All Questions
Tagged with lie-groups sg.symplectic-geometry
72 questions
4
votes
1
answer
333
views
Check symplectomorphism property on infinitesimal generators
I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G \...
CommunityBot
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1
vote
1
answer
233
views
Non Hamiltonian vector field
Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
- 1,301
1
vote
1
answer
399
views
Integrating Poisson groups
Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...
5
votes
1
answer
585
views
Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold
Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-...
- 1,723
4
votes
0
answers
468
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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
2
votes
1
answer
429
views
Kodaira dimension of co-adjoint orbit
Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...
CommunityBot
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8
votes
1
answer
882
views
Formula for the Haar measure in the linear symplectic group
What is (or where can I find) an explicit formula for the Haar measure of the group of linear symplectic transformations of $\mathbb{R}^{2n}$?
Added 13/05/2014.
Some clarifying remarks:
(1) by ...
- 13.5k
1
vote
1
answer
215
views
Relation between volume of reduced space and phase space
Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$...
- 27.8k
1
vote
1
answer
301
views
Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure
I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...
- 12.1k
5
votes
0
answers
363
views
Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
- 6,210
1
vote
0
answers
213
views
Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$
Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on $\...
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0
votes
1
answer
313
views
Symplectic structure on $Sym^kG^{\mathbb{C}} $
Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{...
CommunityBot
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0
votes
1
answer
587
views
fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$
Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in \mathfrak{g^*}$...
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3
votes
1
answer
614
views
Coadjoint orbits and homogeneous symplectic $G$-manifolds
We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
- 8,597
3
votes
1
answer
3k
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Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...
- 108k
2
votes
1
answer
1k
views
Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 40.2k
2
votes
2
answers
721
views
Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure?
Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$).
My questions is: it is always true that we have ...
- 27.8k
3
votes
2
answers
589
views
How to deal with the singular reduction of the Hamiltonian n body problem?
I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.
...
- 1,710
6
votes
3
answers
466
views
Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
- 23k
7
votes
1
answer
721
views
Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups
The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid ...
- 965
6
votes
4
answers
1k
views
Polar decomposition for quaternionic matrices?
A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
- 1,749
7
votes
1
answer
669
views
Is a Poisson Group a group object in the category of Poisson Manifolds?
I realized that I am very confused about a certain sign in the definition of a Poisson group. I will give some definitions, and then point out my confusion.
Definitions
Group objects
Let $\mathcal ...
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