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Modular vector fields and Batalin-Vilkovisky algebras. (English) Zbl 1018.17020
Grabowski, Janusz (ed.) et al., Poisson geometry. Stanisław Zakrzewski in memoriam. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 51, 109-129 (2000).
Summary: We show that a modular class arises from the existence of two generating operators for a Batalin-Vilkovisky algebra. In particular, for every triangular Lie bialgebroid $$(A,P)$$ such that its top exterior power is a trivial line bundle, there is a section of the vector bundle $$A$$ whose $$d_P$$-cohomology class is well-defined. We give simple proofs of its properties. The modular class of an orientable Poisson manifold is an example. We analyse the relationships between generating operators of the Gerstenhaber algebra of a Lie algebroid, right actions on the elements of degree 0, and left actions on the elements of top degree. We show that the modular class of a triangular Lie bialgebroid coincides with the characteristic class of a Lie algebroid with representation on a line bundle.
For the entire collection see [Zbl 0936.00035].

##### MSC:
 17B70 Graded Lie (super)algebras 53D05 Symplectic manifolds (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58A10 Differential forms in global analysis
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