Symplectic realizations

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Γκενεράλης Ιωάννης
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Ιάκωβος Ανδρουλιδάκης Επίκ. Καθηγητής
Original Title:
Symplectic realizations
Translated title:
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
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

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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