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Groupoides symplectiques et troisième théorème de Lie “non linéaire”. (Symplectic groupoids and the “non-linear” third theorem of Lie). (French) Zbl 0702.58023
Géométrie symplectique et mécanique, Colloq. Int. Sémin. Sud- Rhodan. Géom. V, La Grande Motte/Fr. 1988, Lect. Notes Math. 1416, 39-74 (1990).
[For the entire collection see Zbl 0683.00020.]
The article is devoted to the investigation of symplectic groupoids (or Lie pseudogroups), introduced independently by A. Weinstein and M. V. Karasev. This notion is connected with a certain type of generalizations of the third theorem of Lie (the existence of a Lie group corresponding to given finite-dimensional Lie algebra) to infinite dimension. This generalization is called the “non-linear” third theorem of Lie. Using Libermann’s theory of isotopic realizations, the author (in particular) gives an obstruction to the existence of symplectic groupoids over certain regular Poisson manifolds.
Reviewer: Yu.E.Gliklikh

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
22E60 Lie algebras of Lie groups
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties