Homology and modular classes of Lie algebroids

Janusz Grabowski[1]; Giuseppe Marmo[2]; Peter W. Michor[3]

  • [1] Polish Academy of Sciences Mathematical Institute Śniadeckich 8, P.O. Box 21 00-956 Warszawa (Poland)
  • [2] Universitá di Napoli Federico II and INFN, Dipartimento di Scienze Fisice Sezione di Napoli, via Cintia 80126 Napoli (Italy)
  • [3] Universität Wien Institut für Mathematik Nordbergstrasse 15 A-1090 Wien (Austria) and Erwin Schrödinger Institut für Mathematische Physik Boltzmanngasse 9 A-1090 Wien (Austria)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 1, page 69-83
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Grabowski, Janusz, Marmo, Giuseppe, and Michor, Peter W.. "Homology and modular classes of Lie algebroids." Annales de l’institut Fourier 56.1 (2006): 69-83. <http://eudml.org/doc/10143>.

@article{Grabowski2006,
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.},
affiliation = {Polish Academy of Sciences Mathematical Institute Śniadeckich 8, P.O. Box 21 00-956 Warszawa (Poland); Universitá di Napoli Federico II and INFN, Dipartimento di Scienze Fisice Sezione di Napoli, via Cintia 80126 Napoli (Italy); Universität Wien Institut für Mathematik Nordbergstrasse 15 A-1090 Wien (Austria) and Erwin Schrödinger Institut für Mathematische Physik Boltzmanngasse 9 A-1090 Wien (Austria)},
author = {Grabowski, Janusz, Marmo, Giuseppe, Michor, Peter W.},
journal = {Annales de l’institut Fourier},
keywords = {Lie algebroid; de Rham cohomology; Poincaré duality; divergence; homology of Lie algebroid; connection; modular class},
language = {eng},
number = {1},
pages = {69-83},
publisher = {Association des Annales de l’institut Fourier},
title = {Homology and modular classes of Lie algebroids},
url = {http://eudml.org/doc/10143},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Grabowski, Janusz
AU - Marmo, Giuseppe
AU - Michor, Peter W.
TI - Homology and modular classes of Lie algebroids
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 1
SP - 69
EP - 83
AB - 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.
LA - eng
KW - Lie algebroid; de Rham cohomology; Poincaré duality; divergence; homology of Lie algebroid; connection; modular class
UR - http://eudml.org/doc/10143
ER -

References

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  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), 417-436 Zbl0968.58014MR1726784
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  10. Y. Kosmann-Schwarzbach, Modular vector fields and Batalin-Vilkovisky algebras, Poisson Geometry 51 (2000), 109-129, GrabowskiJ.J., Warszawa Zbl1018.17020MR1764439
  11. Y. Kosmann-Schwarzbach, K. Mackenzie, Differential operators and actions of Lie algebroids, Contemp. Math. 315 (2002), 213-233 Zbl1040.17020MR1958838
  12. Y. Kosmann-Schwarzbach, J. Monterde, Divergence operators and odd Poisson brackets, Ann. Inst. Fourier 52 (2002), 419-456 Zbl1054.53094MR1906481
  13. Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Élie Cartan hors série (1985), 257-271, Lyon, 1984 Zbl0615.58029MR837203
  14. K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, (1987), Cambridge University Press Zbl0683.53029MR896907
  15. E. Nelson, Tensor analysis, (1967), Princeton University Press Zbl0152.39001
  16. A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394 Zbl0902.58013MR1484598
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  18. P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560 Zbl0941.17016MR1675117

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