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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.

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
Full Text: DOI Numdam EuDML arXiv
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