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Johannes Huebschmann
Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras
Annales de l'institut Fourier, 48 no. 2 (1998), p. 425-440, doi: 10.5802/aif.1624
Article PDF | Reviews MR 99b:17021 | Zbl 0973.17027 | 3 citations in Cedram

Résumé - Abstract

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 $\Lambda _A^nL$ and right $(A,L)$-connections on $A$ and between left $(A,L)$-module structures on $\Lambda _A^nL$ 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.

Bibliography

[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.
arXiv
[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.
Article |  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.
arXiv |  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.
Article |  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.
arXiv |  Zbl 0941.17016
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