Huebschmann: Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

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Johannes Huebschmann
Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras
Annales de l'institut Fourier, 48 no. 2 (1998), p. 425-440, doi: 10.5802/aif.1624
Article PDF | Reviews MR 99b:17021 | Zbl 0973.17027 | 3 citations in Cedram

Résumé - Abstract

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.

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