Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. (English) Zbl 0973.17027

Summary: 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.


17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
17B63 Poisson algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
Full Text: DOI arXiv Numdam EuDML


[1] [1] and , Quantization of gauge theories with linearly dependent generators, Phys. Rev., D 28 (1983), 2567-2582.
[2] [BHP] , , , The exceptional set for Goldbach’s problem in short intervals - Sieve Methods
[3] [3] and , Existence theorem for gauge algebra, Jour. Math. Phys., 26 (1985), 172-184.
[4] [4] , , and , Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, preprint. · Zbl 0968.58014
[5] [5] , The cohomology structure of an associative ring, Ann. of Math., 78 (1963), 267-288. · Zbl 0131.27302
[6] [6] and , 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. · Zbl 0788.17009
[7] [7] , Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. in Math. Phys., 195 (1994), 265-285. · Zbl 0807.17026
[8] [8] , Relative homological algebra, Trans. Amer. Math. Soc., 82 (1956), 246-269. · Zbl 0070.26903
[9] [Hu] , Linear algebraic groups, GTM 21, Berlin, Heidelberg, New-York, Springer (1975). · Zbl 0699.53037
[10] [10] , Duality for Lie-Rinehart algebras and the modular class, preprint dg-ga/9702008, 1997. · Zbl 1034.53083
[11] [11] , and , Differential homological algebra and homogeneous spaces J. of Pure and Applied Algebra, 5 (1974), 113-185. · Zbl 0364.18008
[12] [12] , Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Mathematicae, 41 (1995), 153-165. · Zbl 0837.17014
[13] [13] , 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] [14] and , New perspectives on the BRST-algebraic structure of string theory, Comm. in Math. Phys., 154 (1993), 613-646. · Zbl 0780.17029
[15] [15] , Differential forms for general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195-222. · Zbl 0113.26204
[16] [16] , 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. · Zbl 1149.81359
[17] [17] , 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] [18] , Gerstenhaber algebras and BV-algebras in Poisson geometry, preprint, 1997. · Zbl 0941.17016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.