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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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