Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 124, 12 pages      arXiv:1805.00542      https://doi.org/10.3842/SIGMA.2018.124

Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids

Pedro Frejlich
UFRGS, Departamento de Matemática Pura e Aplicada, Porto Alegre, Brasil

Received June 18, 2018, in final form November 08, 2018; Published online November 15, 2018

Abstract
In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169].

Key words: Lie algebroids; modular class; characteristic classes; Morita equivalence.

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References

  1. Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. 663 (2012), 91-126, arXiv:0911.2859.
  2. Bursztyn H., Lima H., Meinrenken E., Splitting theorems for Poisson and related structures, J. Reine Angew. Math., to appear, arXiv:1605.05386.
  3. Caseiro R., Fernandes R.L., The modular class of a Poisson map, Ann. Inst. Fourier (Grenoble) 63 (2013), 1285-1329, arXiv:1103.4305.
  4. Crainic M., Chern characters via nonlinear connections, math.DG/0009229.
  5. Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681-721, math.DG/0008064.
  6. Crainic M., Fernandes R.L., Secondary characteristic classes of Lie algebroids, in Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys., Vol. 662, Springer, Berlin, 2005, 157-176.
  7. Crainic M., Fernandes R.L., Martínez Torres D., Poisson manifolds of compact types (PMCT 1), J. Reine Angew. Math., to appear, arXiv:1510.07108.
  8. Damianou P.A., Fernandes R.L., Integrable hierarchies and the modular class, Ann. Inst. Fourier (Grenoble) 58 (2008), 107-137, math.DG/0607784.
  9. del Hoyo M., Ortiz C., Morita equivalences of vector bundles, Int. Math. Res. Not., to appear, arXiv:1612.09289.
  10. Evens S., Lu J.-H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford 50 (1999), 417-436, dg-ga/9610008.
  11. Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179, math.DG/0007132.
  12. Frejlich P., Mărcuţ I., The homology class of a Poisson transversal, Int. Math. Res. Not., to appear, arXiv:1704.04724.
  13. Ginzburg V.L., Grothendieck groups of Poisson vector bundles, J. Symplectic Geom. 1 (2001), 121-169, math.DG/0009124.
  14. Ginzburg V.L., Golubev A., Holonomy on Poisson manifolds and the modular class, Israel J. Math. 122 (2001), 221-242, math.DG/9812153.
  15. Ginzburg V.L., Lu J.-H., Poisson cohomology of Morita-equivalent Poisson manifolds, Int. Math. Res. Not. 1992 (1992), 199-205.
  16. Grabowski J., Modular classes of skew algebroid relations, Transform. Groups 17 (2012), 989-1010, arXiv:1108.2366.
  17. Gracia-Saz A., Mehta R.A., Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math. 223 (2010), 1236-1275, arXiv:0810.0066.
  18. Guillemin V., Miranda E., Pires A.R., Codimension one symplectic foliations and regular Poisson structures, Bull. Braz. Math. Soc. (N.S.) 42 (2011), 607-623, arXiv:1009.1175.
  19. Huebschmann J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103-159, dg-ga/9702008.
  20. Kosmann-Schwarzbach Y., Modular vector fields and Batalin-Vilkovisky algebras, in Poisson Geometry (Warsaw, 1998), Banach Center Publ., Vol. 51, Polish Acad. Sci. Inst. Math., Warsaw, 2000, 109-129.
  21. Kosmann-Schwarzbach Y., Poisson manifolds, Lie algebroids, modular classes: a survey, SIGMA 4 (2008), 005, 30 pages, arXiv:0710.3098.
  22. Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Modular classes of Lie algebroid morphisms, Transform. Groups 13 (2008), 727-755, arXiv:0712.3021.
  23. Kosmann-Schwarzbach Y., Weinstein A., Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris 341 (2005), 509-514, math.DG/0508515.
  24. Koszul J.-L., Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985), 257-271.
  25. Kubarski J., Fibre integral in regular Lie algebroids, in New Developments in Differential Geometry (Budapest 1996), Kluwer Acad. Publ., Dordrecht, 1999, 173-202.
  26. Kubarski J., The Weil algebra and the secondary characteristic homomorphism of regular Lie algebroids, in Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ., Vol. 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 135-173.
  27. Kubarski J., Mishchenko A., Nondegenerate cohomology pairing for transitive Lie algebroids, characterization, Cent. Eur. J. Math. 2 (2004), 663-707.
  28. Mehta R.A., Lie algebroid modules and representations up to homotopy, Indag. Math. (N.S.) 25 (2014), 1122-1134, arXiv:1107.1539.
  29. Stiénon M., Xu P., Modular classes of Loday algebroids, C. R. Math. Acad. Sci. Paris 346 (2008), 193-198, arXiv:0803.2047.
  30. Vaisman I., Characteristic classes of Lie algebroid morphisms, Differential Geom. Appl. 28 (2010), 635-647, arXiv:0812.4658.
  31. Weinstein A., The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394.
  32. Xu P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560, dg-ga/9703001.

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