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Library of the School of Science

Library of the School of Science

2016-02-26

2016

Γκενεράλης Ιωάννης

Ιάκωβος Ανδρουλιδάκης Επίκ. Καθηγητής

Symplectic realizations

English

Συμπλεκτικές υλοποιήσεις

In 1983, Alan Weinstein published a groundbreaking paper, which set

the foundations of the modern treatment of Poisson Geometry and Topology. In

that paper, there are three important ideas.

1. Splitting theorem, which gives the local structure of Poisson manifolds.

Actually, the Splitting theorem is the first instance of the fact that Poisson

manifolds are foliations with symplectic leaves, often presenting

singularities. Also, the Splitting theorem is the beginning of Poisson

Topology, a field which is growing rapidly these days.

2. The Splitting theorem shows that the Poisson manifolds are quite complicated

structures. In the

effort to simplify them, and based in the ideas by Shopus Lie, A.Weinstein

postulated the problem

of symplectic realizations of Poisson manifolds. That is realize a Poisson

manifold (P,Π), as a

quotient of a symplectic manifold (S,ω) under a surjective submersion which

maps S το P, which is a Poisson

map.

Once we have a symplectic realization as such, we may be able to lift problems

from (P,Π) to

(S,ω), which is considerably simpler. Since the submersion which maps S to P is

a Poisson map, the hope is that the solutions at that level of S, will pushed

down to solutions at the level of P. He also proved that locally, symplectic

realizations as such always exist.

3. Looking for global symplectic realizations A.Weinstein examined the example

of the case of Lie-Poisson structures. He showed that such a realization is

T*G, which, moreover, carries a natural Lie groupoid structure. This relates

the symplectic realization problem with the external symmetries of the

structure.

In other words, to find a global symplectic realization, one should be looking

for an appropriate Lie groupoid.

All the above points put Poisson geometry at the context of Lie algebroids and

Lie groupoids. Indeed, it turns out that a Poisson structure on P is the same

as a Lie algebroid structure on the cotangent bundle T*P.

It turns out that global symplectic realizations correspond to an integration

of the Lie algebroid

T*P. So, in fact, the global symplectic realization problemis really a problem

of integrability.

The scope of this dissertation is to present a quick introduction to Poisson

geometry and the role of Lie algebroids and foliations in the theory, and to

present the above results.

the foundations of the modern treatment of Poisson Geometry and Topology. In

that paper, there are three important ideas.

1. Splitting theorem, which gives the local structure of Poisson manifolds.

Actually, the Splitting theorem is the first instance of the fact that Poisson

manifolds are foliations with symplectic leaves, often presenting

singularities. Also, the Splitting theorem is the beginning of Poisson

Topology, a field which is growing rapidly these days.

2. The Splitting theorem shows that the Poisson manifolds are quite complicated

structures. In the

effort to simplify them, and based in the ideas by Shopus Lie, A.Weinstein

postulated the problem

of symplectic realizations of Poisson manifolds. That is realize a Poisson

manifold (P,Π), as a

quotient of a symplectic manifold (S,ω) under a surjective submersion which

maps S το P, which is a Poisson

map.

Once we have a symplectic realization as such, we may be able to lift problems

from (P,Π) to

(S,ω), which is considerably simpler. Since the submersion which maps S to P is

a Poisson map, the hope is that the solutions at that level of S, will pushed

down to solutions at the level of P. He also proved that locally, symplectic

realizations as such always exist.

3. Looking for global symplectic realizations A.Weinstein examined the example

of the case of Lie-Poisson structures. He showed that such a realization is

T*G, which, moreover, carries a natural Lie groupoid structure. This relates

the symplectic realization problem with the external symmetries of the

structure.

In other words, to find a global symplectic realization, one should be looking

for an appropriate Lie groupoid.

All the above points put Poisson geometry at the context of Lie algebroids and

Lie groupoids. Indeed, it turns out that a Poisson structure on P is the same

as a Lie algebroid structure on the cotangent bundle T*P.

It turns out that global symplectic realizations correspond to an integration

of the Lie algebroid

T*P. So, in fact, the global symplectic realization problemis really a problem

of integrability.

The scope of this dissertation is to present a quick introduction to Poisson

geometry and the role of Lie algebroids and foliations in the theory, and to

present the above results.

Poisson manifolds, Symplectic Realizations, Lie algebroids, Symplectic singular foliations, Integrability of Algebroids

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