On linearized Poisson structures. (Q1809972)

The linearizability of a given Poisson structure \(\Psi \) at a non-zero dimensional symplectic leaf B (\(\dim B\geq 1\)) is studied. Two basic structures are introduced: a normal bundle \(E\) to the symplectic leaf \(B\) and the pull-back \(\Psi _{f}\) of the Poisson structure \(\Psi \) to \(E\) via an exponential map \(f\). Using the idea of minimal coupling, it is shown that there exists a linearized Poisson structure \(\Pi _{L}\) of \(B\) that gives a first approximation to \(\Psi _{f}\). This linearized structure \(\Pi _{L}\) is uniquely determined by the choice of a subbundle \(L\) transversal to \(B\). The main result of the paper is that in a neighborhood of the zero section \(B\) in \(E\), the linearized Poisson structure \(\Pi _{L}\) is independent of the choice of a transversal \(L\) up to a diffeomorphism identical on \(B\). It is concluded that this result leads to the natural setting of the linearization problem for Poisson structures near embedded symplectic leaves.