Lie-Rinehart algebras, Gerstenhaber algebras, and B-V algebras

Abstract

For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra \Lambda_A L correspond bijectively to right (A,L)-connections on A in such a way that B-V structures correspond to right (A,L)-module structures on A. When L is projective as an A-module, given an exact generator \partial, the homology of the B-V algebra (\Lambda_A L,\partial) coincides with that of L with coefficients in A with respect 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, for the left (A,L)-module structure determined by \partial. Some applications are discussed.