Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

### Abstract

In this paper, we develop one of the questions raised by the author in the mini-course he gave at the conference Geometry and Physics V held at the University Cheikh Anta Diop, Dakar in May 2007). Let Π be a Poisson tensor on a manifold *M*. We suppose that it is decomposable in a neighborhood *U* of a point *m*, i.e. we have Π=*X*∧*Y* on *U* where *X* and *Y* are two vector fields. We will exhibit examples where every Poisson tensor near enough Π seems to be also decomposable in a neighborhood of a point which can be chosen arbitrarily near *m*; and this works even if *M* has a big dimension. This idea is a consequence of a cohomology calculation which can be interesting by itself.