Classification of Nonresonant Poisson Structures

Abstract

Unless otherwise explicitly stated all mappings and tensors in the paper are C∞. A Poisson structure on a (C∞) manifold M is a bracket operation (f, g) ↦ {f, g}, on the set of functions on M, which gives to this set a Lie algebra structure and which verifies the relation {f,gh}={f,g}h+g{f,h}. An equivalent way to get such a structure is to give a 2-vector (that is, an antisymmetric two times contravariant tensor) P satisfying [P, P]=0 where [,] is the Schouten bracket [7]. We then have P(df,dg)={f,g}. This paper is devoted to the local classification of these structures. The decomposition theorem of A. Weinstein [8, 9] reduces the problem to the case where P is a Poisson structure on Rn which vanishes at zero. In this paper we will denote by PS(n) the set of germs at zero of Poisson structures on Rn vanishing at the origin.