Skip to main content
SpringerLink
Log in

Modular Classes of Lie Algebroid Morphisms

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. T. Baues, G. Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), 187–211.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Bursztyn, A. Weinstein, Picard groups in Poisson geometry, Moscow Math. J. 4 (2004), 39–66.

    MATH  MathSciNet  Google Scholar 

  3. Z. Chen, Z.-J. Liu, On (co-)morphisms of Lie pseudoalgebras and groupoids, J. Algebra 316 (2007), 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Crainic, Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681–721.

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Dolgushev, The Van den Bergh duality and the modular symmmetry of a Poisson variety, available at arXiv:math/0612288.

  6. J.-P. Dufour, N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, Vol. 242, Birkhäuser, Boston, 2005.

    Google Scholar 

  7. S. Evens, J.-H. Lu, A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser. 2 50 (1999), 417–436.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119–179.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. I. Generalov, Relative homological algebra, cohomology of categories, posets and coalgebras, in: Handbook of Algebra, M. Hazewinkel ed., Vol. 1, North-Holland, Amsterdam, 1996, pp. 611–638.

    Chapter  Google Scholar 

  10. V. L. Ginzburg, Grothendieck groups of Poisson vector bundles, J. Symplectic Geom. 1 (2001), 121–169.

    MATH  MathSciNet  Google Scholar 

  11. V. L. Ginzburg, J.-H. Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Int. Math. Res. Not. 10 (1992), 199–205.

    Article  MathSciNet  Google Scholar 

  12. J. Grabowski, G. Marmo, P. W. Michor, Homology and modular classes of Lie algebroids, Ann. Inst. Fourier (Grenoble) 56 (2006), 69–83.

    MATH  MathSciNet  Google Scholar 

  13. P. J. Higgins, K. C. H. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194–230.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Huebschmann, Duality for Lie–Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103–159.

    MATH  MathSciNet  Google Scholar 

  15. Y. Kosmann-Schwarzbach, Modular vector fields and Batalin–Vilkovisky algebras, Banach Center Publications 51 (2000), 109–129.

    MathSciNet  Google Scholar 

  16. Y. Kosmann-Schwarzbach, C. Laurent-Gengoux, The modular class of a twisted Poisson structure, Travaux Mathématiques (Luxembourg) 16 (2005), 315–339.

    MathSciNet  Google Scholar 

  17. Y. Kosmann-Schwarzbach, A. Weinstein, Relative modular classes of Lie algebroids, C. R. Acad. Sci. Paris Ser. I 341 (2005), 509–514.

    MATH  MathSciNet  Google Scholar 

  18. Y. Kosmann-Schwarzbach, M. Yakimov, Modular classes of regular twisted Poisson structures on Lie algebroids, Lett. Math. Phys. 80 (2007), 183–197.

    Article  MATH  MathSciNet  Google Scholar 

  19. J.-L. Koszul, Crochet de Schouten–Nijenhuis et cohomologie, in: The Mathematical Heritage of Élie Cartan, Astérisque, numéro hors série (1985), 257–271.

  20. J. Kubarski, The Weil algebra and the secondary characteristic homomorphism of regular Lie algebroids, Banach Center Publications 54 (2001), 135–173.

    MathSciNet  Google Scholar 

  21. K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.

    Google Scholar 

  22. K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.

    Google Scholar 

  23. I. Moerdijk, J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, Cambridge, 2003.

    Google Scholar 

  24. N. Neumaier, S. Waldmann, Deformation quantization of Poisson structures associated to Lie algebroids, available at arXiv:0708.0516.

  25. А. Вайнтроб, Алгеброиды Ди и гомологияеские векторные поля, УМН 52 (1997), no. 2(314), 161–162. English transl.: A. Vaintrob, Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), no. 2, 428–429.

  26. A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379–394.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545–560.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Mathématiques, Laurent Schwartz, École Polytechnique, 91128, Palaiseau, France

    Y. Kosmann-Schwarzbach

  2. Département de Mathématiques, Université de Poitiers, 86962, Futuroscope Cedex, France

    C. Laurent-Gengoux

  3. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    A. Weinstein

Authors
  1. Y. Kosmann-Schwarzbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Laurent-Gengoux
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Weinstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Kosmann-Schwarzbach.

Additional information

Dedicated to Bertram Kostant on the occasion of his eightieth birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kosmann-Schwarzbach, Y., Laurent-Gengoux, C. & Weinstein, A. Modular Classes of Lie Algebroid Morphisms. Transformation Groups 13, 727–755 (2008). https://doi.org/10.1007/s00031-008-9032-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-008-9032-y

Keywords

Search

Navigation