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On some sequence of graded Lie algebras associated to manifolds. (English) Zbl 0824.58024

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.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B70 Graded Lie (super)algebras
58H15 Deformations of general structures on manifolds
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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
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