# Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

Volume 9, Number 2 (2010), 74 - 81

## Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

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### 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=XY 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.