Abstract
Introducing the notion of an admissible graded Lie subalgebra A of the Nijenhui-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex C A of the Chevalley-Eilenberg complex Λ(C 0 M), extends in a cononical way as a graded cohomology class of weight — 1 of A. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight — 1 of the Schouten graded Lie algebra of M. This allows us to construct explicitly all 1-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.
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Lecomte, P.B.A., Melotte, D. & Roger, C. Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra. Lett Math Phys 18, 275–285 (1989). https://doi.org/10.1007/BF00405259
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DOI: https://doi.org/10.1007/BF00405259