For any Lie-Rinehart algebra , B(atalin)-V(ilkovisky) algebra structures on the exterior -algebra correspond bijectively to right -module structures on ; likewise, generators for the Gerstenhaber algebra correspond bijectively to right -connections on . When is projective as an -module, given a B-V algebra structure on , the homology of the B-V algebra coincides with the homology of with coefficients in with reference to the right -module structure determined by . When is also of finite rank , there are bijective correspondences between -connections on and right -connections on and between left -module structures on and right -module structures on . Hence there are bijective correspondences between -connections on and generators for the Gerstenhaber bracket on and between -module structures on and B-V algebra structures on . The homology of such a B-V algebra coincides with the cohomology of with coefficients in , with reference to the left -module structure determined by . Some applications to Poisson structures and to differential geometry are discussed.
Pour une algèbre de Lie-Rinehart , les liens entre les structures d’algèbre de Batalin-Vilkovisky et de Gerstenhaber sur l’algèbre extérieure et de -module à droite sur ou plus généralement de connexion à droite sur sont établis ainsi que les liens correspondants en homologie. Sous l’hypothèse additionnelle que est projective de rang constant fini en tant que -module, on obtient une description de l’homologie de l’algèbre de Batalin-Vilkovisky correspondante en fonction de la cohomologie de 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}, pages = {425--440}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {48}, number = {2}, year = {1998}, doi = {10.5802/aif.1624}, zbl = {0973.17027}, mrnumber = {99b:17021}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1624/} }
TY - JOUR AU - Huebschmann, Johannes TI - Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras JO - Annales de l'Institut Fourier PY - 1998 SP - 425 EP - 440 VL - 48 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1624/ DO - 10.5802/aif.1624 LA - en ID - AIF_1998__48_2_425_0 ER -
%0 Journal Article %A Huebschmann, Johannes %T Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras %J Annales de l'Institut Fourier %D 1998 %P 425-440 %V 48 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1624/ %R 10.5802/aif.1624 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1624/
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