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

Huebschmann, Johannes

doi:10.5802/aif.1624

Huebschmann, Johannes

Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 425-440.

Résumé
Abstract

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.

For any Lie-Rinehart algebra $(A, L)$ , B(atalin)-V(ilkovisky) algebra structures $\partial$ 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 $\partial$ on $Λ_{A} L$ , the homology of the B-V algebra $(Λ_{A} L, \partial)$ 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 $Λ_{A}^{n} L$ 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 $(Λ_{A} L, \partial)$ coincides with the cohomology of $L$ with coefficients in $Λ_{A}^{n} L$ , with reference to the left $(A, L)$ -module structure determined by $\partial$ . Some applications to Poisson structures and to differential geometry are discussed.

MR Zbl | 3 citations dans Numdam

DOI : 10.5802/aif.1624

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     author = {Huebschmann, Johannes},
     title = {Lie-Rinehart algebras, {Gerstenhaber} algebras and {Batalin-Vilkovisky} algebras},
     journal = {Annales de l'Institut Fourier},
     pages = {425--440},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     doi = {10.5802/aif.1624},
     mrnumber = {99b:17021},
     zbl = {0973.17027},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1624/}
}

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Huebschmann, Johannes. Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 425-440. doi : 10.5802/aif.1624. https://www.numdam.org/articles/10.5802/aif.1624/

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