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Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. (English) Zbl 0973.17027
Summary: For any Lie-Rinehart algebra $$(A,L)$$, B(atalin)-V(ilkovisky) algebra structures $$\partial$$ on the exterior $$A$$-algebra $$\Lambda_A L$$ correspond bijectively to right $$(A,L)$$-module structures on $$A$$; likewise, generators for the Gerstenhaber algebra $$\Lambda_A L$$ correspond bijectively to right $$(A,L)$$-connections on $$A$$. When $$L$$ is projective as an $$A$$-module, given a B-V algebra structure $$\partial$$ on $$\Lambda_A L$$, the homology of the B-V algebra $$(\Lambda_A L,\partial)$$ coincides with the homology of $$L$$ with coefficients in $$A$$ with reference to the right $$(A,L)$$-module structure determined by $$\partial$$. When $$L$$ is also of finite rank $$n$$, there are bijective correspondences between $$(A,L)$$-connections on $$\Lambda_A^nL$$ and right $$(A,L)$$-connections on $$A$$ and between left $$(A,L)$$-module structures on $$\Lambda_A^nL$$ and right $$(A,L)$$-module structures on $$A$$. Hence there are bijective correspondences between $$(A,L)$$-connections on $$\Lambda_A^n L$$ and generators for the Gerstenhaber bracket on $$\Lambda_A L$$ and between $$(A,L)$$-module structures on $$\Lambda_A^n L$$ and B-V algebra structures on $$\Lambda_A L$$. The homology of such a B-V algebra $$(\Lambda_A L,\partial)$$ coincides with the cohomology of $$L$$ with coefficients in $$\Lambda_A^n L$$, with reference to the left $$(A,L)$$-module structure determined by $$\partial$$. Some applications to Poisson structures and to differential geometry are discussed.

MSC:
 17B55 Homological methods in Lie (super)algebras 17B56 Cohomology of Lie (super)algebras 17B66 Lie algebras of vector fields and related (super) algebras 17B63 Poisson algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids
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