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Chevalley cohomology of vector graphs. (Cohomologie de Chevalley des graphes vectoriels.) (French) Zbl 1207.17026

The space of smooth functions and vector fields on \(\mathbb R^d\) is a Lie subalgebra of the (graded) Lie algebra \(T_{\text{poly}}(\mathbb R^d)\), equipped with the Schouten bracket. Here the authors compute the cohomology of this subalgebra for the adjoint representation in \(T_{\text{poly}}(\mathbb R^d)\), restricting themselves to the case of cochains defined with purely aerial Kontsevich graphs, as in D. Arnal, A. Gammella and M. Masmoudi [Pac. J. Math. 218, No. 2, 201–239 (2005; Zbl 1156.53321)]. The results are very similar to the classical result of I. M. Gel’fand and D. B. Fuks [Funct. Anal. Appl. 4, 110–116 (1970); translation from Funkts. Anal. Prilozh. 4, No. 2, 23–31 (1970; Zbl 0208.51401)] and those of M. De Wilde et P. B. A. Lecomte [J. Math. Pures Appl., IX. Sér. 62, 197–214 (1983; Zbl 0481.58032)].

MSC:

17B56 Cohomology of Lie (super)algebras
53D55 Deformation quantization, star products
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
17B66 Lie algebras of vector fields and related (super) algebras
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