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},
     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/

[1] I.A. Batalin and G.S. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev., D 28 (1983), 2567-2582.

[2] I.A. Batalin and G.S. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B, 234 (1984), 106-124.

[3] I.A. Batalin and G.S. Vilkovisky, Existence theorem for gauge algebra, Jour. Math. Phys., 26 (1985), 172-184.

[4] S. Evens, J.-H. Lu, and A. Weinstein, Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, preprint.

[5] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math., 78 (1963), 267-288. | MR 28 #5102 | Zbl 0131.27302

[6] M. Gerstenhaber and Samuel D. Schack, Algebras, bialgebras, quantum groups and algebraic deformations, In: Deformation theory and quantum groups with applications to mathematical physics, M. Gerstenhaber and J. Stasheff, eds. Cont. Math., AMS, Providence, 134 (1992), 51-92. | MR 94b:16045 | Zbl 0788.17009

[7] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. in Math. Phys., 195 (1994), 265-285. | MR 95h:81099 | Zbl 0807.17026

[8] G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc., 82 (1956), 246-269. | MR 18,278a | Zbl 0070.26903

[9] J. Huebschmann, Poisson cohomology and quantization, J. für die Reine und Angew. Math., 408 (1990), 57-113. | MR 92e:17027 | Zbl 0699.53037

[10] J. Huebschmann, Duality for Lie-Rinehart algebras and the modular class, preprint dg-ga/9702008, 1997. | Zbl 01287585

[11] D. Husemoller, J.C. Moore and J.D. Stasheff, Differential homological algebra and homogeneous spaces J. of Pure and Applied Algebra, 5 (1974), 113-185. | MR 51 #1823 | Zbl 0364.18008

[12] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Mathematicae, 41 (1995), 153-165. | MR 97i:17021 | Zbl 0837.17014

[13] J.-L. Koszul, Crochet de Schouten-Nijenhuiset cohomologie, in E. Cartan et les Mathématiciens d'aujourd'hui, Lyon, 25-29 Juin, 1984, Astérisque, hors-série, (1985) 251-271. | Zbl 0615.58029

[14] B.H. Lian and G.J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Comm. in Math. Phys., 154 (1993), 613-646. | MR 94e:81333 | Zbl 0780.17029

[15] G. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195-222. | MR 27 #4850 | Zbl 0113.26204

[16] J.D. Stasheff, Deformation theory and the Batalin-Vilkovisky master equation, in: Deformation Theory and Symplectic Geometry, Proceedings of the Ascona meeting, June 1996, D. Sternheimer, J. Rawnsley, S. Gutt, eds., Mathematical Physics Studies, Vol. 20 Kluwer Academic Publishers, Dordrecht-Boston-London, 1997, 271-284.

[17] A. Weinstein, The modular automorphism group of a Poisson manifold, to appear in: special volume in honor of A. Lichnerowicz, J. of Geometry and Physics. | Zbl 0902.58013

[18] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, preprint, 1997. | Zbl 0941.17016