zbMATH — the first resource for mathematics

Homology and modular classes of Lie algebroids. (English) Zbl 1141.17018
The standard approach to homology theory of Lie algebroids is to define homology via generating operators of low degree (such as flat connection and divergence) for the Schouten bracket. Here the authors propose an alternative approach, based on divergence defined for the so-called “odd-forms”, and interpret in its terms the modular class of Lie algebroids.
The advantage of this approach is that the so defined homology does not depend on the choice of generating operators.

17B56 Cohomology of Lie (super)algebras
17B63 Poisson algebras
18G60 Other (co)homology theories (MSC2010)
53C05 Connections, general theory
Full Text: DOI Numdam EuDML arXiv
[1] Crainic, M., Chern characters via connections up to homotopy
[2] De Rham, G., Variétés différentiables, (1955), Hermann, Paris · Zbl 0065.32401
[3] Evens, S.; Lu, J.-H.; Weinstein, A., Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quarterly J. Math., Oxford Ser. 2, 50, 2, 417-436, (1999) · Zbl 0968.58014
[4] Fernandes, R. L., Lie algebroids, holonomy and characteristic classes, Adv. Math., 170, 119-179, (2000) · Zbl 1007.22007
[5] Grabowski, J., Quasi-derivations and QD-algebroids, Rep. Math. Phys., 52, 445-451, (2003) · Zbl 1051.53067
[6] Grabowski, J.; Marmo, G., Jacobi structures revisited, J. Phys. A: Math. Gen., 34, 10975-10990, (2001) · Zbl 0998.53054
[7] Grabowski, J.; Marmo, G., The graded Jacobi algebras and (co)homology, J. Phys. A: Math. Gen., 36, 161-181, (2003) · Zbl 1039.53090
[8] Hübschmann, J., Lie-rinehart algebras, gerstenhaber algebras, and Batalin-Vilkovisky algebras, Ann. Inst. Fourier, 48, 425-440, (1998) · Zbl 0973.17027
[9] Iglesias, D.; Marrero, J. C., Generalized Lie bialgebroids and Jacobi structures, J. Geom. Phys., 40, 176-199, (2001) · Zbl 1001.17025
[10] Kosmann-Schwarzbach, Y.; Grabowski, J.; Urbański, P., Poisson Geometry, 51, Modular vector fields and Batalin-Vilkovisky algebras, 109-129, (2000), Warszawa · Zbl 1018.17020
[11] Kosmann-Schwarzbach, Y.; Mackenzie, K., Differential operators and actions of Lie algebroids, Contemp. Math., 315, 213-233, (2002) · Zbl 1040.17020
[12] Kosmann-Schwarzbach, Y.; Monterde, J., Divergence operators and odd Poisson brackets, Ann. Inst. Fourier, 52, 419-456, (2002) · Zbl 1054.53094
[13] Koszul, Jean-Louis, Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Élie Cartan, hors série, 257-271, (1985), Lyon, 1984 · Zbl 0615.58029
[14] Mackenzie, K., Lie groupoids and Lie algebroids in differential geometry, (1987), Cambridge University Press · Zbl 0683.53029
[15] Nelson, E., Tensor analysis, (1967), Princeton University Press · Zbl 0152.39001
[16] Weinstein, A., The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23, 379-394, (1997) · Zbl 0902.58013
[17] Witten, E., Supersymmetry and Morse theory, J. Diff. Geom., 17, 661-692, (1982) · Zbl 0499.53056
[18] Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys., 200, 545-560, (1999) · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.