Lecomte, Pierre B. A. On some sequence of graded Lie algebras associated to manifolds. (English) Zbl 0824.58024 Ann. Global Anal. Geom. 12, No. 2, 183-192 (1994). 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 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. Reviewer: J.Monterde (Burjasot) Cited in 6 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B70 Graded Lie (super)algebras 58H15 Deformations of general structures on manifolds Keywords:splitting of short exact sequences; Chevalley cohomology; Nijenhuis- Richardson graded Lie algebra Citations:Zbl 0685.58039; Zbl 0592.55010 PDFBibTeX XMLCite \textit{P. B. A. Lecomte}, Ann. Global Anal. Geom. 12, No. 2, 183--192 (1994; Zbl 0824.58024) Full Text: DOI References: [1] Lecomte, P.B.A.: Sur la suite exacte canonique associée à un fibré principal.Bull. Soc. Math. Fr. 113 (1985), 259–271. · Zbl 0592.55010 [2] De Wilde, M.;Lecomte, P.B.A.: Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations. Deformation Theory of Algebras and Structures and applications.NATO ASI Ser., Ser. C,247 (1988), 897–960. [3] Lecomte, P.B.A.;Michor, P.W.;Schicketanz, H.: The multigraded Nijenhuis-Richardson algebra, its universal property and applications.J. Pure Appl. Algebra 77 (1992), 87–102. · Zbl 0752.17019 · doi:10.1016/0022-4049(92)90032-B [4] Nijenhuis, A.;Richardson, R.: Deformation of Lie algebra structures.J. of Math. Mech. 17 (1967), 89–105. · Zbl 0166.30202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.