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Janusz Grabowski; Giuseppe Marmo; Peter W. Michor
Homology and modular classes of Lie algebroids
(Classes d’homologie et classes modulaire pour les algébroïdes de Lie)
Annales de l'institut Fourier, 56 no. 1 (2006), p. 69-83
Article: subscription required | Reviews MR 2228680 | Zbl 1141.17018 | 1 citation in Cedram
Class. Math.: 17B56, 17B66, 17B70, 53C05
Keywords: Lie algebroid, de Rham cohomology, Poincaré duality, divergence
[1] M. Crainic, “Chern characters via connections up to homotopy”, arXiv: math.DG/0009229
arXiv
[2] G. De Rham, Variétés différentiables, Hermann, 1955 Zbl 0065.32401
[3] S. Evens, J.-H. Lu & A. Weinstein, “Transverse measures, the modular class, and a cohomology pairing for Lie algebroids”, Quarterly J. Math., Oxford Ser. 2 50 (1999), p. 417-436 MR 1726784 | Zbl 0968.58014
[4] R. L. Fernandes, “Lie algebroids, holonomy and characteristic classes”, Adv. Math. 170 (2000), p. 119-179 MR 1929305 | Zbl 1007.22007
[5] J. Grabowski, “Quasi-derivations and QD-algebroids”, Rep. Math. Phys. 52 (2003), p. 445-451 MR 2029773 | Zbl 1051.53067
[6] J. Grabowski & G. Marmo, “Jacobi structures revisited”, J. Phys. A: Math. Gen. 34 (2001), p. 10975-10990 MR 1872975 | Zbl 0998.53054
[7] J. Grabowski & G. Marmo, “The graded Jacobi algebras and (co)homology”, J. Phys. A: Math. Gen. 36 (2003), p. 161-181 MR 1959019 | Zbl 1039.53090
[8] J. Hübschmann, “Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras”, Ann. Inst. Fourier 48 (1998), p. 425-440
Cedram | MR 1625610 | Zbl 0973.17027
[9] D. Iglesias & J.C. Marrero, “Generalized Lie bialgebroids and Jacobi structures”, J. Geom. Phys. 40 (2001), p. 176-199 MR 1862087 | Zbl 1001.17025
[10] Y. Kosmann-Schwarzbach, Modular vector fields and Batalin-Vilkovisky algebras, in J. Grabowski, P. Urbański, ed., Poisson Geometry, 2000, p. 109–129 MR 1764439 | Zbl 1018.17020
[11] Y. Kosmann-Schwarzbach & K. Mackenzie, “Differential operators and actions of Lie algebroids”, Contemp. Math. 315 (2002), p. 213-233 MR 1958838 | Zbl 1040.17020
[12] Y. Kosmann-Schwarzbach & J. Monterde, “Divergence operators and odd Poisson brackets”, Ann. Inst. Fourier 52 (2002), p. 419-456
Cedram | MR 1906481 | Zbl 1054.53094
[13] Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in The mathematical heritage of Élie Cartan, Astérisque, 1985, p. 257-271 MR 837203 | Zbl 0615.58029
[14] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, 1987 MR 896907 | Zbl 0683.53029
[15] E. Nelson, Tensor analysis, Princeton University Press, 1967 Zbl 0152.39001
[16] A. Weinstein, “The modular automorphism group of a Poisson manifold”, J. Geom. Phys. 23 (1997), p. 379-394 MR 1484598 | Zbl 0902.58013
[17] E. Witten, “Supersymmetry and Morse theory”, J. Diff. Geom. 17 (1982), p. 661-692 MR 683171 | Zbl 0499.53056
[18] P. Xu, “Gerstenhaber algebras and BV-algebras in Poisson geometry”, Comm. Math. Phys. 200 (1999), p. 545-560 MR 1675117 | Zbl 0941.17016
Janusz Grabowski; Giuseppe Marmo; Peter W. Michor
Homology and modular classes of Lie algebroids
(Classes d’homologie et classes modulaire pour les algébroïdes de Lie)
Annales de l'institut Fourier, 56 no. 1 (2006), p. 69-83
Article: subscription required | Reviews MR 2228680 | Zbl 1141.17018 | 1 citation in Cedram
Class. Math.: 17B56, 17B66, 17B70, 53C05
Keywords: Lie algebroid, de Rham cohomology, Poincaré duality, divergence
Résumé - Abstract
For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.
Bibliography
arXiv
[2] G. De Rham, Variétés différentiables, Hermann, 1955 Zbl 0065.32401
[3] S. Evens, J.-H. Lu & A. Weinstein, “Transverse measures, the modular class, and a cohomology pairing for Lie algebroids”, Quarterly J. Math., Oxford Ser. 2 50 (1999), p. 417-436 MR 1726784 | Zbl 0968.58014
[4] R. L. Fernandes, “Lie algebroids, holonomy and characteristic classes”, Adv. Math. 170 (2000), p. 119-179 MR 1929305 | Zbl 1007.22007
[5] J. Grabowski, “Quasi-derivations and QD-algebroids”, Rep. Math. Phys. 52 (2003), p. 445-451 MR 2029773 | Zbl 1051.53067
[6] J. Grabowski & G. Marmo, “Jacobi structures revisited”, J. Phys. A: Math. Gen. 34 (2001), p. 10975-10990 MR 1872975 | Zbl 0998.53054
[7] J. Grabowski & G. Marmo, “The graded Jacobi algebras and (co)homology”, J. Phys. A: Math. Gen. 36 (2003), p. 161-181 MR 1959019 | Zbl 1039.53090
[8] J. Hübschmann, “Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras”, Ann. Inst. Fourier 48 (1998), p. 425-440
Cedram | MR 1625610 | Zbl 0973.17027
[9] D. Iglesias & J.C. Marrero, “Generalized Lie bialgebroids and Jacobi structures”, J. Geom. Phys. 40 (2001), p. 176-199 MR 1862087 | Zbl 1001.17025
[10] Y. Kosmann-Schwarzbach, Modular vector fields and Batalin-Vilkovisky algebras, in J. Grabowski, P. Urbański, ed., Poisson Geometry, 2000, p. 109–129 MR 1764439 | Zbl 1018.17020
[11] Y. Kosmann-Schwarzbach & K. Mackenzie, “Differential operators and actions of Lie algebroids”, Contemp. Math. 315 (2002), p. 213-233 MR 1958838 | Zbl 1040.17020
[12] Y. Kosmann-Schwarzbach & J. Monterde, “Divergence operators and odd Poisson brackets”, Ann. Inst. Fourier 52 (2002), p. 419-456
Cedram | MR 1906481 | Zbl 1054.53094
[13] Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in The mathematical heritage of Élie Cartan, Astérisque, 1985, p. 257-271 MR 837203 | Zbl 0615.58029
[14] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, 1987 MR 896907 | Zbl 0683.53029
[15] E. Nelson, Tensor analysis, Princeton University Press, 1967 Zbl 0152.39001
[16] A. Weinstein, “The modular automorphism group of a Poisson manifold”, J. Geom. Phys. 23 (1997), p. 379-394 MR 1484598 | Zbl 0902.58013
[17] E. Witten, “Supersymmetry and Morse theory”, J. Diff. Geom. 17 (1982), p. 661-692 MR 683171 | Zbl 0499.53056
[18] P. Xu, “Gerstenhaber algebras and BV-algebras in Poisson geometry”, Comm. Math. Phys. 200 (1999), p. 545-560 MR 1675117 | Zbl 0941.17016
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