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A cohomology for homogeneous Poisson Lie algebras. (Une cohomologie pour les algèbres de Lie de Poisson homogènes.) (French) Zbl 0841.58027

A Poisson algebra is a commutative algebra \(A\) together with a structure of Lie algebra \(\{\cdot, \cdot\}\), which are compatible in the sense that the bracket behaves as a derivation for the multiplication. The multiplication alone gives rise to the usual Hochschild complex and the Lie algebra structure alone to the Cartan-Chevalley-Eilenberg complex calculating Lie algebra cohomology while the Poisson structure leads to yet another theory which has been called Poisson cohomology. The latter was introduced for smooth Poisson manifolds by A. Lichnerowicz [J. Differ. Geom. 12, No. 2, 253–300 (1977; Zbl 0405.53024)] and for arbitrary Poisson algebras by the reviewer [J. Reine Angew. Math. 408, 57–113 (1990; Zbl 0699.53037)]. The chain complex calculating it is related to those for the other theories mentioned above. In the paper a description of the complex calculating Poisson cohomology for a finitely generated polynomial algebra is given different from the one in the reviewer’s paper in such a way that the links to the Hochschild and Cartan-Chevalley-Eilenberg complexes are more transparent. In the case at hand, this description allows for more structure; in particular, it entails a notion of weight induced by the usual grading of a polynomial algebra. This is then applied to the study of equivalence of deformations of Poisson algebras and in particular Sklyanin algebras. Thereafter, examples of deformations of quadratic Poisson algebras are given. Finally, in the framework of a suitable graded theory involving graded Lie algebras and the Richardson-Nijenhuis bracket, which has been developed by Roger jointly with P. Lecomte, a somewhat different approach to the deformation problem of the relevant Poisson algebras is given in terms of what is called a structure and, furthermore, the problem of extension of Poisson bracket under extension of indeterminates is studied.
{Reviewer’s remark: The grading of Poisson cochains used by the authors is different from the one in the reviewer’s paper}.

MSC:

17B56 Cohomology of Lie (super)algebras
17B63 Poisson algebras
17B70 Graded Lie (super)algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
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