The Poisson Linearization Problem for Sl2(c)

In this paper, we prove a version of Conn's linearization theorem for the Lie algebra sl2(C) c so(3, 1). Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to sl2(C) is linearizable. In the first part, we calculate the Poisson cohomology associated to sl2(C), and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.