Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras
Annales de l'Institut Fourier, Volume 48 (1998) no. 2, p. 425-440
For any Lie-Rinehart algebra (A,L), B(atalin)-V(ilkovisky) algebra structures ∂ on the exterior A-algebra Λ A L correspond bijectively to right (A,L)-module structures on A; likewise, generators for the Gerstenhaber algebra Λ 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 ∂ on Λ A L, the homology of the B-V algebra (Λ A L,∂) coincides with the homology of L with coefficients in A with reference to the right (A,L)-module structure determined by ∂. When L is also of finite rank n, there are bijective correspondences between (A,L)-connections on Λ A n L and right (A,L)-connections on A and between left (A,L)-module structures on Λ A n L and right (A,L)-module structures on A. Hence there are bijective correspondences between (A,L)-connections on Λ A n L and generators for the Gerstenhaber bracket on Λ A L and between (A,L)-module structures on Λ A n L and B-V algebra structures on Λ A L. The homology of such a B-V algebra (Λ A L,∂) coincides with the cohomology of L with coefficients in Λ A n L, with reference to the left (A,L)-module structure determined by ∂. Some applications to Poisson structures and to differential geometry are discussed.
Pour une algèbre de Lie-Rinehart (A,L), les liens entre les structures d’algèbre de Batalin-Vilkovisky et de Gerstenhaber sur l’algèbre extérieure Λ A L et de (A,L)-module à droite sur A ou plus généralement de connexion à droite sur A sont établis ainsi que les liens correspondants en homologie. Sous l’hypothèse additionnelle que L est projective de rang constant fini en tant que A-module, on obtient une description de l’homologie de l’algèbre de Batalin-Vilkovisky correspondante en fonction de la cohomologie de L à valeurs dans un module adapté. Des applications aux structures de Poisson et en géométrie différentielle sont abordées. 
@article{AIF_1998__48_2_425_0,
     author = {Huebschmann, Johannes},
     title = {Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     pages = {425-440},
     doi = {10.5802/aif.1624},
     mrnumber = {99b:17021},
     zbl = {0973.17027},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1998__48_2_425_0}
}

Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 425-440. doi : 10.5802/aif.1624. https://aif.centre-mersenne.org/item/AIF_1998__48_2_425_0/

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