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

Huebschmann, Johannes

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

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 ${\Lambda }_{A}L$ 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 ${\Lambda }_{A}L$ 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 ${\Lambda }_A^{n}L$ and right $(A,L)$-connections on $A$ and between left $(A,L)$-module structures on ${\Lambda }_A^{n}L$ 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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