On some sequence of graded Lie algebras associated to manifolds (Q1892341)

A basic tool in the construction of star-products and of formal deformations of the Poisson bracket on symplectic manifolds is the existence of a preferred family of cohomology classes associated to the Nijenhuis-Richardson graded Lie algebra of the space of functions of a smooth manifold. These classes were constructed by the author and \textit{M. De Wilde} [NATO ASI Ser., Ser. C 247, 897-960 (1988; Zbl 0685.58039)]. In this paper the author gives a new construction of these classes. In fact he shows that they are the obstructions of degree 2 and weight \(-1\) against the splitting of a short exact sequence of \(\mathbb{Z}\)-graded Lie algebras naturally associated to manifolds by means of the dual \(d^*\) of the de Rham differential \(d\). It is shown that the associated sequence is never split. Combined with a sort of algebraic Chern-Weil homomorphism adapted from a previous paper of the author [Bull. Soc. Math. Fr. 113, 259-271 (1985; Zbl 0592.55010)] to the \(\mathbb{Z}^ n\)-graded case, this leads to a family of cohomology classes.