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 $\left(A,L\right)$, B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$-algebra ${\Lambda }_{A}L$ correspond bijectively to right $\left(A,L\right)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra ${\Lambda }_{A}L$ correspond bijectively to right $\left(A,L\right)$-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 $\left({\Lambda }_{A}L,\partial \right)$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $\left(A,L\right)$-module structure determined by $\partial$. When $L$ is also of finite rank $n$, there are bijective correspondences between $\left(A,L\right)$-connections on ${\Lambda }_{A}^{n}L$ and right $\left(A,L\right)$-connections on $A$ and between left $\left(A,L\right)$-module structures on ${\Lambda }_{A}^{n}L$ and right $\left(A,L\right)$-module structures on $A$. Hence there are bijective correspondences between $\left(A,L\right)$-connections on ${\Lambda }_{A}^{n}L$ and generators for the Gerstenhaber bracket on ${\Lambda }_{A}L$ and between $\left(A,L\right)$-module structures on ${\Lambda }_{A}^{n}L$ and B-V algebra structures on ${\Lambda }_{A}L$. The homology of such a B-V algebra $\left({\Lambda }_{A}L,\partial \right)$ coincides with the cohomology of $L$ with coefficients in ${\Lambda }_{A}^{n}L$, with reference to the left $\left(A,L\right)$-module structure determined by $\partial$. Some applications to Poisson structures and to differential geometry are discussed.

Pour une algèbre de Lie-Rinehart $\left(A,L\right)$, les liens entre les structures d’algèbre de Batalin-Vilkovisky et de Gerstenhaber sur l’algèbre extérieure ${\Lambda }_{A}L$ et de $\left(A,L\right)$-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},
zbl = {0973.17027},
mrnumber = {99b:17021},
language = {en},
url = {http://www.numdam.org/item/AIF_1998__48_2_425_0}
}

Huebschmann, Johannes. 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. http://www.numdam.org/item/AIF_1998__48_2_425_0/

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