together with the maxiumum possible number of Poisson commuting functions on the manifold which are almost everywhere independent. By the ...
- 979
5
votes
0
answers
411
views
Lagrangian subgroup of a nonabelian Lie group
My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...
- 10.5k
5
votes
0
answers
572
views
Isn't the quantomorphism group really just the "WKB-quantomorphism" group?
Introduction
In his second-most upvoted post, called "Why quantum mechanics?" (second only to his post on fibre bundles & gauge theory) in the physics SE community, Urs Schreiber, in the setting ...
- 215
5
votes
0
answers
363
views
Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
- 59
4
votes
1
answer
186
views
Explicit formula for complex structure on flag manifold/isospectral matrices?
Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
- 637
4
votes
0
answers
115
views
Examples of non-equivariant momentum maps
What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group ...
- 5,824
4
votes
0
answers
178
views
The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
- 516
4
votes
0
answers
250
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Quotients of Kähler manifolds
Let $X$ be a Kähler manifold and $G$ a complex semisimple Lie group acting freely on $X$ by biholomorphisms and such that the Riemannian metric is preserved by a maximal compact subgroup $K$ of $G$. ...
- 41
4
votes
0
answers
438
views
Symplectic principal bundles
A symplectic principal bundle is a principal bundle $(X,B, G)$ with projection map $q:X\to B$ such that $X$ and $B$ are symplectic manifolds and the right action of $G$ preserves the symplectic ...
- 356
4
votes
0
answers
468
views
Complex symplectic reduction
Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...
- 1,347
3
votes
0
answers
80
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
- 2,527
3
votes
0
answers
74
views
Coordinates for quasiperiodic motion after reconstruction
Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
- 141
3
votes
0
answers
135
views
Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$
Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
- 369
3
votes
0
answers
170
views
Finding generators of equivariant cohomology
Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
- 1,942
3
votes
0
answers
230
views
Possible to express the coadjoint orbits in terms of Kahler reduction?
I have heard for many times that the coadjoint orbits of a compact semi-simple Lie group are Kahler. While I know that the symplectic structure on a coadjoint orbit can be given by the symplectic ...
- 461
3
votes
0
answers
337
views
Quotient space of Grassmannian
The Grassmannian $G(k,2k)$ is equipped with a nice $\mathbb Z_2$ action with respect to a non-degenerate symplectic bilinear form: $1.V=V^{perp}$. Is there a reference where the ring of polynomial ...
- 673
3
votes
0
answers
274
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 $\mathfrak{g}^*$.
...
- 95
2
votes
0
answers
85
views
Coherent states on compact abelian state spaces and complexification
First, to establish notation, let $T^*(M)$ denote the cotangent bundle of a manifold $M$. Let $\widehat{(-)}:= \hom_{\sf LCAbGrp}(-,\mathbb{T}):{\sf LCAbGrp}^{\sf op}\simeq {\sf LCAbGrp}$ denote the ...
- 169
2
votes
0
answers
100
views
Effective actions by non-commutative groups have non-commuting fundamental vector fields?
I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
- 979
2
votes
0
answers
47
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 ...
- 501
1
vote
0
answers
48
views
Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions
Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...
- 1,677
1
vote
0
answers
285
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Coadjoint orbits
I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer!
While I was trying to teach my ...
- 139
1
vote
0
answers
81
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 ...
- 501
1
vote
0
answers
213
views
Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$
Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on $\...
user21574
0
votes
0
answers
101
views
G-invariant functions on manifold for G compact
In a paper I saw the following statement:
Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
- 501