Normal forms for analytic Poisson structures. (English) Zbl 0553.58004

Conn, Jack F.

Ann. Math. (2) 119, 577-601 (1984).

The paper addresses the question of finding local coordinates in a Poisson manifold in which the expression of the Poisson structure is particularly simple. The splitting theorem gives rise to Darboux coordinates plus a third set of coordinates in which the Poisson tensor vanishes at zero. The paper proves that if the Poisson structure is real analytic or holomorphic and if the Lie algebra whose structure constants are the first derivatives of the Poisson tensor with respect to the third set of variables is semisimple, then locally, the Poisson tensor is the sum of a symplectic part plus a Lie-Poisson part. The key techniques of the paper are Lie algebra cohomology and a KAM-type accelerated convergence method.

Reviewer: T.Ratiu

MSC:

58A10	Differential forms in global analysis
58A15	Exterior differential systems (Cartan theory)
58H99	Pseudogroups, differentiable groupoids and general structures on manifolds
17B56	Cohomology of Lie (super)algebras

Keywords:

linearization of Poisson structures; Poisson manifold; splitting theorem; Poisson tensor; Lie algebra cohomology